{
  "schemaVersion": 1,
  "generator": "readable-paper 0.1.4",
  "paperId": "0a77d9b162cc",
  "version": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
  "sha256": "eb1a535a1d8a9dc4c87d9eba7f861d5eda65531b75ac473eb5a5cb3ab7d3f727",
  "title": "The critical dimension for one-phase Bernoulli minimizers",
  "status": "Unverified reading map",
  "nodes": [
    {
      "id": "theorem-1",
      "kind": "theorem",
      "label": "Theorem 1.1",
      "source": {
        "blockId": "p1-4296a3c860",
        "page": 1,
        "quote": "Theorem 1.1. Every nonzero one-homogeneous global minimizer of (1.1) in Rd, 1 ≤d ≤6, is flat.\nThere is a nonflat one-homogeneous global minimizer in R7. Consequently,"
      },
      "statement": [
        {
          "blockId": "p1-4296a3c860",
          "page": 1,
          "quote": "Theorem 1.1. Every nonzero one-homogeneous global minimizer of (1.1) in Rd, 1 ≤d ≤6, is flat.\nThere is a nonflat one-homogeneous global minimizer in R7. Consequently,"
        },
        {
          "blockId": "p1-9a4a5d76b1",
          "page": 1,
          "quote": "d∗= 7."
        }
      ],
      "proof": [],
      "guide": [
        {
          "text": "The claim combines flatness in dimensions one through six with a nonflat minimizing cone in dimension seven. Both parts are needed for the stated critical dimension. The definition of global minimizer allows arbitrary Sobolev competitors with the prescribed trace.",
          "sources": [
            {
              "blockId": "p1-1ef6120946",
              "page": 1,
              "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
            },
            {
              "blockId": "p1-4296a3c860",
              "page": 1,
              "quote": "Theorem 1.1. Every nonzero one-homogeneous global minimizer of (1.1) in Rd, 1 ≤d ≤6, is flat.\nThere is a nonflat one-homogeneous global minimizer in R7. Consequently,"
            },
            {
              "blockId": "p1-9a4a5d76b1",
              "page": 1,
              "quote": "d∗= 7."
            }
          ]
        },
        {
          "text": "Keep the cone classification separate from the regularity consequence for local minimizers. Corollary 1.2 states the singular-set bound and its sharpness; the paragraph immediately before it defines the blowups and singular points used there.",
          "sources": [
            {
              "blockId": "p1-1398459c75",
              "page": 1,
              "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
            },
            {
              "blockId": "p1-63eea2c5f4",
              "page": 1,
              "quote": "Corollary 1.2 (Sharp singular-set bound). Let n ≥2, let D ⊂Rn be open, and let u ∈H1\nloc(D) be\na nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is\nsmooth outside Sing(u). This set is empty when n ≤6, is locally finite when n = 7, and satisfies"
            },
            {
              "blockId": "p1-43352643db",
              "page": 1,
              "quote": "dimH Sing(u) ≤n −7\n(n ≥7)."
            },
            {
              "blockId": "p1-3a49c267fd",
              "page": 1,
              "quote": "For every n ≥7, a global minimizer attains this dimension bound."
            }
          ]
        }
      ],
      "guideStatus": "Agent-authored reading guide; not independently verified."
    },
    {
      "id": "corollary-2",
      "kind": "corollary",
      "label": "Corollary 1.2",
      "source": {
        "blockId": "p1-63eea2c5f4",
        "page": 1,
        "quote": "Corollary 1.2 (Sharp singular-set bound). Let n ≥2, let D ⊂Rn be open, and let u ∈H1\nloc(D) be\na nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is\nsmooth outside Sing(u). This set is empty when n ≤6, is locally finite when n = 7, and satisfies"
      },
      "statement": [
        {
          "blockId": "p1-63eea2c5f4",
          "page": 1,
          "quote": "Corollary 1.2 (Sharp singular-set bound). Let n ≥2, let D ⊂Rn be open, and let u ∈H1\nloc(D) be\na nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is\nsmooth outside Sing(u). This set is empty when n ≤6, is locally finite when n = 7, and satisfies"
        },
        {
          "blockId": "p1-43352643db",
          "page": 1,
          "quote": "dimH Sing(u) ≤n −7\n(n ≥7)."
        },
        {
          "blockId": "p1-3a49c267fd",
          "page": 1,
          "quote": "For every n ≥7, a global minimizer attains this dimension bound."
        }
      ],
      "proof": [
        {
          "blockId": "p35-69e0697d40",
          "page": 35,
          "quote": "Proof of Corollary 1.2. Apply the cited dimension-reduction theorem with volume coefficient one\non balls compactly contained in D. Its inputs are uniform flatness regularity, linear nondegeneracy,\ncompactness of blowups, and the fact that all blowup limits are homogeneous global minimizers.\nThese are the constant-coefficient results used in Proposition 2.1; flatness regularity is [17, Theo-\nrem 8.1]. The theorem gives an empty singular set below d∗, local finiteness in dimension d∗, and\nHausdorff dimension at most n −d∗above it. Theorem 1.1 sets d∗= 7. Smoothness at each regular\npoint follows from Proposition 2.1(iii). A countable cover by interior balls gives the assertions on D.\nFor sharpness, let U be the nonflat global minimizer in R7 of De Silva and Jerison [7, Theorem 1.1],\nand set Un(x, y) = U(x) on R7 × Rn−7. Repeated application of Proposition 2.6 shows that Un is\nglobally minimizing. At each point (0, y0), every rescaling of Un is Un itself, by homogeneity and\ntranslation invariance in y. It cannot be flat: a half-space normal would have zero y-component,\nforcing U to be flat. Thus {0} × Rn−7 ⊂Sing(Un). Its dimension is n −7, and the proved upper\nbound gives equality."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-3",
      "kind": "proposition",
      "label": "Proposition 2.1",
      "source": {
        "blockId": "p4-5aee8e3401",
        "page": 4,
        "quote": "Proposition 2.1 (Regularity and blowups of minimizers). Let d ≥2 and let u ≥0 be a local\nminimizer of the one-phase energy with volume coefficient 1 in an open set D ⊂Rd."
      },
      "statement": [
        {
          "blockId": "p4-5aee8e3401",
          "page": 4,
          "quote": "Proposition 2.1 (Regularity and blowups of minimizers). Let d ≥2 and let u ≥0 be a local\nminimizer of the one-phase energy with volume coefficient 1 in an open set D ⊂Rd."
        },
        {
          "blockId": "p4-87d718bcb3",
          "page": 4,
          "quote": "(i) There is a locally Lipschitz representative of u, and it is harmonic in {u > 0}."
        },
        {
          "blockId": "p4-cd834ef0a2",
          "page": 4,
          "quote": "(ii) If x0 ∈D ∩∂{u > 0}, every sequence rj ↓0 has a subsequence for which"
        },
        {
          "blockId": "p4-6b51763d0e",
          "page": 4,
          "quote": "ux0,rj(y) := u(x0 + rjy)"
        },
        {
          "blockId": "p4-656ba12637",
          "page": 4,
          "quote": "rj\n−→U(y)\n(2.1)"
        },
        {
          "blockId": "p4-3a8780641d",
          "page": 4,
          "quote": "locally uniformly in Rd. The limit is a nonzero, nonnegative, one-homogeneous global minimizer.\nAlong a further subsequence the convergence is strong in H1\nloc, and the positivity indicators\nconverge in L1\nloc."
        },
        {
          "blockId": "p4-d4e305255d",
          "page": 4,
          "quote": "(iii) If one such limit equals (y · e)+ with |e| = 1, then near x0 the free boundary is a smooth\nhypersurface and {u > 0} is exactly one of its sides. The function u is smooth up to that\nhypersurface from the positive side, and"
        },
        {
          "blockId": "p4-e807330307",
          "page": 4,
          "quote": "u = 0,\n∇u = NE\non the free boundary,\n(2.2)"
        },
        {
          "blockId": "p4-377923367a",
          "page": 4,
          "quote": "where NE is the Euclidean unit normal pointing into {u > 0}."
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "lemma-4",
      "kind": "lemma",
      "label": "Lemma 2.2",
      "source": {
        "blockId": "p5-7b73106fd2",
        "page": 5,
        "quote": "Lemma 2.2 (Translation invariance of an off-origin blowup). Let u be a one-homogeneous global\nminimizer, let 0 ̸= x0 ∈∂{u > 0}, and let U be any limit in (2.1). Then"
      },
      "statement": [
        {
          "blockId": "p5-7b73106fd2",
          "page": 5,
          "quote": "Lemma 2.2 (Translation invariance of an off-origin blowup). Let u be a one-homogeneous global\nminimizer, let 0 ̸= x0 ∈∂{u > 0}, and let U be any limit in (2.1). Then"
        },
        {
          "blockId": "p5-16fb8ef603",
          "page": 5,
          "quote": "U(y + tx0) = U(y)\n(y ∈Rd, t ∈R)."
        }
      ],
      "proof": [
        {
          "blockId": "p5-19acdad6af",
          "page": 5,
          "quote": "Proof. Write ur = ux0,r. For fixed t ∈R and small r > 0, set r′ = r/(1 + rt) > 0. Homogeneity\ngives the exact identity"
        },
        {
          "blockId": "p5-a73e49f286",
          "page": 5,
          "quote": "ur(y + tx0) = u\n\n(1 + rt)(x0 + r′y)\n"
        },
        {
          "blockId": "p5-16a211c14c",
          "page": 5,
          "quote": "r\n= ur′(y).\n(2.3)"
        },
        {
          "blockId": "p6-474378faca",
          "page": 6,
          "quote": "Let L be a Lipschitz constant for u on a fixed neighborhood of x0. Since u(x0) = 0, for |y| ≤M\nand small r,"
        },
        {
          "blockId": "p6-909e57afa6",
          "page": 6,
          "quote": "|ur′(y) −ur(y)| ≤L|r −r′|M"
        },
        {
          "blockId": "p6-8cd3300b16",
          "page": 6,
          "quote": "r"
        },
        {
          "blockId": "p6-964333ec96",
          "page": 6,
          "quote": "r′\n+ LrM\n\n1\nr′ −1"
        },
        {
          "blockId": "p6-3302d0be5e",
          "page": 6,
          "quote": "= 2LM |r −r′|"
        },
        {
          "blockId": "p6-7fe9a0c698",
          "page": 6,
          "quote": "r′\n= 2LM|t|r."
        },
        {
          "blockId": "p6-42c2898c77",
          "page": 6,
          "quote": "Thus the r′ rescalings along the same chosen sequence have the same locally uniform limit as the r\nrescalings. Passing to that limit in (2.3) proves the assertion for every fixed y, t. No uniqueness of\nblowups is required."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-5",
      "kind": "lemma",
      "label": "Lemma 2.3",
      "source": {
        "blockId": "p6-35c36a36fe",
        "page": 6,
        "quote": "Lemma 2.3 (Minimality of a translation-invariant factor). Suppose U(y, s) = v(y) is a global\nminimizer in Rk × R, where v ≥0 belongs to H1\nloc(Rk). Then v is a global minimizer in Rk."
      },
      "statement": [
        {
          "blockId": "p6-35c36a36fe",
          "page": 6,
          "quote": "Lemma 2.3 (Minimality of a translation-invariant factor). Suppose U(y, s) = v(y) is a global\nminimizer in Rk × R, where v ≥0 belongs to H1\nloc(Rk). Then v is a global minimizer in Rk."
        }
      ],
      "proof": [
        {
          "blockId": "p6-5a86e5b19a",
          "page": 6,
          "quote": "Proof. Suppose that a ball B ⊂Rk admits a nonnegative q ∈H1(B) with q −v ∈H1\n0(B) and"
        },
        {
          "blockId": "p6-e4883ffbed",
          "page": 6,
          "quote": "J(v; B) −J(q; B) = δ > 0."
        },
        {
          "blockId": "p6-9d760e4b18",
          "page": 6,
          "quote": "Let ψ be the zero extension of q −v. For L > 1, choose ηL : R →[0, 1] equal to 1 on [−L, L], equal\nto 0 outside [−L −1, L + 1], and linear on each intervening interval. Define"
        },
        {
          "blockId": "p6-2f894589b6",
          "page": 6,
          "quote": "VL(y, s) = U(y, s) + ηL(s)ψ(y)."
        },
        {
          "blockId": "p6-0789ceed61",
          "page": 6,
          "quote": "This function is nonnegative: on B × R it is the convex combination (1 −ηL)v + ηLq. Its difference\nfrom U belongs to H1 and has compact support in a sufficiently large ball BL ⊂Rk+1, so it belongs\nto H1\n0(BL).\nThe energy difference on B × [−L, L] is −2Lδ. On either transition cylinder, convexity gives"
        },
        {
          "blockId": "p6-40fb45e6d6",
          "page": 6,
          "quote": "|∇yVL|2 ≤(1 −ηL)|∇v|2 + ηL|∇q|2,\n|∂sVL|2 ≤|q −v|2."
        },
        {
          "blockId": "p6-eb384bfd4a",
          "page": 6,
          "quote": "The difference of the positivity indicators is at most 1. Hence the sum of the two transition\ncontributions is bounded above by"
        },
        {
          "blockId": "p6-d5fd8317f7",
          "page": 6,
          "quote": "C = 2\nZ"
        },
        {
          "blockId": "p6-a280a08341",
          "page": 6,
          "quote": "\n|∇q|2 + |q −v|2 + 1\n\n,"
        },
        {
          "blockId": "p6-459382c6fd",
          "page": 6,
          "quote": "B"
        },
        {
          "blockId": "p6-0c9d12b210",
          "page": 6,
          "quote": "independently of L. Outside these three cylinders the energy is unchanged. Thus"
        },
        {
          "blockId": "p6-1a6a7a5e80",
          "page": 6,
          "quote": "J(VL; BL) −J(U; BL) ≤−2Lδ + C < 0"
        },
        {
          "blockId": "p6-1035b8716b",
          "page": 6,
          "quote": "for large L, contradicting minimality of U."
        },
        {
          "blockId": "p6-5aff418e48",
          "page": 6,
          "quote": "In dimension one, every nonnegative one-homogeneous Sobolev function has the form u(x) =\nb+x+ + b−(−x)+ with b+, b−≥0. If both slopes are positive, a nonnegative smooth bump ϕ\nsupported in (−R, R) with ϕ(0) > 0 preserves positivity volume and gives"
        },
        {
          "blockId": "p6-7aa4a96b63",
          "page": 6,
          "quote": "J(u + τϕ; (−R, R)) −J(u; (−R, R)) = −2τ(b+ + b−)ϕ(0) + τ 2\nZ R"
        },
        {
          "blockId": "p6-83bb31912e",
          "page": 6,
          "quote": "−R\n|ϕ′|2 < 0"
        },
        {
          "blockId": "p6-05b288455c",
          "page": 6,
          "quote": "for small τ > 0. Thus a nonzero minimizing cone has just one positive slope. After reflection write\nu(x) = bx+, b > 0. The competitors vt(x) = bR(x −t)+/(R −t) on (−R, R), |t| < R, have the"
        },
        {
          "blockId": "p7-d328f4a8f3",
          "page": 7,
          "quote": "same trace and energy b2R2/(R −t) + (R −t). Its derivative at the minimizing parameter t = 0 is\nb2 −1 = 0, so b = 1. This proves flatness in dimension one directly.\nThe lower-dimensional classification of Jerison–Savin [13, Theorem 1.1 and Corollary 1.2] and\nthe dimension-seven minimizing example of De Silva–Jerison [7, Theorem 1.1] give 5 ≤d∗≤7. De\nSilva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors.\nFor the exclusion argument, suppose from now on that"
        },
        {
          "blockId": "p7-7f67ab3c19",
          "page": 7,
          "quote": "d = d∗∈{5, 6},\n(2.4)"
        },
        {
          "blockId": "p7-8843a63db7",
          "page": 7,
          "quote": "and fix a nonzero nonflat one-homogeneous global minimizer u in Rd."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-6",
      "kind": "proposition",
      "label": "Proposition 2.4",
      "source": {
        "blockId": "p7-36823df07f",
        "page": 7,
        "quote": "Proposition 2.4 (Smooth original link). Every point of ∂{u > 0} \\ {0} is regular. Put mo = d −1\nand go = u|Smo. The set {go > 0} is nonempty and proper, has finitely many connected components,\nand their closures are pairwise disjoint compact smooth manifolds with boundary. On each such\ncomponent Ωo, the restriction of go is smooth up to the boundary from within Ωo and satisfies"
      },
      "statement": [
        {
          "blockId": "p7-36823df07f",
          "page": 7,
          "quote": "Proposition 2.4 (Smooth original link). Every point of ∂{u > 0} \\ {0} is regular. Put mo = d −1\nand go = u|Smo. The set {go > 0} is nonempty and proper, has finitely many connected components,\nand their closures are pairwise disjoint compact smooth manifolds with boundary. On each such\ncomponent Ωo, the restriction of go is smooth up to the boundary from within Ωo and satisfies"
        },
        {
          "blockId": "p7-616c575fdc",
          "page": 7,
          "quote": "∆Smogo = −mogo\nin Ωo,\ngo = 0,\n∇Smogo = No\non ∂Ωo,\n(2.5)"
        },
        {
          "blockId": "p7-f9e57cbfaa",
          "page": 7,
          "quote": "where No is the inward spherical unit normal. In particular, ∂Ωo ̸= ∅."
        }
      ],
      "proof": [
        {
          "blockId": "p7-df0b192be6",
          "page": 7,
          "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
        },
        {
          "blockId": "p7-5c054ff718",
          "page": 7,
          "quote": "∆Rdu(ry) = r−1\n∆Smogo(y) + mogo(y)\n\n(2.6)"
        },
        {
          "blockId": "p7-7608cd04c9",
          "page": 7,
          "quote": "in the positive phase. If go > 0 on the entire sphere, integrating the harmonicity identity over the\nsphere would give mo\nR\ngo = 0, a contradiction. Thus its positive set is proper.\nThe conical free boundary has the radial direction tangent to it at every nonzero point: the\ncurve r 7→rx lies in that free boundary. Consequently its smooth hypersurface is transverse to the\nunit sphere, and their intersection is a smooth hypersurface of Smo. Its regular charts have the\nspherical positive phase exactly on one side. Each point of the sphere therefore has a neighborhood\nwhose intersection with {go > 0} is either connected or empty (using a small ball for points away\nfrom the free boundary). A finite subcover of the sphere shows that there are only finitely many\npositive components. If two component closures met, their common point would be a free-boundary\npoint; the connected positive side of a regular chart there would meet both components, forcing\nthem to coincide. The same charts show that each closure is a smooth compact domain. Each\ncomponent has nonempty boundary because the sphere is connected and the positive set is proper.\nSmoothness of u from its positive side gives smoothness of go on these closures. Equation (2.6)\ngives the interior equation in (2.5). On the boundary, the radial derivative of u is go = 0, so the\nEuclidean gradient is tangent to the sphere. Equation (2.2) then gives the remaining boundary\nconditions."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-7",
      "kind": "lemma",
      "label": "Lemma 2.5",
      "source": {
        "blockId": "p7-425de8ddc1",
        "page": 7,
        "quote": "Lemma 2.5 (Selection of a nonflat component). There is a component Ωo in Proposition 2.4 such\nthat\nTo := ∇2\nSmogo + go Id ̸≡0\non Ωo.\n(2.7)"
      },
      "statement": [
        {
          "blockId": "p7-425de8ddc1",
          "page": 7,
          "quote": "Lemma 2.5 (Selection of a nonflat component). There is a component Ωo in Proposition 2.4 such\nthat\nTo := ∇2\nSmogo + go Id ̸≡0\non Ωo.\n(2.7)"
        },
        {
          "blockId": "p7-4a3d32fa84",
          "page": 7,
          "quote": "Here Id denotes the spherical metric, viewed as a symmetric two-tensor."
        }
      ],
      "proof": [
        {
          "blockId": "p8-8930ff9fba",
          "page": 8,
          "quote": "Proof. For y in the spherical positive set and r > 0,"
        },
        {
          "blockId": "p8-4aa778ebb5",
          "page": 8,
          "quote": "∇Rdu(ry) = go(y)y + ∇Smogo(y)."
        },
        {
          "blockId": "p8-ffe8311d4a",
          "page": 8,
          "quote": "Differentiating this formula, or using Euler’s identity for the radial entries, gives"
        },
        {
          "blockId": "p8-81876620ea",
          "page": 8,
          "quote": "D2u(ry)[y, ·] = 0,\nD2u(ry)[v1, v2] = r−1To(y)[v1, v2]\n(v1, v2 ⊥y).\n(2.8)"
        },
        {
          "blockId": "p8-5eb064b940",
          "page": 8,
          "quote": "Suppose that To vanished identically on every positive component.\nThen D2u = 0 on each\ncorresponding connected open cone Di, so u(x) = bi · x + ci there. Homogeneity forces ci = 0, and\npositivity forces bi ̸= 0. Thus Di ⊂Hi := {bi · x > 0}. The set Di is also closed relative to Hi: if\nx ∈Di ∩Hi, continuity gives u(x) = bi · x > 0; a small positive neighborhood of x meets Di, hence\nbelongs to the same component. Since Hi is connected, Di = Hi. The Bernoulli condition at any\nnonzero point of its boundary gives |bi| = 1.\nTwo disjoint open half-spaces through the origin must have opposite unit normals: if their normals\nare e, f with f ̸= −e, the vector e + f lies in both. But opposite half-spaces have spherical closures\nmeeting along the equator e⊥∩Smo, contrary to the pairwise disjoint closures in Proposition 2.4.\nThus there is exactly one positive component, and u = (x · e)+, contrary to nonflatness. This proves\n(2.7)."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-8",
      "kind": "proposition",
      "label": "Proposition 2.6",
      "source": {
        "blockId": "p8-05cf6278e3",
        "page": 8,
        "quote": "Proposition 2.6 (Suspension preserves global minimality). If u ≥0 is a global minimizer in Rk,\nthen eu(x, s) = u(x) is a global minimizer in Rk+1. If u is the cone in (2.4) with d = 5, the singular\nfree-boundary set of eu is exactly {0}×R. For the component chosen in Lemma 2.5, the corresponding\npositive component on S5 is"
      },
      "statement": [
        {
          "blockId": "p8-05cf6278e3",
          "page": 8,
          "quote": "Proposition 2.6 (Suspension preserves global minimality). If u ≥0 is a global minimizer in Rk,\nthen eu(x, s) = u(x) is a global minimizer in Rk+1. If u is the cone in (2.4) with d = 5, the singular\nfree-boundary set of eu is exactly {0}×R. For the component chosen in Lemma 2.5, the corresponding\npositive component on S5 is"
        },
        {
          "blockId": "p8-7722e556c0",
          "page": 8,
          "quote": "eΩ=\n\nΦ(t, y) := (sin t y, cos t) : 0 < t < π, y ∈Ωo\n\t\n,\neg(Φ(t, y)) = sin t go(y).\n(2.9)"
        },
        {
          "blockId": "p8-9d96cf4b31",
          "page": 8,
          "quote": "Its boundary consists of the smooth side Φ((0, π) × ∂Ωo) and the two singular tips p+ = (0, 1) and\np−= (0, −1)."
        }
      ],
      "proof": [
        {
          "blockId": "p8-fdfcec216b",
          "page": 8,
          "quote": "Proof. Fix a ball B = BR((xc, sc)) ⊂Rk+1 and a nonnegative V ∈H1(B) with V −eu ∈H1\n0(B).\nExtend that difference by zero to F ∈H1(Rk+1). For |s −sc| < R, let"
        },
        {
          "blockId": "p8-367d182055",
          "page": 8,
          "quote": "Bs = B√"
        },
        {
          "blockId": "p8-c3e3c2d6dd",
          "page": 8,
          "quote": "R2−(s−sc)2(xc)"
        },
        {
          "blockId": "p8-26bd66ccb6",
          "page": 8,
          "quote": "be the horizontal slice. For almost every such s, F(·, s)|Bs ∈H1\n0(Bs). To see the zero-boundary\nassertion without a trace regularity assumption on V , approximate F in H1(Rk+1) by the zero\nextensions of functions in C∞\nc (B). After passing to a subsequence, Fubini’s Theorem gives convergence\nof their slices to F(·, s) in H1(Rk) for almost every s. Each approximating slice lies in C∞\nc (Bs), so\nthe assertion follows by closure.\nFor almost every s, Vs = V (·, s) is therefore an admissible nonnegative competitor for u in Bs.\nIntegrating its minimizing inequality and using the nonnegativity of the additional derivative energy,\nwe obtain"
        },
        {
          "blockId": "p8-6ae0fb328c",
          "page": 8,
          "quote": "J(eu; B) =\nZ sc+R"
        },
        {
          "blockId": "p8-f86a1e4e3c",
          "page": 8,
          "quote": "sc−R\nJ(u; Bs) ds"
        },
        {
          "blockId": "p8-cf7551d8db",
          "page": 8,
          "quote": "≤\nZ sc+R"
        },
        {
          "blockId": "p8-1e92aa4b2f",
          "page": 8,
          "quote": "sc−R\nJ(Vs; Bs) ds ≤J(V ; B)."
        },
        {
          "blockId": "p9-bf3a0756e6",
          "page": 9,
          "quote": "This proves the full ballwise minimizing property, for arbitrary centers and arbitrary admissible\nSobolev competitors. One-homogeneity and nontriviality are plainly preserved. Flatness of eu would\nforce its half-space normal to have zero last coordinate, and would therefore force u to be flat.\nIn the conical case, u(0) = 0 and a positive ray approaches the origin, so 0 ∈∂{u > 0}. The\nproduct identity\n∂{eu > 0} = ∂{u > 0} × R"
        },
        {
          "blockId": "p9-6271b34b61",
          "page": 9,
          "quote": "shows that all its off-axis points are regular: their positive phase is the product of a smooth one-sided\nregular chart for u and an interval. At any axis point z0 = (0, s0), every rescaling is exactly"
        },
        {
          "blockId": "p9-a95f4394b0",
          "page": 9,
          "quote": "eu(z0 + r(x, s))"
        },
        {
          "blockId": "p9-535468eefc",
          "page": 9,
          "quote": "r\n= u(rx)"
        },
        {
          "blockId": "p9-251ecacf7c",
          "page": 9,
          "quote": "r\n= eu(x, s)."
        },
        {
          "blockId": "p9-6632f13d2e",
          "page": 9,
          "quote": "This is nonflat, so no axis point is regular. The axis meets S5 only in p+ and p−.\nAway from these two points, Φ is a smooth product coordinate map from (0, π) × S4 to\nS5 \\ {p+, p−}. Positivity in these coordinates is precisely go(y) > 0, so connected components are\nas in (2.9). Since Ωo is compact, its suspended closure is obtained by adding the two endpoints of\nthe interval, with each endpoint collapsed to its tip. Its remaining boundary is exactly the stated\nsmooth side."
        },
        {
          "blockId": "p9-b613bfd01d",
          "page": 9,
          "quote": "Notation for the remaining argument.\nFix the original component Ωo ⊂Smo selected in\nLemma 2.5, where mo ∈{4, 5}. If mo = 5, set Ω= Ωo, g = go, and Σ = ∅. If mo = 4, set Ω= eΩ,\ng = eg, and Σ = {p+, p−}. In either case Ω⊂S5 is connected, and g comes from a nonnegative\none-homogeneous global minimizer in R6. The closure is smooth away from Σ, with positive phase\non one side of its regular boundary\n∂regΩ:= ∂Ω\\ Σ."
        },
        {
          "blockId": "p9-6c6e1e3c68",
          "page": 9,
          "quote": "All spherical boundary values below are taken from the positive side. Tests on Ωwill be smooth and\nsupported away from Σ. Every compact portion of the selected Euclidean cone’s closure that avoids\nthe origin and, in the suspended case, the axis is separated from all other positive components. This\nfollows from the disjoint compact original closures, and in the suspension from the same fact on any\nclosed subinterval of (0, π). Consequently the local variations on this component can be made as\nvariations of the global minimizer. No minimizing property of the selected component extended by\nzero is needed."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-9",
      "kind": "proposition",
      "label": "Proposition 3.1",
      "source": {
        "blockId": "p10-ce852626e7",
        "page": 10,
        "quote": "Proposition 3.1 (Interior identities). The tensor T is symmetric and trace-free, and B is fully\nsymmetric and trace-free in every pair of slots. In particular div T = 0. The following identities\nhold:"
      },
      "statement": [
        {
          "blockId": "p10-ce852626e7",
          "page": 10,
          "quote": "Proposition 3.1 (Interior identities). The tensor T is symmetric and trace-free, and B is fully\nsymmetric and trace-free in every pair of slots. In particular div T = 0. The following identities\nhold:"
        },
        {
          "blockId": "p10-ead43e8dd0",
          "page": 10,
          "quote": "∇lBijk −∇iBljk = δljTik + δlkTij −δijTlk −δikTlj,\n(3.3)"
        },
        {
          "blockId": "p10-4f905adc86",
          "page": 10,
          "quote": "∆T = mT,\n∆w2 = 2mw2 + 2|B|2,\n(3.4)"
        },
        {
          "blockId": "p10-671ef66fb9",
          "page": 10,
          "quote": "∇lpi = Tli −gδli,\n∇lg = pl,\n(3.5)"
        },
        {
          "blockId": "p10-9cd42a53f7",
          "page": 10,
          "quote": "∇s2 = 2Tp,\n∆s2 = 2w2.\n(3.6)"
        },
        {
          "blockId": "p10-fb891dff39",
          "page": 10,
          "quote": "On the open set {w > 0} one also has"
        },
        {
          "blockId": "p10-39b3f272e5",
          "page": 10,
          "quote": "∇lw = w−1TjkBljk,\n|∇w| ≤|B|.\n(3.7)"
        },
        {
          "blockId": "p10-6d18a15113",
          "page": 10,
          "quote": "The inequality |∇(w2)|2 ≤4w2|B|2 holds everywhere, including at w = 0."
        }
      ],
      "proof": [
        {
          "blockId": "p10-12ceae48c3",
          "page": 10,
          "quote": "Proof. The trace equation is tr T = ∆g + mg = 0. On the unit sphere the commutation rule for the\ncovector p gives\n∇i(∇2g)jk −∇j(∇2g)ik = δikpj −δjkpi."
        },
        {
          "blockId": "p10-3fb8cef4c5",
          "page": 10,
          "quote": "Adding the derivatives of g Id cancels the right side. Thus T is Codazzi: ∇iTjk = ∇jTik. Together\nwith symmetry in j, k, this proves full symmetry of B. Differentiating tr T = 0 shows Bijj = 0; full\nsymmetry gives all other traces and hence div T = 0.\nFor clarity, the same curvature convention on a covariant two-tensor is"
        },
        {
          "blockId": "p10-b7b15cdbf0",
          "page": 10,
          "quote": "([∇l, ∇i]T)jk = δljTik + δlkTij −δijTlk −δikTlj."
        },
        {
          "blockId": "p10-3a51ae8b53",
          "page": 10,
          "quote": "This proves (3.3). Substitute (l, i, j, k) = (i, j, i, k) there and sum over i. By symmetry and the zero\ntraces of B, the left side is ∆Tjk, whereas the right side is"
        },
        {
          "blockId": "p10-d0dac569b1",
          "page": 10,
          "quote": "mTjk + Tjk −Tjk −δjk tr T = mTjk."
        },
        {
          "blockId": "p10-685261eec2",
          "page": 10,
          "quote": "The product rule for |T|2 now proves (3.4). The definitions give (3.5) and"
        },
        {
          "blockId": "p10-3dd350e865",
          "page": 10,
          "quote": "∇ls2 = 2gpl + 2pi(Tli −gδli) = 2Tlipi."
        },
        {
          "blockId": "p10-df9c6a96db",
          "page": 10,
          "quote": "Taking the divergence and using div T = 0 and tr T = 0 gives ∆s2 = 2Tli(Tli −gδli) = 2w2. Finally,\ndifferentiating w2 and applying Cauchy–Schwarz proves both gradient estimates."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-10",
      "kind": "lemma",
      "label": "Lemma 3.2",
      "source": {
        "blockId": "p11-79aa63b2b7",
        "page": 11,
        "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
      },
      "statement": [
        {
          "blockId": "p11-79aa63b2b7",
          "page": 11,
          "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
        },
        {
          "blockId": "p11-901a03e7df",
          "page": 11,
          "quote": "Then\nHo > 0,\nTopo = −Hopo,\nwo > 0\non ∂Ωo.\n(3.9)"
        },
        {
          "blockId": "p11-17796fd435",
          "page": 11,
          "quote": "Consequently wo has a positive minimum on ∂Ωo, and its zero set is a compact subset of Ωo."
        }
      ],
      "proof": [
        {
          "blockId": "p11-48cf81081f",
          "page": 11,
          "quote": "Proof. The boundary conditions give s2\no = 1 on ∂Ωo. By (3.6), s2\no is subharmonic. The maximum\nprinciple on the compact closure gives s2\no ≤1. If equality held at an interior point, the strong\nmaximum principle would make s2\no constant on the connected component, and then ∆s2\no = 2w2\no\nwould imply To ≡0. Thus s2\no < 1 inside.\nThe smooth boundary has the interior sphere property at each point. The Hopf boundary\nlemma therefore gives ∂Nos2\no < 0. Since po = No there, (3.6) identifies this derivative as 2To(No, No),\nproving Ho > 0. For a tangent vector τ to ∂Ωo, tangential differentiation of s2\no = 1 similarly gives\nTo(τ, No) = 0. This proves the eigenvector assertion and wo ≥Ho > 0. Continuity and compactness\nof the boundary give a positive minimum of wo and a neighborhood of the boundary disjoint from\nits zero set."
        },
        {
          "blockId": "p11-dbdd6650ef",
          "page": 11,
          "quote": "The remaining boundary formulas are local. Thus they apply on the original boundary and,\nonce the strict sign has been transported, on the regular side of the suspension."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-11",
      "kind": "proposition",
      "label": "Proposition 3.3",
      "source": {
        "blockId": "p11-fdd2a71ec4",
        "page": 11,
        "quote": "Proposition 3.3 (Boundary jets). Suppose g = 0, p = N, and T(N, N) = −H < 0 on a smooth\nboundary portion, where N is inward. Let Greek indices be tangent to this boundary and put"
      },
      "statement": [
        {
          "blockId": "p11-fdd2a71ec4",
          "page": 11,
          "quote": "Proposition 3.3 (Boundary jets). Suppose g = 0, p = N, and T(N, N) = −H < 0 on a smooth\nboundary portion, where N is inward. Let Greek indices be tangent to this boundary and put"
        },
        {
          "blockId": "p11-4388b4b044",
          "page": 11,
          "quote": "hαβ = ⟨∇αN, eβ⟩."
        },
        {
          "blockId": "p11-c696783e4a",
          "page": 11,
          "quote": "Then TNα = 0, Tαβ = hαβ, and H = tr h; in particular H is a sum of principal curvatures with the\nspecified convention, without division by m −1. If ∇∂is the boundary connection, all components\nof B are described, up to permutation of slots, by"
        },
        {
          "blockId": "p11-064ff4fb38",
          "page": 11,
          "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
        },
        {
          "blockId": "p11-d8ff12d046",
          "page": 11,
          "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
        },
        {
          "blockId": "p11-a243f0a817",
          "page": 11,
          "quote": "The components with an odd number of normal slots are thus determined by the pointwise tensor T;\nthe other components involve tangential derivatives of curvature. The tensor V below encodes this\nodd-normal part after normalization. Here w > 0. Define the normalized tensors and eigenvalue"
        },
        {
          "blockId": "p11-a3924f6d0b",
          "page": 11,
          "quote": "A = T/w,\nλ = TNN/w = −H/w,\nS0 = λA −A2,"
        },
        {
          "blockId": "p11-a37ebd7ab3",
          "page": 11,
          "quote": "and the symmetric tensor"
        },
        {
          "blockId": "p11-ee96e4a03b",
          "page": 11,
          "quote": "Vijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(3.12)"
        },
        {
          "blockId": "p11-7580df6f09",
          "page": 11,
          "quote": "Then |p| = 1, Ap = λp, λ < 0, S0p = 0, tr S0 = −1, and V is trace-free. The tensors B/w2 and V\nagree in every component with an odd number of normal slots. Moreover,"
        },
        {
          "blockId": "p11-f6ddc57c80",
          "page": 11,
          "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
        }
      ],
      "proof": [
        {
          "blockId": "p12-7bd615b1c8",
          "page": 12,
          "quote": "Proof. On the boundary, tangential differentiation of |p|2 = 1 gives TNα = 0, and ∇αp = Teα\nbecause g = 0. Since p = N along the boundary, this proves Tαβ = hαβ. The trace equation for\nT gives tr h = −TNN = H. In differentiating three tangential entries of T, the normal connection\nterms pair with T(N, eα) = 0. This yields the first formula in (3.10). Likewise,"
        },
        {
          "blockId": "p12-8755606744",
          "page": 12,
          "quote": "∂αT(N, N) = BαNN + 2T(∇αN, N) = BαNN,"
        },
        {
          "blockId": "p12-e7c20dd423",
          "page": 12,
          "quote": "which proves the second formula by symmetry of B.\nFor a tangent field eα, differentiate T(N, eα) = 0 along eβ. Since ⟨N, ∇βeα⟩= −hβα, this gives"
        },
        {
          "blockId": "p12-84d5476660",
          "page": 12,
          "quote": "0 = BβNα + (T 2)βα −TNNTβα."
        },
        {
          "blockId": "p12-390df1acf6",
          "page": 12,
          "quote": "This proves the first formula in (3.11). Taking its tangential trace gives\nX"
        },
        {
          "blockId": "p12-ecab27bc42",
          "page": 12,
          "quote": "α\nBNαα = TNN\nX"
        },
        {
          "blockId": "p12-80f71c8f3c",
          "page": 12,
          "quote": "α\nTαα −\nX"
        },
        {
          "blockId": "p12-b900895217",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −w2."
        },
        {
          "blockId": "p12-b912b80307",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −T 2\nNN −\nX"
        },
        {
          "blockId": "p12-4584ce2868",
          "page": 12,
          "quote": "The zero trace of B then gives BNNN = w2.\nNormalization gives S0p = 0 and tr S0 = −|A|2 = −1. Consequently Vijj = pi tr S0 + 2(S0p)i +\npi = 0. In an orthonormal frame with first vector p = N, one has VNαβ = (S0)αβ and VNNN = 1.\nThese are exactly the normalized identities (3.11), and VNjk = S0,jk + pjpk. Finally, T has only its\nnormal-normal and tangential-tangential blocks, so the odd-normal formulas account for the entire\ncontraction TjkBNjk. Thus"
        },
        {
          "blockId": "p12-a1dcbd3ddc",
          "page": 12,
          "quote": "∂Nw = w−1TjkBNjk = w2Ajk(S0 + p ⊗p)jk = w2(2λ −tr A3),"
        },
        {
          "blockId": "p12-ec22d1dead",
          "page": 12,
          "quote": "as asserted."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-12",
      "kind": "proposition",
      "label": "Proposition 3.4",
      "source": {
        "blockId": "p12-6e66e62020",
        "page": 12,
        "quote": "Proposition 3.4 (Suspended data). If mo = 4, write ρ = sin t, c = cos t and parameterize the\nregular part of the suspended component by"
      },
      "statement": [
        {
          "blockId": "p12-6e66e62020",
          "page": 12,
          "quote": "Proposition 3.4 (Suspended data). If mo = 4, write ρ = sin t, c = cos t and parameterize the\nregular part of the suspended component by"
        },
        {
          "blockId": "p12-9d3f73e267",
          "page": 12,
          "quote": "(t, y) 7−→(ρy, c),\n0 < t < π,\ny ∈Ωo."
        },
        {
          "blockId": "p12-31d748db9a",
          "page": 12,
          "quote": "Its metric is dt2 + ρ2ds2\nS4, and eg = ρgo. With E0 = ∂t and Eα = ρ−1eα for a lifted orthonormal\nframe on S4, the data satisfy"
        },
        {
          "blockId": "p12-b9af30c473",
          "page": 12,
          "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
        },
        {
          "blockId": "p12-f4f4514465",
          "page": 12,
          "quote": "The only possibly nonzero components of eB are"
        },
        {
          "blockId": "p12-fe39066800",
          "page": 12,
          "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
        },
        {
          "blockId": "p12-bcec913ecf",
          "page": 12,
          "quote": "In particular,"
        },
        {
          "blockId": "p12-8b2672b64e",
          "page": 12,
          "quote": "ew = ρ−1wo,\n| eB|2 = ρ−4\n|Bo|2 + 3c2w2\no\n\n,\neH = ρ−1Ho\non the regular side.\n(3.17)"
        },
        {
          "blockId": "p12-a27512c3aa",
          "page": 12,
          "quote": "The volume and side-boundary measures are"
        },
        {
          "blockId": "p12-8a8302c5ce",
          "page": 12,
          "quote": "dV = ρ4 dt dVo,\ndσ = ρ3 dt dσo.\n(3.18)"
        },
        {
          "blockId": "p12-7c1a267d69",
          "page": 12,
          "quote": "Every local identity of Proposition 3.1 holds with m = 5, and Proposition 3.3 holds on the regular\nside. Also es2 < 1 inside and eH, ew > 0 on that side."
        }
      ],
      "proof": [
        {
          "blockId": "p13-b107b6910b",
          "page": 13,
          "quote": "Proof. At a point where the base frame is normal, the warped metric has connection"
        },
        {
          "blockId": "p13-1921e32005",
          "page": 13,
          "quote": "∇E0E0 = ∇E0Eα = 0,\n∇EαE0 = (c/ρ)Eα,\n∇EαEβ = −(c/ρ)δαβE0."
        },
        {
          "blockId": "p13-61831a7400",
          "page": 13,
          "quote": "The horizontal part of the last expression at other points is the base connection with factor ρ−1.\nDifferentiating eg = ρgo gives (3.14). Its Hessian has entries"
        },
        {
          "blockId": "p13-13d5fe94da",
          "page": 13,
          "quote": "(∇2eg)00 = −ρgo,\n(∇2eg)0α = 0,\n(∇2eg)αβ = ρ−1(∇2\nogo)αβ + (c2/ρ)goδαβ."
        },
        {
          "blockId": "p13-aca19449d5",
          "page": 13,
          "quote": "Adding eg Id proves (3.15). Differentiating that tensor with the displayed connection gives (3.16);\nentries with two or three zero slots vanish. Each term with exactly one zero slot occurs in three\npositions, which proves the factor 3 in (3.17).\nThe inward normal to the side is the horizontal lift of No. Thus the normal-normal entry of\n(3.15) gives eH = ρ−1Ho. The metric on the side is dt2 + ρ2ds2\n∂Ωo, proving (3.18). The trace of (3.15)\nis zero, so ∆eg = −5eg. The interior identities follow locally from Proposition 3.1. The bounds for s2,\nH and w follow from the original compact component and (3.14)–(3.17); no maximum principle at a\ntip is used. The local boundary hypotheses and hence Proposition 3.3 hold on the regular side."
        },
        {
          "blockId": "p13-6249f48ba3",
          "page": 13,
          "quote": "We now use the unadorned notation Ω, g, p, T, B, w, H for the working component in S5 defined\nat the end of Section 2. It is either Ωo when mo = 5, or the above suspension when mo = 4. All\nboundary integrals mean integrals over the regular boundary, with the two tips omitted in the\nsuspended case. The original data retain the subscript o when needed."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-13",
      "kind": "proposition",
      "label": "Proposition 3.5",
      "source": {
        "blockId": "p13-2b3a07bdc3",
        "page": 13,
        "quote": "Proposition 3.5 (Euclidean stability on a component). Let D be the selected positivity component\nof a global minimizing cone in Rm+1. Let ϕ be any real smooth function up to the regular boundary\nof D, with compact support in that regular closure away from the vertex and from any other singular\npoints. Then\nZ"
      },
      "statement": [
        {
          "blockId": "p13-2b3a07bdc3",
          "page": 13,
          "quote": "Proposition 3.5 (Euclidean stability on a component). Let D be the selected positivity component\nof a global minimizing cone in Rm+1. Let ϕ be any real smooth function up to the regular boundary\nof D, with compact support in that regular closure away from the vertex and from any other singular\npoints. Then\nZ"
        },
        {
          "blockId": "p13-4c29be1598",
          "page": 13,
          "quote": "D\n|∇ϕ|2 dx ≥\nZ"
        },
        {
          "blockId": "p13-39ac08043a",
          "page": 13,
          "quote": "∂D\nHEϕ2 dσE,\nHE = −D2u(NE, NE),\n(3.19)"
        },
        {
          "blockId": "p13-979cdf02af",
          "page": 13,
          "quote": "where NE is the inward Euclidean unit normal. No restriction is imposed on the boundary trace or\nnormal derivative of ϕ."
        }
      ],
      "proof": [
        {
          "blockId": "p13-a694bf1a3d",
          "page": 13,
          "quote": "Proof. Every regular free-boundary point has a neighborhood where the positive phase is the single\nside belonging to D. Compactness of the boundary portion meeting supp ϕ gives finitely many such\nneighborhoods and a collar separated from every other positivity component and every singular\npoint. Choose the collar over a slightly larger boundary portion, so that the extension of ϕ can\nvanish near its lateral edge. Extend ϕ smoothly across this boundary with compact support in these\nneighborhoods. Also extend the positive side of u smoothly to a function F on the collar, with its\ninterior jets unchanged. Since u = 0 and ∇u = NE on the boundary, shrinking the collar makes\nF negative on its exterior side and positive on its interior side. Such an extension need only be\nsmooth to finite order sufficient for the expansions below; smooth boundary regularity supplies this,\nand a partition of unity joins the local extensions while preserving the interior function and its jets.\nFor a real parameter ε of either sufficiently small sign, use (F + εϕ)+ in the collar, u + εϕ on the\nrest of the affected portion of D, and u elsewhere. These definitions agree: F = u on the interior of\nthe collar, and on the compact interior support outside a smaller collar u has a positive minimum."
        },
        {
          "blockId": "p14-66c1ab50cc",
          "page": 14,
          "quote": "Near the exterior and lateral edges of the modification the function agrees with u. The resulting\ncompetitor uε is nonnegative, belongs to H1\nloc, and has compactly supported difference from u in a\ncontaining ball. It is therefore admissible for the original global minimizer. This construction uses\nno separate minimizing assertion for D.\nWe compute the change in energy in such a ball. First integrate the perturbed density over the\nfixed original domain D, and then account for the signed boundary layer where its positivity set\nchanges. The fixed-domain contribution is"
        },
        {
          "blockId": "p14-5471f5971d",
          "page": 14,
          "quote": "2ε\nZ"
        },
        {
          "blockId": "p14-72f4405762",
          "page": 14,
          "quote": "D\n⟨∇u, ∇ϕ⟩dx + ε2\nZ"
        },
        {
          "blockId": "p14-194d0e1184",
          "page": 14,
          "quote": "D\n|∇ϕ|2 dx"
        },
        {
          "blockId": "p14-1cf2fec128",
          "page": 14,
          "quote": "= −2ε\nZ"
        },
        {
          "blockId": "p14-9b869eba9c",
          "page": 14,
          "quote": "∂D\nϕ dσE + ε2\nZ"
        },
        {
          "blockId": "p14-563124ccb1",
          "page": 14,
          "quote": "D\n|∇ϕ|2 dx.\n(3.20)"
        },
        {
          "blockId": "p14-193e7fe7bf",
          "page": 14,
          "quote": "Here harmonicity eliminates the interior term and the outward normal is −NE, so ∂−NEu = −1.\nAll integrations by parts have compact support on the regular portion.\nUse coordinates (y, ξ) obtained by flowing from the original boundary under"
        },
        {
          "blockId": "p14-5dd6d92075",
          "page": 14,
          "quote": "VF =\n∇F\n|∇F|2 ,\nξ = F."
        },
        {
          "blockId": "p14-534134ce25",
          "page": 14,
          "quote": "The denominator is nonzero on a sufficiently small compact collar. At ξ = 0, VF = NE. If j(y, ξ) is\nthe volume Jacobian relative to dσE(y) dξ, then"
        },
        {
          "blockId": "p14-9f21210952",
          "page": 14,
          "quote": "j(y, 0) = 1,\n∂ξj(y, 0) = div VF (y, 0) = ∆F −2FNENE = 2HE.\n(3.21)"
        },
        {
          "blockId": "p14-443810c310",
          "page": 14,
          "quote": "The equality ∆F = 0 on the original boundary follows from matching the interior second derivatives;\nharmonicity outside D is unnecessary. Moreover,"
        },
        {
          "blockId": "p14-ee2a00e317",
          "page": 14,
          "quote": "∂ξ|∇F|2\nξ=0 = 2FNENE = −2HE,"
        },
        {
          "blockId": "p14-7772931587",
          "page": 14,
          "quote": "and hence\n∂ξ\n\n(1 + |∇F|2)j\n\nξ=0 = 2HE.\n(3.22)"
        },
        {
          "blockId": "p14-85993fe6a5",
          "page": 14,
          "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
        },
        {
          "blockId": "p14-e7b31ba882",
          "page": 14,
          "quote": "Throughout the layer |ξ| ≤C|ε|, its perturbed energy density is"
        },
        {
          "blockId": "p14-0461573878",
          "page": 14,
          "quote": "(1 + |∇(F + εϕ)|2)j = 2 + 2εϕN + 2HEξ + O(ε2 + |εξ| + ξ2).\n(3.24)"
        },
        {
          "blockId": "p14-a047bf202f",
          "page": 14,
          "quote": "Integrating this density from bε to 0 is the signed correction, whether the phase expands or contracts.\nThe terms of order at most two are\nZ 0"
        },
        {
          "blockId": "p14-10f6d62795",
          "page": 14,
          "quote": "bε\n(1 + |∇(F + εϕ)|2)j dξ = −2bε −2εϕNbε −HEb2\nε + O(|ε|3)"
        },
        {
          "blockId": "p14-59fd4da02e",
          "page": 14,
          "quote": "= 2εϕ0 + ε2\n−2ϕ0ϕN + 2ϕ0ϕN −HEϕ2\n0\n\n+ O(|ε|3)"
        },
        {
          "blockId": "p14-b2fb2c5ad6",
          "page": 14,
          "quote": "= 2εϕ0 −ε2HEϕ2\n0 + O(|ε|3).\n(3.25)"
        },
        {
          "blockId": "p15-01dace12df",
          "page": 15,
          "quote": "This displays the cancellation of the normal-derivative terms. The uniform remainder is integrable\nover the compact affected boundary. Adding (3.20) also cancels the linear terms, leaving"
        },
        {
          "blockId": "p15-e8738a7c4a",
          "page": 15,
          "quote": "\n+ o(ε2)."
        },
        {
          "blockId": "p15-d3a05d2d6f",
          "page": 15,
          "quote": "J(uε) −J(u) = ε2\nZ"
        },
        {
          "blockId": "p15-b4d5841cf2",
          "page": 15,
          "quote": "D\n|∇ϕ|2 dx −\nZ"
        },
        {
          "blockId": "p15-9f647298da",
          "page": 15,
          "quote": "∂D\nHEϕ2 dσE"
        },
        {
          "blockId": "p15-d0552c7cd2",
          "page": 15,
          "quote": "The energies here are in a ball containing the modification. Minimality and division by ε2 prove\n(3.19)."
        },
        {
          "blockId": "p15-5286d7c1ec",
          "page": 15,
          "quote": "The angular constant comes from the sharp radial Hardy threshold, which also enters the cone\ninstability criteria in [13, Propositions 2.1–2.2]. We include the radial calculation to specify the\ncompact supports and the order of limits."
        }
      ],
      "guide": []
    },
    {
      "id": "corollary-14",
      "kind": "corollary",
      "label": "Corollary 3.6",
      "source": {
        "blockId": "p15-e7b7ce4464",
        "page": 15,
        "quote": "Corollary 3.6 (Spherical stability). For every real function v smooth up to the regular boundary of\nthe working component Ω⊂S5, with compact support in its closure away from the tips if present,\nZ"
      },
      "statement": [
        {
          "blockId": "p15-e7b7ce4464",
          "page": 15,
          "quote": "Corollary 3.6 (Spherical stability). For every real function v smooth up to the regular boundary of\nthe working component Ω⊂S5, with compact support in its closure away from the tips if present,\nZ"
        },
        {
          "blockId": "p15-9bcb9d1c60",
          "page": 15,
          "quote": "\n|∇v|2 + 4v2\ndV.\n(3.26)"
        },
        {
          "blockId": "p15-8f2c317ea2",
          "page": 15,
          "quote": "∂Ω\nHv2 dσ ≤\nZ"
        },
        {
          "blockId": "p15-a039742004",
          "page": 15,
          "quote": "Ω"
        },
        {
          "blockId": "p15-e09c2fcd7c",
          "page": 15,
          "quote": "The trace of v on the regular boundary is arbitrary."
        }
      ],
      "proof": [
        {
          "blockId": "p15-aabfeea988",
          "page": 15,
          "quote": "Proof. More generally, consider a link component Ω⊂Sm with angular tests supported in its regular\nclosure, and write D = {ry : r > 0, y ∈Ω}. On its regular side the normal is tangent to Sm, and\n(3.2) gives HE(ry) = r−1H(y). Polar volume has factor rm dr and side area has factor rm−1 dr. For\nζ ∈C∞\nc ((0, ∞)) and an allowed angular test v, put ϕ(ry) = ζ(r)v(y) in Proposition 3.5. It gives\nZ ∞"
        },
        {
          "blockId": "p15-d356107f3e",
          "page": 15,
          "quote": "0\nrm−2ζ2 dr\n Z"
        },
        {
          "blockId": "p15-3575cea404",
          "page": 15,
          "quote": "∂Ω\nHv2 dσ ≤\nZ ∞"
        },
        {
          "blockId": "p15-a298907820",
          "page": 15,
          "quote": "0\nrm−2ζ2 dr\n Z"
        },
        {
          "blockId": "p15-b2ecf62b93",
          "page": 15,
          "quote": "Ω\n|∇v|2 dV"
        },
        {
          "blockId": "p15-8bd8f07404",
          "page": 15,
          "quote": "+\nZ ∞"
        },
        {
          "blockId": "p15-88096af21e",
          "page": 15,
          "quote": "0\nrm|ζ′|2 dr\n Z"
        },
        {
          "blockId": "p15-7de63b4ca0",
          "page": 15,
          "quote": "Ω\nv2 dV."
        },
        {
          "blockId": "p15-294693052d",
          "page": 15,
          "quote": "For each fixed ζ and v, the Euclidean support lies in a finite annulus and is separated from the\nsingular set and other components, so the preceding second variation applies.\nSet β = (m −1)/2. For fζ(r) = rβζ(r), integration of the cross term gives the radial Hardy\nidentity\nZ ∞"
        },
        {
          "blockId": "p15-22db447160",
          "page": 15,
          "quote": "0\nrm|ζ′|2 dr −β2\nZ ∞"
        },
        {
          "blockId": "p15-00dbd6f43a",
          "page": 15,
          "quote": "0\nrm−2ζ2 dr =\nZ ∞"
        },
        {
          "blockId": "p15-071055b17e",
          "page": 15,
          "quote": "0\nr|f′\nζ(r)|2 dr ≥0."
        },
        {
          "blockId": "p15-4984ad8b0d",
          "page": 15,
          "quote": "R"
        },
        {
          "blockId": "p15-888fe1eb00",
          "page": 15,
          "quote": "R |ψ′|2 dτ\nR"
        },
        {
          "blockId": "p15-db8eb916e0",
          "page": 15,
          "quote": "k2"
        },
        {
          "blockId": "p15-51cc79bd7d",
          "page": 15,
          "quote": "R ψ2 dτ .\n(3.27)"
        },
        {
          "blockId": "p15-12704680e3",
          "page": 15,
          "quote": "To approach its constant, choose nonzero ψ ∈C∞\nc (R) and set ζk(r) = r−βψ((log r)/k).\nThe\nsubstitution τ = (log r)/k yields\nR ∞\n0 rm|ζ′\nk|2 dr\nR ∞\n0 rm−2ζ2\nk dr = β2 + 1"
        },
        {
          "blockId": "p15-643ccf95b5",
          "page": 15,
          "quote": "Indeed the numerator before division is k\nR"
        },
        {
          "blockId": "p15-91cc1d2aa7",
          "page": 15,
          "quote": "R(−βψ + k−1ψ′)2 dτ, and\nR\nψψ′ = 0 by compact support.\nDivide the separated stability inequality by its positive radial factor and let k →∞. In tangent\ndimension m = 5, one has β2 = 4, proving (3.26). The same computation on an original link with\nmo = 4 would give 9/4; the working link always has m = 5.\nThe order of limits is fixed: for each angular test and each finite k, first take ε →0 in the second\nvariation, and then take k →∞. Angular cutoffs used later are removed only after (3.26) has been\nobtained for each fixed smooth test. The collar and the allowable size of ε may depend on these\nfixed supports."
        },
        {
          "blockId": "p16-e2bbbf27a2",
          "page": 16,
          "quote": "4\nA universal algebraic certificate"
        },
        {
          "blockId": "p16-eb9722a177",
          "page": 16,
          "quote": "The comparison needed for the stability inequality is a statement about finite-dimensional tensors.\nWe state it with free algebraic data before deriving its geometric consequences. All tensor spaces\nhere are over R5, with the Euclidean inner product and full tensor norm; repeated indices range\nfrom 1 to 5 and are summed. A symmetric tensor is called trace-free if contraction of any two slots\nis zero. We write STFr(R5) for the space of symmetric trace-free tensors of rank r. Fix"
        },
        {
          "blockId": "p16-95bb697f36",
          "page": 16,
          "quote": "a = 29"
        },
        {
          "blockId": "p16-0026491c88",
          "page": 16,
          "quote": "50,\nQ = 250000,\nϵ∗=\n438907\n12000000,\nδ∗= 6984137359"
        },
        {
          "blockId": "p16-e7c1bb9818",
          "page": 16,
          "quote": "31250000000.\n(4.1)"
        },
        {
          "blockId": "p16-6e69ee52c4",
          "page": 16,
          "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
        },
        {
          "blockId": "p16-4f79bc733f",
          "page": 16,
          "quote": "where p, ℓ∈R5 and g, w, z ∈R are otherwise free. In particular, w and z need not be positive. On\ntensor polynomials in A, p, g, δ, define Dl by the product rule and"
        },
        {
          "blockId": "p16-89211ce53b",
          "page": 16,
          "quote": "DlAij = Xlij −qlAij,\nDlpi = wAli −gzδli,\nDlg = zpl,\nDlδij = 0.\n(4.2)"
        },
        {
          "blockId": "p16-194cc9dd41",
          "page": 16,
          "quote": "This is a first derivative with values in the enlarged polynomial algebra containing X, w, z; no\nderivative of these new variables is used."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-15",
      "kind": "proposition",
      "label": "Proposition 4.1",
      "source": {
        "blockId": "p16-bae9fc97ae",
        "page": 16,
        "quote": "Proposition 4.1 (Universal inequalities). There are rational tensor polynomials f, Ujk, Klijk, Ml, Pl\nin A, p, g, δ, with the following properties.\nThey are equivariant under orthogonal changes of\ncoordinates, with f invariant. Their coefficients are specified by Table I in Appendix A, using the\ndecoding in Section 6.1. Put"
      },
      "statement": [
        {
          "blockId": "p16-bae9fc97ae",
          "page": 16,
          "quote": "Proposition 4.1 (Universal inequalities). There are rational tensor polynomials f, Ujk, Klijk, Ml, Pl\nin A, p, g, δ, with the following properties.\nThey are equivariant under orthogonal changes of\ncoordinates, with f invariant. Their coefficients are specified by Table I in Appendix A, using the\ndecoding in Section 6.1. Put"
        },
        {
          "blockId": "p16-b848250caa",
          "page": 16,
          "quote": "Clijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk."
        },
        {
          "blockId": "p16-5a9c71c879",
          "page": 16,
          "quote": "For every choice of the data above with χ ≥0,"
        },
        {
          "blockId": "p16-af17922025",
          "page": 16,
          "quote": "I := z2 [5UjkAjk + Clijk(δljAik + δlkAij) −4f]"
        },
        {
          "blockId": "p16-7e646fc07e",
          "page": 16,
          "quote": "+ [DlGlijk + (2a −1)qlGlijk] Xijk\n+ w [DlMl + (2a + 1)qlMl] + z [DlPl + 2aqlPl]"
        },
        {
          "blockId": "p16-7cb9687645",
          "page": 16,
          "quote": "+ f|ℓ|2 −(Dlf + 2aqlf)ℓl ≥ϵ∗(z2 + |ℓ|2).\n(4.3)"
        },
        {
          "blockId": "p16-102340751e",
          "page": 16,
          "quote": "In particular, f ≥ϵ∗whenever g2 + |p|2 ≤1.\nFor every symmetric trace-free unit matrix A and vector p satisfying"
        },
        {
          "blockId": "p16-57f3a15ba4",
          "page": 16,
          "quote": "g = 0,\n|p| = 1,\nAp = λp,\nλ < 0,"
        },
        {
          "blockId": "p16-cc99c86d04",
          "page": 16,
          "quote": "define\nS0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk."
        },
        {
          "blockId": "p16-12ac49d162",
          "page": 16,
          "quote": "Then\nS := −λf + Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl ≥δ∗.\n(4.4)"
        },
        {
          "blockId": "p16-5719369278",
          "page": 16,
          "quote": "Moreover, U and K are even in p, and P = 0 when g = 0."
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "lemma-16",
      "kind": "lemma",
      "label": "Lemma 5.1",
      "source": {
        "blockId": "p17-e955da3b7c",
        "page": 17,
        "quote": "Lemma 5.1. At the data (5.2), the derivation Dl of (4.2), evaluated at z = 1, equals covariant\ndifferentiation on every coefficient polynomial in A, p, g, δ. In particular,"
      },
      "statement": [
        {
          "blockId": "p17-e955da3b7c",
          "page": 17,
          "quote": "Lemma 5.1. At the data (5.2), the derivation Dl of (4.2), evaluated at z = 1, equals covariant\ndifferentiation on every coefficient polynomial in A, p, g, δ. In particular,"
        },
        {
          "blockId": "p17-ea70133cef",
          "page": 17,
          "quote": "∇lw = wql,\n∇lAij = Xlij −qlAij,\n∇lpi = wAli −gδli,\n∇lg = pl.\n(5.3)"
        },
        {
          "blockId": "p17-4e99f8fde8",
          "page": 17,
          "quote": "Furthermore,\n1\nwGlijk∇lBijk = 5UjkAjk + Clijk(δljAik + δlkAij).\n(5.4)"
        }
      ],
      "proof": [
        {
          "blockId": "p17-19ff65de4b",
          "page": 17,
          "quote": "Proof. Differentiating w2 = TjkTjk gives ∇lw = TjkBljk/w = wql. The quotient rule then gives the\nformula for ∇A, and T = ∇2g +g Id gives the formula for ∇p. In a covariantly constant orthonormal\nframe at the point under consideration these are exactly the generator rules for Dl. The product\nrule and compatibility of the connection with contractions prove the assertion for every coefficient\npolynomial. The rules also preserve the normalization constraints:"
        },
        {
          "blockId": "p17-391bd90f4b",
          "page": 17,
          "quote": "Dl(tr A) = 0,\nDl(|A|2) = 2ql −2ql = 0"
        },
        {
          "blockId": "p17-2efa1d3509",
          "page": 17,
          "quote": "on the admissible data.\nHere Dl is applied only to coefficient polynomials in A, p, g, δ. Differentiation of the factors w\nand X, which are not coefficient variables, is performed separately. In particular, no rule for DlX\nor Dlq is being assumed.\nFor the U part of (5.4),"
        },
        {
          "blockId": "p17-d84708e421",
          "page": 17,
          "quote": "1\nwδliUjk∇lBijk = 1"
        },
        {
          "blockId": "p17-7403db8f4d",
          "page": 17,
          "quote": "wUjk∆Tjk = 5UjkAjk."
        },
        {
          "blockId": "p18-17071beaaf",
          "page": 18,
          "quote": "For the skew part, the curvature identity in Proposition 3.1 yields"
        },
        {
          "blockId": "p18-8f990c9974",
          "page": 18,
          "quote": "1\nwClijk∇lBijk = 1"
        },
        {
          "blockId": "p18-d64b0d54c9",
          "page": 18,
          "quote": "2wClijk(∇lBijk −∇iBljk)"
        },
        {
          "blockId": "p18-03c8107517",
          "page": 18,
          "quote": "= 1"
        },
        {
          "blockId": "p18-e9ec2e0d65",
          "page": 18,
          "quote": "2Clijk(δljAik + δlkAij −δijAlk −δikAlj)"
        },
        {
          "blockId": "p18-a658af75df",
          "page": 18,
          "quote": "= Clijk(δljAik + δlkAij)."
        },
        {
          "blockId": "p18-3f9b6dca97",
          "page": 18,
          "quote": "The last equality follows by interchanging the summed indices l, i in the last two terms and using\nCiljk = −Clijk. Equivalently, Clijk∇lBijk equals Klijk(∇lBijk −∇iBljk); thus the factor 1/2 belongs\nto the formula with C, not to the formula with K.\nThis calculation eliminates every undetermined fourth derivative of g. Indeed, if two possible\ntensors ∇lBijk have the same Laplacian contraction and the same skew difference in l, i, their\ndifference has zero contraction with δliUjk and with Clijk. No constraint on the remaining fourth\nderivative entries is required."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-17",
      "kind": "proposition",
      "label": "Proposition 5.2",
      "source": {
        "blockId": "p18-dba8c2d892",
        "page": 18,
        "quote": "Proposition 5.2. The scalar h and vector field Z satisfy"
      },
      "statement": [
        {
          "blockId": "p18-dba8c2d892",
          "page": 18,
          "quote": "Proposition 5.2. The scalar h and vector field Z satisfy"
        },
        {
          "blockId": "p18-9950cf5a0f",
          "page": 18,
          "quote": "div Z ≥4h2 + |∇h|2 + ϵ∗w2a\non Ω+.\n(5.7)"
        }
      ],
      "proof": [
        {
          "blockId": "p18-2d48b3fdd2",
          "page": 18,
          "quote": "Proof. First,"
        },
        {
          "blockId": "p18-a2c86df539",
          "page": 18,
          "quote": "∇lXijk = 1"
        },
        {
          "blockId": "p18-8cfc89e479",
          "page": 18,
          "quote": "w∇lBijk −qlXijk."
        },
        {
          "blockId": "p18-676c2987e7",
          "page": 18,
          "quote": "Applying the product rule to each of the three summands of Z, and then using (5.4), gives the exact\nidentity"
        },
        {
          "blockId": "p18-4ac9acb3fe",
          "page": 18,
          "quote": "w−2a div Z = 5UjkAjk + Clijk(δljAik + δlkAij)"
        },
        {
          "blockId": "p18-1a4e499ddb",
          "page": 18,
          "quote": "+\n\nDlGlijk + (2a −1)qlGlijk\n\nXijk\n+ w\n\nDlMl + (2a + 1)qlMl\n\n+ DlPl + 2aqlPl.\n(5.8)"
        },
        {
          "blockId": "p18-842e7465ba",
          "page": 18,
          "quote": "Every Dl in Equation (5.8) is evaluated at z = 1. The three coefficients of q come respectively from\ndifferentiating w2aX, w2a+1, and w2a; in the first expression the quotient X = B/w contributes\n−qlX. Thus their values are exactly"
        },
        {
          "blockId": "p18-8fe9f8056a",
          "page": 18,
          "quote": "2a −1 = 4"
        },
        {
          "blockId": "p18-1248ccf181",
          "page": 18,
          "quote": "25,\n2a + 1 = 54"
        },
        {
          "blockId": "p18-7d3ea03621",
          "page": 18,
          "quote": "25,\n2a = 29"
        },
        {
          "blockId": "p18-558d442c25",
          "page": 18,
          "quote": "25."
        },
        {
          "blockId": "p19-fb5f4cbc43",
          "page": 19,
          "quote": "Put\nFl = Dlf + 2aqlf,\nℓ∗= F"
        },
        {
          "blockId": "p19-d1706f0609",
          "page": 19,
          "quote": "2f ."
        },
        {
          "blockId": "p19-d28ab51ebc",
          "page": 19,
          "quote": "Differentiating h and completing the square give, respectively,"
        },
        {
          "blockId": "p19-d318feaef0",
          "page": 19,
          "quote": "∇lh = wa"
        },
        {
          "blockId": "p19-8023154c00",
          "page": 19,
          "quote": "4f |F|2,\n(5.9)"
        },
        {
          "blockId": "p19-4886bf6683",
          "page": 19,
          "quote": "2√f Fl,\n|∇h|2 = w2a"
        },
        {
          "blockId": "p19-73294d56b4",
          "page": 19,
          "quote": "and"
        },
        {
          "blockId": "p19-47d57055e3",
          "page": 19,
          "quote": "f|ℓ|2 −F · ℓ= f|ℓ−ℓ∗|2 −|F|2"
        },
        {
          "blockId": "p19-ae41fd4430",
          "page": 19,
          "quote": "4f ."
        },
        {
          "blockId": "p19-67cf63d7b4",
          "page": 19,
          "quote": "In particular, substituting (5.8) into the interior polynomial I of (4.3) gives the identity, for every ℓ,"
        },
        {
          "blockId": "p19-7492df17f5",
          "page": 19,
          "quote": "I\n\nz=1 = w−2a\ndiv Z −4h2 −|∇h|2\n+ f|ℓ−ℓ∗|2.\n(5.10)"
        },
        {
          "blockId": "p19-9475943626",
          "page": 19,
          "quote": "The division by f is justified uniformly by (5.5). Evaluating (4.3) at ℓ= ℓ∗therefore gives the\nslightly stronger estimate"
        },
        {
          "blockId": "p19-6d735e7032",
          "page": 19,
          "quote": "div Z −4h2 −|∇h|2 ≥ϵ∗w2a(1 + |ℓ∗|2),"
        },
        {
          "blockId": "p19-976b6fc63d",
          "page": 19,
          "quote": "which implies (5.7)."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-18",
      "kind": "lemma",
      "label": "Lemma 5.3",
      "source": {
        "blockId": "p19-b9cba26141",
        "page": 19,
        "quote": "Lemma 5.3. At such a point,"
      },
      "statement": [
        {
          "blockId": "p19-b9cba26141",
          "page": 19,
          "quote": "Lemma 5.3. At such a point,"
        },
        {
          "blockId": "p19-9565de44e5",
          "page": 19,
          "quote": "plGlijk\nBijk"
        },
        {
          "blockId": "p19-7966fef82a",
          "page": 19,
          "quote": "w2 = plGlijkVijk = Ujk(S0 + p ⊗p)jk + ClijkplVijk.\n(5.12)"
        }
      ],
      "proof": [
        {
          "blockId": "p19-1b7b8d3435",
          "page": 19,
          "quote": "Proof. Let R = Id −2p ⊗p be reflection in p⊥. Since A is symmetric and Ap = λp, it preserves p⊥,\nand hence RART = A. Also Rp = −p. Orthogonal equivariance and the even parity in p of U, K,\nand therefore of G, give"
        },
        {
          "blockId": "p19-70a91aa13d",
          "page": 19,
          "quote": "R⊗4G(A, p, 0) = G(RART , Rp, 0) = G(A, −p, 0) = G(A, p, 0)."
        },
        {
          "blockId": "p19-a5f8674b3a",
          "page": 19,
          "quote": "For the three-tensor Fijk = plGlijk, contraction commutes with orthogonal transformations, so"
        },
        {
          "blockId": "p19-95d85cb07c",
          "page": 19,
          "quote": "R⊗3F = (Rp)l(R⊗4G)lijk = −F."
        },
        {
          "blockId": "p19-99832b79b9",
          "page": 19,
          "quote": "Choose any orthonormal basis with its last vector p. A component of a three-tensor with k normal\nslots is multiplied by (−1)k under R⊗3. It follows that F has zero components whenever the number\nof normal slots is even. This assertion concerns the full unsymmetrized tensor F; no permutation of\nits slots, and no extra symmetry of U or K, has been used."
        },
        {
          "blockId": "p20-044918f5ba",
          "page": 20,
          "quote": "For tangent indices α, β, the boundary jet identities give"
        },
        {
          "blockId": "p20-9e3441a439",
          "page": 20,
          "quote": "BNαβ"
        },
        {
          "blockId": "p20-b941ebff60",
          "page": 20,
          "quote": "w2\n= λAαβ −(A2)αβ = (S0)αβ,\nBNNN"
        },
        {
          "blockId": "p20-a4b8ca16d8",
          "page": 20,
          "quote": "w2\n= 1."
        },
        {
          "blockId": "p20-f585b9ba9f",
          "page": 20,
          "quote": "Because S0p = 0, these are precisely the components of V with one or three normal slots, including\nall their permutations. Thus the part of B/w2 with an odd number of normal slots equals V ; V\nitself has no even-normal components. The even-normal components of B, which are not determined\nby A, p, g, w, have zero contraction with F. This proves the first equality in (5.12).\nFinally, S0p = 0 and |p| = 1 imply"
        },
        {
          "blockId": "p20-e5286ce9bf",
          "page": 20,
          "quote": "piVijk = (S0)jk + pjpk."
        },
        {
          "blockId": "p20-330dcb5f2c",
          "page": 20,
          "quote": "Splitting G according to (5.1) proves the second equality. The argument used only the orthogonal\nsplitting R5 = p⊥⊕Rp. It therefore applies without any assumption on multiplicities, signs, or\nnonvanishing of the tangential eigenvalues of A."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-19",
      "kind": "proposition",
      "label": "Proposition 5.4",
      "source": {
        "blockId": "p20-9e624b4119",
        "page": 20,
        "quote": "Proposition 5.4. On the regular side boundary,"
      },
      "statement": [
        {
          "blockId": "p20-9e624b4119",
          "page": 20,
          "quote": "Proposition 5.4. On the regular side boundary,"
        },
        {
          "blockId": "p20-c8cb318614",
          "page": 20,
          "quote": "N · Z + Hh2 = w2a+1S ≥δ∗w2a+1 > 0,\nδ∗= 6984137359"
        },
        {
          "blockId": "p20-7818ef941d",
          "page": 20,
          "quote": "31250000000,\n(5.13)"
        },
        {
          "blockId": "p20-6b50dd4cfe",
          "page": 20,
          "quote": "where S is the boundary polynomial in (4.4). In particular N · Z + Hh2 ≥0."
        }
      ],
      "proof": [
        {
          "blockId": "p20-cf8d16908c",
          "page": 20,
          "quote": "Proof. Proposition 4.1 gives P = 0 at the boundary, where g = 0. Since X = B/w, Lemma 5.3 gives"
        },
        {
          "blockId": "p20-eea01575b0",
          "page": 20,
          "quote": "N · Z = w2a+1 (Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl) ,"
        },
        {
          "blockId": "p20-333aedd965",
          "page": 20,
          "quote": "Hh2 = −λw2a+1f."
        },
        {
          "blockId": "p20-bd7ab24c52",
          "page": 20,
          "quote": "Their sum is exactly w2a+1S. The boundary data satisfy all hypotheses of (4.4), proving the stated\nlower bound. The sign is expressed using the inward normal N; the outward normal used by the\ndivergence theorem is −N.\nOnly the boundary value of P vanishes. Its derivative term DlPl remains in (5.8). For example,\na monomial Pl = gQl has DiPl = piQl + gDiQl at z = 1, so its derivative need not vanish when\ng = 0."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-20",
      "kind": "lemma",
      "label": "Lemma 5.5",
      "source": {
        "blockId": "p20-5d81fbc074",
        "page": 20,
        "quote": "Lemma 5.5. There is a finite constant C, depending only on the coefficient polynomials, the\ndimension, a, and ϵ∗, such that on Ω+"
      },
      "statement": [
        {
          "blockId": "p20-5d81fbc074",
          "page": 20,
          "quote": "Lemma 5.5. There is a finite constant C, depending only on the coefficient polynomials, the\ndimension, a, and ϵ∗, such that on Ω+"
        },
        {
          "blockId": "p20-451469844c",
          "page": 20,
          "quote": "|h| ≤Cwa,\n|∇h| ≤Cwa(1 + w + |X|),\n|Z| ≤Cw2a(1 + w + |X|).\n(5.14)"
        },
        {
          "blockId": "p20-f7d8142cce",
          "page": 20,
          "quote": "The constant is independent of w, |X|, and all subsequent cutoff parameters."
        }
      ],
      "proof": [
        {
          "blockId": "p20-f402f245eb",
          "page": 20,
          "quote": "Proof. The coefficient domain"
        },
        {
          "blockId": "p20-db19f2de9c",
          "page": 20,
          "quote": "{(A, p, g) : A ∈STF2(R5), |A| = 1, |p|2 + g2 ≤1}"
        },
        {
          "blockId": "p20-cba39cc8cd",
          "page": 20,
          "quote": "is compact. The coefficient polynomials and their first derivatives with respect to their coefficient\nvariables are therefore uniformly bounded there. Also |q| ≤|X| by Cauchy–Schwarz. The rules (5.3)\nconsequently imply\n|Df| + |q| |f| ≤C(1 + w + |X|)."
        },
        {
          "blockId": "p20-c1ad42d75d",
          "page": 20,
          "quote": "Use (5.5) and (5.9) for the estimates on h and ∇h, and use (5.6) and the boundedness of G, M, P\nfor the estimate on Z."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-21",
      "kind": "lemma",
      "label": "Lemma 6.1",
      "source": {
        "blockId": "p23-0a22271950",
        "page": 23,
        "quote": "Lemma 6.1. All these operators are self-adjoint and positive semidefinite on their entire tensor\nspaces. In particular, Γ ≥0 whenever χ ≥0."
      },
      "statement": [
        {
          "blockId": "p23-0a22271950",
          "page": 23,
          "quote": "Lemma 6.1. All these operators are self-adjoint and positive semidefinite on their entire tensor\nspaces. In particular, Γ ≥0 whenever χ ≥0."
        }
      ],
      "proof": [
        {
          "blockId": "p23-2c341b9f8d",
          "page": 23,
          "quote": "Proof. Each slot permutation is orthogonal, with adjoint its inverse. Averaging over all permutations,\nwith or without their signs, gives the orthogonal projection onto symmetric or alternating tensors.\nThus anti is r! times an orthogonal projection. For a symmetric matrix S, S −δ tr S/5 is its\northogonal trace-free projection. Since F ′ = 2 Sym F in (6.4), that operator is ten times the\nsymmetric trace-free projection.\nFor rank three define\n(J v)ijk = δijvk + δikvj + δjkvi."
        },
        {
          "blockId": "p23-d4a2399d42",
          "page": 23,
          "quote": "Then Tr12 J v = 7v, and for symmetric S, ⟨S, J v⟩= 3⟨Tr12 S, v⟩. It follows that S −J (Tr12 S)/7 is\nthe orthogonal projection onto symmetric trace-free tensors. Since now F ′ = 6 Sym F, (6.5) is 42\ntimes that projection.\nFor completeness, the trace correction in (6.6) can be checked without any representation-\ntheoretic assumptions. Let"
        },
        {
          "blockId": "p23-9b4ec61465",
          "page": 23,
          "quote": "X"
        },
        {
          "blockId": "p23-4996dba2c5",
          "page": 23,
          "quote": "S12 = Id +(12)"
        },
        {
          "blockId": "p23-2914490705",
          "page": 23,
          "quote": "2\n,\nS = 1"
        },
        {
          "blockId": "p23-22c24c946d",
          "page": 23,
          "quote": "6"
        },
        {
          "blockId": "p23-e7f9509e86",
          "page": 23,
          "quote": "π∈S3\nπ,\nP0 = S12 −S."
        },
        {
          "blockId": "p23-b11a3fe1a3",
          "page": 23,
          "quote": "The identities S12S = SS12 = S show that P0 is an orthogonal projection. Its image W consists of\nfirst-pair symmetric tensors orthogonal to fully symmetric tensors. For H ∈W,"
        },
        {
          "blockId": "p23-191b6ff9d2",
          "page": 23,
          "quote": "Hijk = Hjik,\nHijk + Hikj + Hjki = 0."
        },
        {
          "blockId": "p23-b7b89ef4b8",
          "page": 23,
          "quote": "If v = Tr12 H, contraction of the second identity gives Tr13 H = Tr23 H = −v/2. Define"
        },
        {
          "blockId": "p23-00b7c735c4",
          "page": 23,
          "quote": "(Bv)ijk = 2δijvk −δikvj −δjkvi."
        },
        {
          "blockId": "p23-df3ee0c80c",
          "page": 23,
          "quote": "Direct substitution gives Bv ∈W, Tr12 Bv = 8v, and"
        },
        {
          "blockId": "p23-0137625f81",
          "page": 23,
          "quote": "⟨H, Bz⟩= 2⟨v, z⟩−⟨Tr13 H, z⟩−⟨Tr23 H, z⟩= 3⟨v, z⟩."
        },
        {
          "blockId": "p23-c0d985c4d1",
          "page": 23,
          "quote": "Therefore Q0H = H −B(Tr12 H)/8 has all traces zero, and H −Q0H is orthogonal to every trace-free\ntensor in W. Thus Q0 is the orthogonal projection in W onto its trace-free part. Since F ′ = 6P0F,\nthe hook operator is 48Q0P0; Q0P0 is the orthogonal projection in the full tensor space onto that\nsubspace. In particular\n⟨F, ΠF⟩= 48|Q0P0F|2 ≥0."
        },
        {
          "blockId": "p23-dfe2b8a1c5",
          "page": 23,
          "quote": "Finally the weights in (6.2) are nonnegative under the stated constraint."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-22",
      "kind": "lemma",
      "label": "Lemma 6.2",
      "source": {
        "blockId": "p24-a934f8e471",
        "page": 24,
        "quote": "Lemma 6.2 (Closure of the graph calculation). The preceding rules evaluate all terms of I and Γ\nby a finite sum of scalar graphs. In every intermediate graph a free slot has exactly one incidence,\neach cubic tensor has exactly three incidences, and there are at most two cubic tensors. Every\nresulting scalar graph has degree two in (w, z, ℓ, X)."
      },
      "statement": [
        {
          "blockId": "p24-a934f8e471",
          "page": 24,
          "quote": "Lemma 6.2 (Closure of the graph calculation). The preceding rules evaluate all terms of I and Γ\nby a finite sum of scalar graphs. In every intermediate graph a free slot has exactly one incidence,\neach cubic tensor has exactly three incidences, and there are at most two cubic tensors. Every\nresulting scalar graph has degree two in (w, z, ℓ, X)."
        }
      ],
      "proof": [
        {
          "blockId": "p24-508d7d7d45",
          "page": 24,
          "quote": "Proof. The incidence assertions hold for each table row by its decoding. A disjoint union preserves\nthem. A permutation only renames free slots. Contracting two free slots removes their unique\nincidences and joins the remaining ends, or closes a trace; it does not change the number of slots of\nany cubic tensor. The derivative insertion described above replaces one edge by two incidences of a\nnew X and supplies its third incidence as a new free slot. Trace differentiation does the same with\na self-edge. The ql term supplies two incidences on its self-edge and the third at l. Derivatives of p\nand g supply a free derivative slot without introducing a cubic tensor. These statements prove the\nincidence assertions by induction over the operations.\nTable I has degree zero and its derivative has degree one in the indicated variables. The formula\nfor I multiplies such derivatives by one further factor, or a degree-zero coefficient by two factors.\nEach Table II row has degree one; projection, contraction, and multiplication by χ preserve that\ndegree. Squaring produces degree two. In particular no operation used here needs three copies of X.\nTermination is immediate from the finite tables, finite permutation sums, and the finite product\nrule; every contraction reduces the number of free slots by two."
        },
        {
          "blockId": "p25-225c20d6c9",
          "page": 25,
          "quote": "The linear contribution of each Table I row is obtained by substitution into (4.3), forming\nC = K −K(12) for a K row. One check on the slot order and coefficients is the single row U = A\nwith coefficient one. Its interior contribution is"
        },
        {
          "blockId": "p25-c211f17c8c",
          "page": 25,
          "quote": "5z2 + |X|2 + (2a −2)|q|2,"
        },
        {
          "blockId": "p25-616b2bfb28",
          "page": 25,
          "quote": "and its boundary contribution is 2λ −tr A3. To expand a Gram block, sum over unordered pairs\nof rows s, t: the coefficient is P\nc cscctc/Q2, doubled when s ̸= t, and the graph polynomial is the\ncontraction of row t with Π applied to row s, multiplied by ρ. This uses self-adjointness of Π and\naccounts for every term of (6.2) exactly once."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-23",
      "kind": "lemma",
      "label": "Lemma 6.3",
      "source": {
        "blockId": "p25-66881bf618",
        "page": 25,
        "quote": "Lemma 6.3. For the constrained matrix A and symmetric trace-free tensor X,"
      },
      "statement": [
        {
          "blockId": "p25-66881bf618",
          "page": 25,
          "quote": "Lemma 6.3. For the constrained matrix A and symmetric trace-free tensor X,"
        },
        {
          "blockId": "p25-26685d701f",
          "page": 25,
          "quote": "∥A∥2\nop ≤θ,\n|Ak|2 ≤θk−1\n(k ≥1),\n(6.9)"
        },
        {
          "blockId": "p25-887c380beb",
          "page": 25,
          "quote": "| tr Ak|2 ≤θk−2\n(k ≥2),\n| tr A3|2 ≤9"
        },
        {
          "blockId": "p25-34df8deab7",
          "page": 25,
          "quote": "20,\n(6.10)"
        },
        {
          "blockId": "p25-256d74d024",
          "page": 25,
          "quote": "|X : A|2 ≤5"
        },
        {
          "blockId": "p25-3033ab8bae",
          "page": 25,
          "quote": "7|X|2,\n(X : A)i = XijkAjk.\n(6.11)"
        },
        {
          "blockId": "p25-5f06dbaca2",
          "page": 25,
          "quote": "If h, b > 0 and g2 + |p|2 ≤1, then"
        },
        {
          "blockId": "p25-6744ff863c",
          "page": 25,
          "quote": "b\n.\n(6.12)"
        },
        {
          "blockId": "p25-138422542a",
          "page": 25,
          "quote": "|g|2h|p|2b ≤\n\nh\nh + b"
        },
        {
          "blockId": "p25-599aa1d9bf",
          "page": 25,
          "quote": "h \nb\nh + b"
        }
      ],
      "proof": [
        {
          "blockId": "p25-2eaaddba0c",
          "page": 25,
          "quote": "Proof. Let the eigenvalues of A be α1, . . . , α5. For each i,"
        },
        {
          "blockId": "p25-8ea936db5d",
          "page": 25,
          "quote": "2"
        },
        {
          "blockId": "p25-0fa9155184",
          "page": 25,
          "quote": ""
        },
        {
          "blockId": "p25-bb4f67a0ee",
          "page": 25,
          "quote": ""
        },
        {
          "blockId": "p25-233900bf37",
          "page": 25,
          "quote": "≤4\nX"
        },
        {
          "blockId": "p25-435108ca7b",
          "page": 25,
          "quote": "α2\ni ="
        },
        {
          "blockId": "p25-fe681386c2",
          "page": 25,
          "quote": "X"
        },
        {
          "blockId": "p25-648affb1a0",
          "page": 25,
          "quote": ""
        },
        {
          "blockId": "p25-538050d6b6",
          "page": 25,
          "quote": "j̸=i\nαj"
        },
        {
          "blockId": "p25-338a35ec9b",
          "page": 25,
          "quote": "j̸=i\nα2\nj = 4(1 −α2\ni )."
        },
        {
          "blockId": "p25-4e4810cb21",
          "page": 25,
          "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
        },
        {
          "blockId": "p25-569fb47a2b",
          "page": 25,
          "quote": "s"
        },
        {
          "blockId": "p25-12294063e6",
          "page": 25,
          "quote": "5 −j"
        },
        {
          "blockId": "p25-2fd48b6216",
          "page": 25,
          "quote": "5j ,\n−"
        },
        {
          "blockId": "p25-83174b2251",
          "page": 25,
          "quote": "j\n5(5 −j)."
        },
        {
          "blockId": "p25-c201706156",
          "page": 25,
          "quote": "Their cubic trace has absolute value |5 −2j|/\np"
        },
        {
          "blockId": "p25-7670809c37",
          "page": 25,
          "quote": "5j(5 −j). Its square is 9/20 for j = 1, 4 and 1/30\nfor j = 2, 3, proving (6.10).\nTo prove (6.11), fix a vector ζ and symmetrize ζ ⊗A:"
        },
        {
          "blockId": "p25-488080adb9",
          "page": 25,
          "quote": "Bijk = ζiAjk + ζjAik + ζkAij"
        },
        {
          "blockId": "p25-b8d0b6bd8e",
          "page": 25,
          "quote": "3\n,\nv = Tr12 B = 2"
        },
        {
          "blockId": "p25-80847c5c58",
          "page": 25,
          "quote": "3Aζ."
        },
        {
          "blockId": "p26-8d1ed535fc",
          "page": 26,
          "quote": "Expanding the nine terms in |B|2 gives |B|2 = |ζ|2/3+2|Aζ|2/3. The symmetric trace-free projection\nis F = B −J v/7, in the notation of Lemma 6.1. Since |J v|2 = 21|v|2 and ⟨B, J v⟩= 3|v|2,"
        },
        {
          "blockId": "p26-93993a34d3",
          "page": 26,
          "quote": "|F|2 = |B|2 −3"
        },
        {
          "blockId": "p26-91f31ba23f",
          "page": 26,
          "quote": "7|v|2 = 1"
        },
        {
          "blockId": "p26-bde1b7401b",
          "page": 26,
          "quote": "3|ζ|2 + 10"
        },
        {
          "blockId": "p26-f6156ce9a4",
          "page": 26,
          "quote": "21|Aζ|2 ≤5"
        },
        {
          "blockId": "p26-d7432d582e",
          "page": 26,
          "quote": "7|ζ|2."
        },
        {
          "blockId": "p26-b8f9e5a948",
          "page": 26,
          "quote": "Symmetry and trace-freeness of X give ⟨X : A, ζ⟩= ⟨X, F⟩; Cauchy–Schwarz and duality prove the\nclaim.\nFinally put s = g2, t = |p|2. For h, b > 0, the maximum of shtb on s, t ≥0, s + t ≤1 lies on\ns + t = 1. Differentiating h log s + b log(1 −s) gives the unique interior maximum s = h/(h + b),\nproving (6.12). If either exponent is zero, the corresponding bound is simply 1."
        },
        {
          "blockId": "p26-35ad311d0d",
          "page": 26,
          "quote": "For a nonzero scalar graph key t of degree two, let"
        },
        {
          "blockId": "p26-e9685695ac",
          "page": 26,
          "quote": "y = (|w|, |z|, |ℓ|, |X|)T"
        },
        {
          "blockId": "p26-63ab49ec44",
          "page": 26,
          "quote": "and let i, j ∈{1, 2, 3, 4} be its two degree-one types, counted with multiplicity and ordered so i ≤j.\nLet h be its g exponent and b its number of p endpoints. Define L2\nt as a product of the following\nrational factors:"
        },
        {
          "blockId": "p26-f07bbce1de",
          "page": 26,
          "quote": "1. for each trace tr Ak, use θk−2, replacing this by 9/20 when k = 3;"
        },
        {
          "blockId": "p26-c3cc386307",
          "page": 26,
          "quote": "2. for an edge of power k between distinct vertices, use θk;"
        },
        {
          "blockId": "p26-98f6bbf972",
          "page": 26,
          "quote": "3. for an edge between two slots of the same X, use 5/7 when k = 1 and θk−1 when k ≥2;"
        },
        {
          "blockId": "p26-f181572fe6",
          "page": 26,
          "quote": "4. if h, b > 0, include the right side of (6.12); otherwise include 1."
        },
        {
          "blockId": "p26-b5c71e485e",
          "page": 26,
          "quote": "Here separate occurrences of a vector endpoint count as distinct vertices; thus a p–p edge is governed\nby Item 2, not Item 3. Power-zero self-edges at X have already vanished in the collection rules."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-24",
      "kind": "lemma",
      "label": "Lemma 6.4",
      "source": {
        "blockId": "p26-e79d4771d9",
        "page": 26,
        "quote": "Lemma 6.4 (Complete graph bound). Every scalar graph retained by Lemma 6.2 satisfies"
      },
      "statement": [
        {
          "blockId": "p26-e79d4771d9",
          "page": 26,
          "quote": "Lemma 6.4 (Complete graph bound). Every scalar graph retained by Lemma 6.2 satisfies"
        },
        {
          "blockId": "p26-51839e719e",
          "page": 26,
          "quote": "|M(t)| ≤Ltyiyj\n(6.13)"
        },
        {
          "blockId": "p26-331d076a7f",
          "page": 26,
          "quote": "on g2 + |p|2 ≤1."
        }
      ],
      "proof": [
        {
          "blockId": "p26-93c0366bec",
          "page": 26,
          "quote": "Proof. First separate all scalar trace factors and use Lemma 6.3. A cubic tensor has at most one\nself-edge, since such an edge consumes two of its three incidences. An edge of power k ≥2 reduces\nthat tensor to the vector vi = Xijk(Ak)jk, for which |v| ≤|X| |Ak| ≤|X|θ(k−1)/2. For k = 1 use\n(6.11). These reductions leave at most two cubic vertices and no self-edge on either.\nWe list all possible connected components of the remaining graph. With zero cubic vertices, a\ncomponent is one edge joining two vectors; its contraction is bounded by the operator norm of its\nmatrix times the two vector norms. With one cubic vertex, all its three slots end at vector leaves.\nMove each edge matrix onto its vector, and use tensor Cauchy–Schwarz, obtaining the norm of X\ntimes the three vector norms and the product of the three operator norms.\nIf a component contains two cubic vertices, they are joined by r = 1, 2, or 3 edges; every other\nslot ends at a vector leaf. Contract these leaves first, obtaining rank-r tensors Y, Z with the same\npreceding norm bounds. Up to a permutation of their slots, the remaining contraction is\nD\nY, (Ak1 ⊗· · · ⊗Akr)Z\nE\n."
        },
        {
          "blockId": "p27-a55e4f662c",
          "page": 27,
          "quote": "Its absolute value is at most |Y ||Z| Qr\ns=1 ∥Aks∥op. This includes all three parallel edges when r = 3,\nwithout a dimension factor. If the two cubic vertices have no edge between them, they lie in two\ncomponents of the one-vertex type already treated. These cases are exhaustive: all other vertices\nhave valence one, so cannot occur internally on a path, and there are no further cubic vertices. The\nreduced vectors from self-edges are ordinary vector leaves in this classification.\nMultiply the estimates over components. Each original ℓor X norm appears once per occurrence,\nall ordinary edge matrices contribute θk/2, and the p endpoints contribute |p|b. The scalar factors\ngh and |p|b are bounded jointly by (6.12). Finally the total degree is two, so all remaining w, z, ℓ, X\nfactors give exactly yiyj."
        },
        {
          "blockId": "p27-b55bc756c0",
          "page": 27,
          "quote": "We can now bound every term left after subtracting the nonnegative comparison forms. The\ngraph estimates reduce this remainder to a quadratic form in four magnitudes. The next calculation\nshows that its error is smaller than the reserved positive diagonal terms."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-25",
      "kind": "lemma",
      "label": "Lemma 7.1",
      "source": {
        "blockId": "p30-79ef5f2437",
        "page": 30,
        "quote": "Lemma 7.1 (Weighted Bochner estimate). The set Zo is a compact subset of Ωo, and\nZ"
      },
      "statement": [
        {
          "blockId": "p30-79ef5f2437",
          "page": 30,
          "quote": "Lemma 7.1 (Weighted Bochner estimate). The set Zo is a compact subset of Ωo, and\nZ"
        },
        {
          "blockId": "p30-6b3ec9e1fd",
          "page": 30,
          "quote": "Ωo,+\nw2a−2\no\n|Bo|2 dVo < ∞.\n(7.1)"
        },
        {
          "blockId": "p31-76b76c6117",
          "page": 31,
          "quote": "Consequently the function"
        },
        {
          "blockId": "p31-d6886de210",
          "page": 31,
          "quote": "Do(y) ="
        },
        {
          "blockId": "p31-c40865070c",
          "page": 31,
          "quote": "(\nwo(y)2a\n1 + |Bo(y)|/wo(y)\n2,\ny ∈Ωo,+,\n0,\ny ∈Zo,\n(7.2)"
        },
        {
          "blockId": "p31-08a0ccbfd4",
          "page": 31,
          "quote": "belongs to L1(Ωo)."
        }
      ],
      "proof": [
        {
          "blockId": "p31-441dea3d25",
          "page": 31,
          "quote": "Proof. The boundary conclusion of Lemma 3.2 gives wo > 0 at every point of ∂Ωo. Compactness\nand continuity therefore give a boundary neighborhood on which wo has a positive lower bound.\nThe closed set Zo is thus compactly contained in the interior. If it is empty, wo has a positive\nminimum on the whole compact closure, and (7.1) follows immediately from the boundedness of Bo.\nOtherwise choose η ∈C∞\nc (Ωo) with 0 ≤η ≤1, equal to one on an open neighborhood of Zo.\nFor 0 < δ ≤1 put Wδ = w2\no + δ. Although wo need not be smooth at its zeros, Wδ is smooth and\nstrictly positive. The identities in Proposition 3.1 give"
        },
        {
          "blockId": "p31-f94ef2c4a7",
          "page": 31,
          "quote": "∆Wδ = 2mow2\no + 2|Bo|2 ≥2|Bo|2,\n|∇Wδ|2 = 4|⟨To, Bo⟩|2 ≤4Wδ|Bo|2."
        },
        {
          "blockId": "p31-6e0e9f4b56",
          "page": 31,
          "quote": "Set Iδ =\nR"
        },
        {
          "blockId": "p31-a9e55efa94",
          "page": 31,
          "quote": "Ωo ηW a−1\nδ\n|Bo|2 dVo. Applying the chain rule to W a\nδ and integrating twice by parts against\nthe compactly supported η yields"
        },
        {
          "blockId": "p31-c770c65917",
          "page": 31,
          "quote": "2Iδ ≤\nZ"
        },
        {
          "blockId": "p31-cfddc1d416",
          "page": 31,
          "quote": "Ωo\nηW a−1\nδ\n∆Wδ dVo"
        },
        {
          "blockId": "p31-1c1c90b019",
          "page": 31,
          "quote": "Z"
        },
        {
          "blockId": "p31-9f617c1428",
          "page": 31,
          "quote": "= 1"
        },
        {
          "blockId": "p31-ecd6bd75dd",
          "page": 31,
          "quote": "a"
        },
        {
          "blockId": "p31-31447e09ef",
          "page": 31,
          "quote": "Ωo\n(∆η)W a\nδ dVo + (1 −a)\nZ"
        },
        {
          "blockId": "p31-c544ea662e",
          "page": 31,
          "quote": "Ωo\nηW a−2\nδ\n|∇Wδ|2 dVo"
        },
        {
          "blockId": "p31-82c42b4ab2",
          "page": 31,
          "quote": "Z"
        },
        {
          "blockId": "p31-9e15cc867a",
          "page": 31,
          "quote": "≤1"
        },
        {
          "blockId": "p31-bb5c20b7d9",
          "page": 31,
          "quote": "a"
        },
        {
          "blockId": "p31-8d32359266",
          "page": 31,
          "quote": "Ωo\n(∆η)W a\nδ dVo + 4(1 −a)Iδ."
        },
        {
          "blockId": "p31-43df15cf87",
          "page": 31,
          "quote": "Since 2 −4(1 −a) = 4a −2 = 8/25, this proves the explicit uniform estimate"
        },
        {
          "blockId": "p31-546b8ca7f7",
          "page": 31,
          "quote": "8\n25Iδ ≤1"
        },
        {
          "blockId": "p31-a5342e9d9d",
          "page": 31,
          "quote": "a(M2\no + 1)a∥∆η∥L1(Ωo).\n(7.3)"
        },
        {
          "blockId": "p31-fd4079e419",
          "page": 31,
          "quote": "As δ ↓0, the nonnegative integrands restricted to Ωo,+ increase to ηw2a−2\no\n|Bo|2 because a < 1.\nMonotone convergence and (7.3) control that integral. The compact set supp(1 −η) ⊂Ωo is disjoint\nfrom Zo; writing bη > 0 for the minimum of wo there, the remaining integral is bounded by\nZ"
        },
        {
          "blockId": "p31-da05d9263f",
          "page": 31,
          "quote": "Ωo,+\n(1 −η)w2a−2\no\n|Bo|2 dVo ≤b2a−2\nη\n∥Bo∥2\nL2(Ωo)."
        },
        {
          "blockId": "p31-6e1b1f928c",
          "page": 31,
          "quote": "Thus all constants in this argument depend only on the fixed original component and its smooth\ndata, on a, and on the chosen interior cutoff η, and are independent of δ. Finally, on Ωo,+,"
        },
        {
          "blockId": "p31-db8667919e",
          "page": 31,
          "quote": "Do ≤2w2a\no + 2w2a−2\no\n|Bo|2."
        },
        {
          "blockId": "p31-383c2aa8e9",
          "page": 31,
          "quote": "The first term is integrable by compactness, and the second by (7.1). No measure, dimension, or\nregularity property of Zo has been used."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-26",
      "kind": "lemma",
      "label": "Lemma 7.2",
      "source": {
        "blockId": "p32-0c22fefac6",
        "page": 32,
        "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
      },
      "statement": [
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        }
      ],
      "proof": [
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        }
      ],
      "guide": []
    },
    {
      "id": "theorem-27",
      "kind": "theorem",
      "label": "Theorem 7.3",
      "source": {
        "blockId": "p34-1efddb603e",
        "page": 34,
        "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
      },
      "statement": [
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        }
      ],
      "proof": [
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        }
      ],
      "guide": []
    },
    {
      "id": "external-28",
      "kind": "external",
      "label": "Reference [1]",
      "source": {
        "blockId": "p49-f275f04025",
        "page": 49,
        "quote": "[1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free\nboundary, J. Reine Angew. Math. 325 (1981), 105–144. doi:10.1515/crll.1981.325.105."
      },
      "statement": [
        {
          "blockId": "p49-f275f04025",
          "page": 49,
          "quote": "[1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free\nboundary, J. Reine Angew. Math. 325 (1981), 105–144. doi:10.1515/crll.1981.325.105."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-29",
      "kind": "external",
      "label": "Reference [2]",
      "source": {
        "blockId": "p49-75b31f99b3",
        "page": 49,
        "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
      },
      "statement": [
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-30",
      "kind": "external",
      "label": "Reference [3]",
      "source": {
        "blockId": "p49-e9f5f2bb45",
        "page": 49,
        "quote": "[3] L. A. Caffarelli, D. Jerison, and C. E. Kenig, Global energy minimizers for free boundary\nproblems and full regularity in three dimensions, in Noncompact Problems at the Intersection of\nGeometry, Analysis, and Topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence,\nRI, 2004, 83–97. doi:10.1090/conm/350/06339."
      },
      "statement": [
        {
          "blockId": "p49-e9f5f2bb45",
          "page": 49,
          "quote": "[3] L. A. Caffarelli, D. Jerison, and C. E. Kenig, Global energy minimizers for free boundary\nproblems and full regularity in three dimensions, in Noncompact Problems at the Intersection of\nGeometry, Analysis, and Topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence,\nRI, 2004, 83–97. doi:10.1090/conm/350/06339."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-31",
      "kind": "external",
      "label": "Reference [4]",
      "source": {
        "blockId": "p49-689109c677",
        "page": 49,
        "quote": "[4] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound.\n13 (2011), no. 2, 223–238. doi:10.4171/IFB/255."
      },
      "statement": [
        {
          "blockId": "p49-689109c677",
          "page": 49,
          "quote": "[4] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound.\n13 (2011), no. 2, 223–238. doi:10.4171/IFB/255."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-32",
      "kind": "external",
      "label": "Reference [5]",
      "source": {
        "blockId": "p49-cc3b900f63",
        "page": 49,
        "quote": "[5] D. De Silva, F. Ferrari, and S. Salsa, Two-phase problems with distributed sources: regularity\nof the free boundary, Anal. PDE 7 (2014), no. 2, 267–310. doi:10.2140/apde.2014.7.267."
      },
      "statement": [
        {
          "blockId": "p49-cc3b900f63",
          "page": 49,
          "quote": "[5] D. De Silva, F. Ferrari, and S. Salsa, Two-phase problems with distributed sources: regularity\nof the free boundary, Anal. PDE 7 (2014), no. 2, 267–310. doi:10.2140/apde.2014.7.267."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-33",
      "kind": "external",
      "label": "Reference [6]",
      "source": {
        "blockId": "p49-6cc65e0bb3",
        "page": 49,
        "quote": "[6] D. De Silva, F. Ferrari, and S. Salsa, Regularity of higher order in two-phase free boundary\nproblems, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3691–3720. doi:10.1090/tran/7550. Author\nmanuscript: arXiv:1701.07868v2."
      },
      "statement": [
        {
          "blockId": "p49-6cc65e0bb3",
          "page": 49,
          "quote": "[6] D. De Silva, F. Ferrari, and S. Salsa, Regularity of higher order in two-phase free boundary\nproblems, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3691–3720. doi:10.1090/tran/7550. Author\nmanuscript: arXiv:1701.07868v2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-34",
      "kind": "external",
      "label": "Reference [7]",
      "source": {
        "blockId": "p49-650460370e",
        "page": 49,
        "quote": "[7] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math.\n635 (2009), 1–21. doi:10.1515/CRELLE.2009.074."
      },
      "statement": [
        {
          "blockId": "p49-650460370e",
          "page": 49,
          "quote": "[7] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math.\n635 (2009), 1–21. doi:10.1515/CRELLE.2009.074."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-35",
      "kind": "external",
      "label": "Reference [8]",
      "source": {
        "blockId": "p49-7ba8a4402f",
        "page": 49,
        "quote": "[8] X. Fern´andez-Real and J. Serra, The stability conjecture for the Bernoulli problem in dimension\nfour, preprint (2026), arXiv:2609.05669v1."
      },
      "statement": [
        {
          "blockId": "p49-7ba8a4402f",
          "page": 49,
          "quote": "[8] X. Fern´andez-Real and J. Serra, The stability conjecture for the Bernoulli problem in dimension\nfour, preprint (2026), arXiv:2609.05669v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-36",
      "kind": "external",
      "label": "Reference [9]",
      "source": {
        "blockId": "p49-3f781fe3d6",
        "page": 49,
        "quote": "[9] X. Fern´andez-Real and H. Yu, Linearized equation and generic regularity in the Alt-Caffarelli\nproblem, preprint (2025), arXiv:2510.18330v1."
      },
      "statement": [
        {
          "blockId": "p49-3f781fe3d6",
          "page": 49,
          "quote": "[9] X. Fern´andez-Real and H. Yu, Linearized equation and generic regularity in the Alt-Caffarelli\nproblem, preprint (2025), arXiv:2510.18330v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-37",
      "kind": "external",
      "label": "Reference [10]",
      "source": {
        "blockId": "p49-3a92df49cd",
        "page": 49,
        "quote": "[10] B. Firester, R. Tsiamis, and Y. Wang, Stability inequalities for one-phase cones, preprint (2026),\narXiv:2601.16966v1."
      },
      "statement": [
        {
          "blockId": "p49-3a92df49cd",
          "page": 49,
          "quote": "[10] B. Firester, R. Tsiamis, and Y. Wang, Stability inequalities for one-phase cones, preprint (2026),\narXiv:2601.16966v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-38",
      "kind": "external",
      "label": "Reference [11]",
      "source": {
        "blockId": "p49-e382e47a48",
        "page": 49,
        "quote": "[11] B. Firester, R. Tsiamis, and Y. Wang, Area-minimizing capillary cones, preprint (2026),\narXiv:2601.18794v1."
      },
      "statement": [
        {
          "blockId": "p49-e382e47a48",
          "page": 49,
          "quote": "[11] B. Firester, R. Tsiamis, and Y. Wang, Area-minimizing capillary cones, preprint (2026),\narXiv:2601.18794v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-39",
      "kind": "external",
      "label": "Reference [12]",
      "source": {
        "blockId": "p49-c80cc7dc40",
        "page": 49,
        "quote": "[12] J. K. Hunter, Notes on Partial Differential Equations, University of California, Davis, revised\nJune 18, 2014. Author’s lecture notes."
      },
      "statement": [
        {
          "blockId": "p49-c80cc7dc40",
          "page": 49,
          "quote": "[12] J. K. Hunter, Notes on Partial Differential Equations, University of California, Davis, revised\nJune 18, 2014. Author’s lecture notes."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-40",
      "kind": "external",
      "label": "Reference [13]",
      "source": {
        "blockId": "p49-cb9f1d7d25",
        "page": 49,
        "quote": "[13] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary\nproblem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. doi:10.1007/s00039-015-0335-6.\nAuthor manuscript: arXiv:1410.7463v1."
      },
      "statement": [
        {
          "blockId": "p49-cb9f1d7d25",
          "page": 49,
          "quote": "[13] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary\nproblem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. doi:10.1007/s00039-015-0335-6.\nAuthor manuscript: arXiv:1410.7463v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-41",
      "kind": "external",
      "label": "Reference [14]",
      "source": {
        "blockId": "p49-17da114e8f",
        "page": 49,
        "quote": "[14] D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm.\nSup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 373–391. Numdam archive."
      },
      "statement": [
        {
          "blockId": "p49-17da114e8f",
          "page": 49,
          "quote": "[14] D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm.\nSup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 373–391. Numdam archive."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-42",
      "kind": "external",
      "label": "Reference [15]",
      "source": {
        "blockId": "p49-4be75fffc1",
        "page": 49,
        "quote": "[15] P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program.,\nSer. B 96 (2003), 293–320. doi:10.1007/s10107-003-0387-5."
      },
      "statement": [
        {
          "blockId": "p49-4be75fffc1",
          "page": 49,
          "quote": "[15] P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program.,\nSer. B 96 (2003), 293–320. doi:10.1007/s10107-003-0387-5."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-43",
      "kind": "external",
      "label": "Reference [16]",
      "source": {
        "blockId": "p49-dd8618eea7",
        "page": 49,
        "quote": "[16] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), no. 1,\n62–105. doi:10.2307/1970556."
      },
      "statement": [
        {
          "blockId": "p49-dd8618eea7",
          "page": 49,
          "quote": "[16] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), no. 1,\n62–105. doi:10.2307/1970556."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-44",
      "kind": "external",
      "label": "Reference [17]",
      "source": {
        "blockId": "p50-e40156af8d",
        "page": 50,
        "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
      },
      "statement": [
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-45",
      "kind": "external",
      "label": "Reference [18]",
      "source": {
        "blockId": "p50-7e8dbb57e4",
        "page": 50,
        "quote": "[18] G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9\n(1999), no. 2, 317–326. doi:10.1007/BF02921941."
      },
      "statement": [
        {
          "blockId": "p50-7e8dbb57e4",
          "page": 50,
          "quote": "[18] G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9\n(1999), no. 2, 317–326. doi:10.1007/BF02921941."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "equation-46",
      "kind": "equation",
      "label": "Equation (1.1)",
      "source": {
        "blockId": "p1-7458d4594f",
        "page": 1,
        "quote": "\n|∇v|2 + 1{v>0}\n\ndx.\n(1.1)"
      },
      "statement": [
        {
          "blockId": "p1-7458d4594f",
          "page": 1,
          "quote": "\n|∇v|2 + 1{v>0}\n\ndx.\n(1.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-47",
      "kind": "equation",
      "label": "Equation (2.1)",
      "source": {
        "blockId": "p4-656ba12637",
        "page": 4,
        "quote": "rj\n−→U(y)\n(2.1)"
      },
      "statement": [
        {
          "blockId": "p4-656ba12637",
          "page": 4,
          "quote": "rj\n−→U(y)\n(2.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-48",
      "kind": "equation",
      "label": "Equation (2.2)",
      "source": {
        "blockId": "p4-e807330307",
        "page": 4,
        "quote": "u = 0,\n∇u = NE\non the free boundary,\n(2.2)"
      },
      "statement": [
        {
          "blockId": "p4-e807330307",
          "page": 4,
          "quote": "u = 0,\n∇u = NE\non the free boundary,\n(2.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-49",
      "kind": "equation",
      "label": "Equation (2.3)",
      "source": {
        "blockId": "p5-16a211c14c",
        "page": 5,
        "quote": "r\n= ur′(y).\n(2.3)"
      },
      "statement": [
        {
          "blockId": "p5-16a211c14c",
          "page": 5,
          "quote": "r\n= ur′(y).\n(2.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-50",
      "kind": "equation",
      "label": "Equation (2.4)",
      "source": {
        "blockId": "p7-7f67ab3c19",
        "page": 7,
        "quote": "d = d∗∈{5, 6},\n(2.4)"
      },
      "statement": [
        {
          "blockId": "p7-7f67ab3c19",
          "page": 7,
          "quote": "d = d∗∈{5, 6},\n(2.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-51",
      "kind": "equation",
      "label": "Equation (2.5)",
      "source": {
        "blockId": "p7-616c575fdc",
        "page": 7,
        "quote": "∆Smogo = −mogo\nin Ωo,\ngo = 0,\n∇Smogo = No\non ∂Ωo,\n(2.5)"
      },
      "statement": [
        {
          "blockId": "p7-616c575fdc",
          "page": 7,
          "quote": "∆Smogo = −mogo\nin Ωo,\ngo = 0,\n∇Smogo = No\non ∂Ωo,\n(2.5)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-52",
      "kind": "equation",
      "label": "Equation (2.6)",
      "source": {
        "blockId": "p7-5c054ff718",
        "page": 7,
        "quote": "∆Rdu(ry) = r−1\n∆Smogo(y) + mogo(y)\n\n(2.6)"
      },
      "statement": [
        {
          "blockId": "p7-5c054ff718",
          "page": 7,
          "quote": "∆Rdu(ry) = r−1\n∆Smogo(y) + mogo(y)\n\n(2.6)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-53",
      "kind": "equation",
      "label": "Equation (2.7)",
      "source": {
        "blockId": "p7-425de8ddc1",
        "page": 7,
        "quote": "Lemma 2.5 (Selection of a nonflat component). There is a component Ωo in Proposition 2.4 such\nthat\nTo := ∇2\nSmogo + go Id ̸≡0\non Ωo.\n(2.7)"
      },
      "statement": [
        {
          "blockId": "p7-425de8ddc1",
          "page": 7,
          "quote": "Lemma 2.5 (Selection of a nonflat component). There is a component Ωo in Proposition 2.4 such\nthat\nTo := ∇2\nSmogo + go Id ̸≡0\non Ωo.\n(2.7)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-54",
      "kind": "equation",
      "label": "Equation (2.8)",
      "source": {
        "blockId": "p8-81876620ea",
        "page": 8,
        "quote": "D2u(ry)[y, ·] = 0,\nD2u(ry)[v1, v2] = r−1To(y)[v1, v2]\n(v1, v2 ⊥y).\n(2.8)"
      },
      "statement": [
        {
          "blockId": "p8-81876620ea",
          "page": 8,
          "quote": "D2u(ry)[y, ·] = 0,\nD2u(ry)[v1, v2] = r−1To(y)[v1, v2]\n(v1, v2 ⊥y).\n(2.8)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-55",
      "kind": "equation",
      "label": "Equation (2.9)",
      "source": {
        "blockId": "p8-7722e556c0",
        "page": 8,
        "quote": "eΩ=\n\nΦ(t, y) := (sin t y, cos t) : 0 < t < π, y ∈Ωo\n\t\n,\neg(Φ(t, y)) = sin t go(y).\n(2.9)"
      },
      "statement": [
        {
          "blockId": "p8-7722e556c0",
          "page": 8,
          "quote": "eΩ=\n\nΦ(t, y) := (sin t y, cos t) : 0 < t < π, y ∈Ωo\n\t\n,\neg(Φ(t, y)) = sin t go(y).\n(2.9)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-56",
      "kind": "equation",
      "label": "Equation (3.1)",
      "source": {
        "blockId": "p10-ae1fb87ac5",
        "page": 10,
        "quote": "p = ∇g,\nT = ∇2g + g Id,\nBijk = ∇iTjk,\nw = |T|,\ns2 = g2 + |p|2.\n(3.1)"
      },
      "statement": [
        {
          "blockId": "p10-ae1fb87ac5",
          "page": 10,
          "quote": "p = ∇g,\nT = ∇2g + g Id,\nBijk = ∇iTjk,\nw = |T|,\ns2 = g2 + |p|2.\n(3.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-57",
      "kind": "equation",
      "label": "Equation (3.2)",
      "source": {
        "blockId": "p10-d4e56f92b7",
        "page": 10,
        "quote": "\n.\n(3.2)"
      },
      "statement": [
        {
          "blockId": "p10-d4e56f92b7",
          "page": 10,
          "quote": "\n.\n(3.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-58",
      "kind": "equation",
      "label": "Equation (3.3)",
      "source": {
        "blockId": "p10-ead43e8dd0",
        "page": 10,
        "quote": "∇lBijk −∇iBljk = δljTik + δlkTij −δijTlk −δikTlj,\n(3.3)"
      },
      "statement": [
        {
          "blockId": "p10-ead43e8dd0",
          "page": 10,
          "quote": "∇lBijk −∇iBljk = δljTik + δlkTij −δijTlk −δikTlj,\n(3.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-59",
      "kind": "equation",
      "label": "Equation (3.4)",
      "source": {
        "blockId": "p10-4f905adc86",
        "page": 10,
        "quote": "∆T = mT,\n∆w2 = 2mw2 + 2|B|2,\n(3.4)"
      },
      "statement": [
        {
          "blockId": "p10-4f905adc86",
          "page": 10,
          "quote": "∆T = mT,\n∆w2 = 2mw2 + 2|B|2,\n(3.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-60",
      "kind": "equation",
      "label": "Equation (3.5)",
      "source": {
        "blockId": "p10-671ef66fb9",
        "page": 10,
        "quote": "∇lpi = Tli −gδli,\n∇lg = pl,\n(3.5)"
      },
      "statement": [
        {
          "blockId": "p10-671ef66fb9",
          "page": 10,
          "quote": "∇lpi = Tli −gδli,\n∇lg = pl,\n(3.5)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-61",
      "kind": "equation",
      "label": "Equation (3.6)",
      "source": {
        "blockId": "p10-9cd42a53f7",
        "page": 10,
        "quote": "∇s2 = 2Tp,\n∆s2 = 2w2.\n(3.6)"
      },
      "statement": [
        {
          "blockId": "p10-9cd42a53f7",
          "page": 10,
          "quote": "∇s2 = 2Tp,\n∆s2 = 2w2.\n(3.6)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-62",
      "kind": "equation",
      "label": "Equation (3.7)",
      "source": {
        "blockId": "p10-39b3f272e5",
        "page": 10,
        "quote": "∇lw = w−1TjkBljk,\n|∇w| ≤|B|.\n(3.7)"
      },
      "statement": [
        {
          "blockId": "p10-39b3f272e5",
          "page": 10,
          "quote": "∇lw = w−1TjkBljk,\n|∇w| ≤|B|.\n(3.7)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-63",
      "kind": "equation",
      "label": "Equation (3.8)",
      "source": {
        "blockId": "p11-79aa63b2b7",
        "page": 11,
        "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
      },
      "statement": [
        {
          "blockId": "p11-79aa63b2b7",
          "page": 11,
          "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-64",
      "kind": "equation",
      "label": "Equation (3.9)",
      "source": {
        "blockId": "p11-901a03e7df",
        "page": 11,
        "quote": "Then\nHo > 0,\nTopo = −Hopo,\nwo > 0\non ∂Ωo.\n(3.9)"
      },
      "statement": [
        {
          "blockId": "p11-901a03e7df",
          "page": 11,
          "quote": "Then\nHo > 0,\nTopo = −Hopo,\nwo > 0\non ∂Ωo.\n(3.9)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-65",
      "kind": "equation",
      "label": "Equation (3.10)",
      "source": {
        "blockId": "p11-064ff4fb38",
        "page": 11,
        "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
      },
      "statement": [
        {
          "blockId": "p11-064ff4fb38",
          "page": 11,
          "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-66",
      "kind": "equation",
      "label": "Equation (3.11)",
      "source": {
        "blockId": "p11-d8ff12d046",
        "page": 11,
        "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
      },
      "statement": [
        {
          "blockId": "p11-d8ff12d046",
          "page": 11,
          "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-67",
      "kind": "equation",
      "label": "Equation (3.12)",
      "source": {
        "blockId": "p11-ee96e4a03b",
        "page": 11,
        "quote": "Vijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(3.12)"
      },
      "statement": [
        {
          "blockId": "p11-ee96e4a03b",
          "page": 11,
          "quote": "Vijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(3.12)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-68",
      "kind": "equation",
      "label": "Equation (3.13)",
      "source": {
        "blockId": "p11-f6ddc57c80",
        "page": 11,
        "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
      },
      "statement": [
        {
          "blockId": "p11-f6ddc57c80",
          "page": 11,
          "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-69",
      "kind": "equation",
      "label": "Equation (3.14)",
      "source": {
        "blockId": "p12-b9af30c473",
        "page": 12,
        "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
      },
      "statement": [
        {
          "blockId": "p12-b9af30c473",
          "page": 12,
          "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-70",
      "kind": "equation",
      "label": "Equation (3.15)",
      "source": {
        "blockId": "p12-b9af30c473",
        "page": 12,
        "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
      },
      "statement": [
        {
          "blockId": "p12-b9af30c473",
          "page": 12,
          "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-71",
      "kind": "equation",
      "label": "Equation (3.16)",
      "source": {
        "blockId": "p12-fe39066800",
        "page": 12,
        "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
      },
      "statement": [
        {
          "blockId": "p12-fe39066800",
          "page": 12,
          "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-72",
      "kind": "equation",
      "label": "Equation (3.17)",
      "source": {
        "blockId": "p12-8b2672b64e",
        "page": 12,
        "quote": "ew = ρ−1wo,\n| eB|2 = ρ−4\n|Bo|2 + 3c2w2\no\n\n,\neH = ρ−1Ho\non the regular side.\n(3.17)"
      },
      "statement": [
        {
          "blockId": "p12-8b2672b64e",
          "page": 12,
          "quote": "ew = ρ−1wo,\n| eB|2 = ρ−4\n|Bo|2 + 3c2w2\no\n\n,\neH = ρ−1Ho\non the regular side.\n(3.17)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-73",
      "kind": "equation",
      "label": "Equation (3.18)",
      "source": {
        "blockId": "p12-8a8302c5ce",
        "page": 12,
        "quote": "dV = ρ4 dt dVo,\ndσ = ρ3 dt dσo.\n(3.18)"
      },
      "statement": [
        {
          "blockId": "p12-8a8302c5ce",
          "page": 12,
          "quote": "dV = ρ4 dt dVo,\ndσ = ρ3 dt dσo.\n(3.18)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-74",
      "kind": "equation",
      "label": "Equation (3.19)",
      "source": {
        "blockId": "p13-39ac08043a",
        "page": 13,
        "quote": "∂D\nHEϕ2 dσE,\nHE = −D2u(NE, NE),\n(3.19)"
      },
      "statement": [
        {
          "blockId": "p13-39ac08043a",
          "page": 13,
          "quote": "∂D\nHEϕ2 dσE,\nHE = −D2u(NE, NE),\n(3.19)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-75",
      "kind": "equation",
      "label": "Equation (3.20)",
      "source": {
        "blockId": "p14-563124ccb1",
        "page": 14,
        "quote": "D\n|∇ϕ|2 dx.\n(3.20)"
      },
      "statement": [
        {
          "blockId": "p14-563124ccb1",
          "page": 14,
          "quote": "D\n|∇ϕ|2 dx.\n(3.20)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-76",
      "kind": "equation",
      "label": "Equation (3.21)",
      "source": {
        "blockId": "p14-9f21210952",
        "page": 14,
        "quote": "j(y, 0) = 1,\n∂ξj(y, 0) = div VF (y, 0) = ∆F −2FNENE = 2HE.\n(3.21)"
      },
      "statement": [
        {
          "blockId": "p14-9f21210952",
          "page": 14,
          "quote": "j(y, 0) = 1,\n∂ξj(y, 0) = div VF (y, 0) = ∆F −2FNENE = 2HE.\n(3.21)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-77",
      "kind": "equation",
      "label": "Equation (3.22)",
      "source": {
        "blockId": "p14-7772931587",
        "page": 14,
        "quote": "and hence\n∂ξ\n\n(1 + |∇F|2)j\n\nξ=0 = 2HE.\n(3.22)"
      },
      "statement": [
        {
          "blockId": "p14-7772931587",
          "page": 14,
          "quote": "and hence\n∂ξ\n\n(1 + |∇F|2)j\n\nξ=0 = 2HE.\n(3.22)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-78",
      "kind": "equation",
      "label": "Equation (3.23)",
      "source": {
        "blockId": "p14-85993fe6a5",
        "page": 14,
        "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
      },
      "statement": [
        {
          "blockId": "p14-85993fe6a5",
          "page": 14,
          "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-79",
      "kind": "equation",
      "label": "Equation (3.24)",
      "source": {
        "blockId": "p14-0461573878",
        "page": 14,
        "quote": "(1 + |∇(F + εϕ)|2)j = 2 + 2εϕN + 2HEξ + O(ε2 + |εξ| + ξ2).\n(3.24)"
      },
      "statement": [
        {
          "blockId": "p14-0461573878",
          "page": 14,
          "quote": "(1 + |∇(F + εϕ)|2)j = 2 + 2εϕN + 2HEξ + O(ε2 + |εξ| + ξ2).\n(3.24)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-80",
      "kind": "equation",
      "label": "Equation (3.25)",
      "source": {
        "blockId": "p14-b2fb2c5ad6",
        "page": 14,
        "quote": "= 2εϕ0 −ε2HEϕ2\n0 + O(|ε|3).\n(3.25)"
      },
      "statement": [
        {
          "blockId": "p14-b2fb2c5ad6",
          "page": 14,
          "quote": "= 2εϕ0 −ε2HEϕ2\n0 + O(|ε|3).\n(3.25)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-81",
      "kind": "equation",
      "label": "Equation (3.26)",
      "source": {
        "blockId": "p15-9bcb9d1c60",
        "page": 15,
        "quote": "\n|∇v|2 + 4v2\ndV.\n(3.26)"
      },
      "statement": [
        {
          "blockId": "p15-9bcb9d1c60",
          "page": 15,
          "quote": "\n|∇v|2 + 4v2\ndV.\n(3.26)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-82",
      "kind": "equation",
      "label": "Equation (3.27)",
      "source": {
        "blockId": "p15-51cc79bd7d",
        "page": 15,
        "quote": "R ψ2 dτ .\n(3.27)"
      },
      "statement": [
        {
          "blockId": "p15-51cc79bd7d",
          "page": 15,
          "quote": "R ψ2 dτ .\n(3.27)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-83",
      "kind": "equation",
      "label": "Equation (4.1)",
      "source": {
        "blockId": "p16-e7c1bb9818",
        "page": 16,
        "quote": "31250000000.\n(4.1)"
      },
      "statement": [
        {
          "blockId": "p16-e7c1bb9818",
          "page": 16,
          "quote": "31250000000.\n(4.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-84",
      "kind": "equation",
      "label": "Equation (4.2)",
      "source": {
        "blockId": "p16-89211ce53b",
        "page": 16,
        "quote": "DlAij = Xlij −qlAij,\nDlpi = wAli −gzδli,\nDlg = zpl,\nDlδij = 0.\n(4.2)"
      },
      "statement": [
        {
          "blockId": "p16-89211ce53b",
          "page": 16,
          "quote": "DlAij = Xlij −qlAij,\nDlpi = wAli −gzδli,\nDlg = zpl,\nDlδij = 0.\n(4.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-85",
      "kind": "equation",
      "label": "Equation (4.3)",
      "source": {
        "blockId": "p16-7cb9687645",
        "page": 16,
        "quote": "+ f|ℓ|2 −(Dlf + 2aqlf)ℓl ≥ϵ∗(z2 + |ℓ|2).\n(4.3)"
      },
      "statement": [
        {
          "blockId": "p16-7cb9687645",
          "page": 16,
          "quote": "+ f|ℓ|2 −(Dlf + 2aqlf)ℓl ≥ϵ∗(z2 + |ℓ|2).\n(4.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-86",
      "kind": "equation",
      "label": "Equation (4.4)",
      "source": {
        "blockId": "p16-12ac49d162",
        "page": 16,
        "quote": "Then\nS := −λf + Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl ≥δ∗.\n(4.4)"
      },
      "statement": [
        {
          "blockId": "p16-12ac49d162",
          "page": 16,
          "quote": "Then\nS := −λf + Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl ≥δ∗.\n(4.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-87",
      "kind": "equation",
      "label": "Equation (5.1)",
      "source": {
        "blockId": "p17-19ab409168",
        "page": 17,
        "quote": "All assertions up to the side boundary use derivatives from the positive side. In the suspended case\nthe two tips are excluded throughout Section 5. We apply the universal inequalities of Proposition 4.1,\nwhose proof follows in Section 6. For its coefficient polynomials f, U, K, M, P, with their stated slot\norders, set\nClijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk.\n(5.1)"
      },
      "statement": [
        {
          "blockId": "p17-19ab409168",
          "page": 17,
          "quote": "All assertions up to the side boundary use derivatives from the positive side. In the suspended case\nthe two tips are excluded throughout Section 5. We apply the universal inequalities of Proposition 4.1,\nwhose proof follows in Section 6. For its coefficient polynomials f, U, K, M, P, with their stated slot\norders, set\nClijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk.\n(5.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-88",
      "kind": "equation",
      "label": "Equation (5.2)",
      "source": {
        "blockId": "p17-064052db40",
        "page": 17,
        "quote": "w ,\nql = XljkAjk.\n(5.2)"
      },
      "statement": [
        {
          "blockId": "p17-064052db40",
          "page": 17,
          "quote": "w ,\nql = XljkAjk.\n(5.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-89",
      "kind": "equation",
      "label": "Equation (5.3)",
      "source": {
        "blockId": "p17-ea70133cef",
        "page": 17,
        "quote": "∇lw = wql,\n∇lAij = Xlij −qlAij,\n∇lpi = wAli −gδli,\n∇lg = pl.\n(5.3)"
      },
      "statement": [
        {
          "blockId": "p17-ea70133cef",
          "page": 17,
          "quote": "∇lw = wql,\n∇lAij = Xlij −qlAij,\n∇lpi = wAli −gδli,\n∇lg = pl.\n(5.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-90",
      "kind": "equation",
      "label": "Equation (5.4)",
      "source": {
        "blockId": "p17-4e99f8fde8",
        "page": 17,
        "quote": "Furthermore,\n1\nwGlijk∇lBijk = 5UjkAjk + Clijk(δljAik + δlkAij).\n(5.4)"
      },
      "statement": [
        {
          "blockId": "p17-4e99f8fde8",
          "page": 17,
          "quote": "Furthermore,\n1\nwGlijk∇lBijk = 5UjkAjk + Clijk(δljAik + δlkAij).\n(5.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-91",
      "kind": "equation",
      "label": "Equation (5.5)",
      "source": {
        "blockId": "p18-686fa7a72d",
        "page": 18,
        "quote": "f(A, p, g) ≥ϵ∗=\n438907\n12000000 > 0\n(5.5)"
      },
      "statement": [
        {
          "blockId": "p18-686fa7a72d",
          "page": 18,
          "quote": "f(A, p, g) ≥ϵ∗=\n438907\n12000000 > 0\n(5.5)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-92",
      "kind": "equation",
      "label": "Equation (5.6)",
      "source": {
        "blockId": "p18-758531bc91",
        "page": 18,
        "quote": "f,\nZl = w2a\nGlijkXijk + wMl + Pl\n\n.\n(5.6)"
      },
      "statement": [
        {
          "blockId": "p18-758531bc91",
          "page": 18,
          "quote": "f,\nZl = w2a\nGlijkXijk + wMl + Pl\n\n.\n(5.6)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-93",
      "kind": "equation",
      "label": "Equation (5.7)",
      "source": {
        "blockId": "p18-9950cf5a0f",
        "page": 18,
        "quote": "div Z ≥4h2 + |∇h|2 + ϵ∗w2a\non Ω+.\n(5.7)"
      },
      "statement": [
        {
          "blockId": "p18-9950cf5a0f",
          "page": 18,
          "quote": "div Z ≥4h2 + |∇h|2 + ϵ∗w2a\non Ω+.\n(5.7)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-94",
      "kind": "equation",
      "label": "Equation (5.8)",
      "source": {
        "blockId": "p18-1a4e499ddb",
        "page": 18,
        "quote": "+\n\nDlGlijk + (2a −1)qlGlijk\n\nXijk\n+ w\n\nDlMl + (2a + 1)qlMl\n\n+ DlPl + 2aqlPl.\n(5.8)"
      },
      "statement": [
        {
          "blockId": "p18-1a4e499ddb",
          "page": 18,
          "quote": "+\n\nDlGlijk + (2a −1)qlGlijk\n\nXijk\n+ w\n\nDlMl + (2a + 1)qlMl\n\n+ DlPl + 2aqlPl.\n(5.8)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-95",
      "kind": "equation",
      "label": "Equation (5.9)",
      "source": {
        "blockId": "p19-8023154c00",
        "page": 19,
        "quote": "4f |F|2,\n(5.9)"
      },
      "statement": [
        {
          "blockId": "p19-8023154c00",
          "page": 19,
          "quote": "4f |F|2,\n(5.9)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-96",
      "kind": "equation",
      "label": "Equation (5.10)",
      "source": {
        "blockId": "p19-7492df17f5",
        "page": 19,
        "quote": "I\n\nz=1 = w−2a\ndiv Z −4h2 −|∇h|2\n+ f|ℓ−ℓ∗|2.\n(5.10)"
      },
      "statement": [
        {
          "blockId": "p19-7492df17f5",
          "page": 19,
          "quote": "I\n\nz=1 = w−2a\ndiv Z −4h2 −|∇h|2\n+ f|ℓ−ℓ∗|2.\n(5.10)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-97",
      "kind": "equation",
      "label": "Equation (5.11)",
      "source": {
        "blockId": "p19-d3b87440c1",
        "page": 19,
        "quote": "S0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(5.11)"
      },
      "statement": [
        {
          "blockId": "p19-d3b87440c1",
          "page": 19,
          "quote": "S0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(5.11)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-98",
      "kind": "equation",
      "label": "Equation (5.12)",
      "source": {
        "blockId": "p19-7966fef82a",
        "page": 19,
        "quote": "w2 = plGlijkVijk = Ujk(S0 + p ⊗p)jk + ClijkplVijk.\n(5.12)"
      },
      "statement": [
        {
          "blockId": "p19-7966fef82a",
          "page": 19,
          "quote": "w2 = plGlijkVijk = Ujk(S0 + p ⊗p)jk + ClijkplVijk.\n(5.12)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-99",
      "kind": "equation",
      "label": "Equation (5.13)",
      "source": {
        "blockId": "p20-7818ef941d",
        "page": 20,
        "quote": "31250000000,\n(5.13)"
      },
      "statement": [
        {
          "blockId": "p20-7818ef941d",
          "page": 20,
          "quote": "31250000000,\n(5.13)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-100",
      "kind": "equation",
      "label": "Equation (5.14)",
      "source": {
        "blockId": "p20-451469844c",
        "page": 20,
        "quote": "|h| ≤Cwa,\n|∇h| ≤Cwa(1 + w + |X|),\n|Z| ≤Cw2a(1 + w + |X|).\n(5.14)"
      },
      "statement": [
        {
          "blockId": "p20-451469844c",
          "page": 20,
          "quote": "|h| ≤Cwa,\n|∇h| ≤Cwa(1 + w + |X|),\n|Z| ≤Cw2a(1 + w + |X|).\n(5.14)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-101",
      "kind": "equation",
      "label": "Equation (6.1)",
      "source": {
        "blockId": "p21-0293f58fa7",
        "page": 21,
        "quote": "(F,s,c) in Table I\nc MrF (s).\n(6.1)"
      },
      "statement": [
        {
          "blockId": "p21-0293f58fa7",
          "page": 21,
          "quote": "(F,s,c) in Table I\nc MrF (s).\n(6.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-102",
      "kind": "equation",
      "label": "Equation (6.2)",
      "source": {
        "blockId": "p22-1fe5a2118c",
        "page": 22,
        "quote": "(\n1,\nmode = I,\nχ,\nmode = B.\n(6.2)"
      },
      "statement": [
        {
          "blockId": "p22-1fe5a2118c",
          "page": 22,
          "quote": "(\n1,\nmode = I,\nχ,\nmode = B.\n(6.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-103",
      "kind": "equation",
      "label": "Equation (6.3)",
      "source": {
        "blockId": "p22-1a1224c4bd",
        "page": 22,
        "quote": "(\n1,\nmode = I,\nx1 + x2 + x3 + x4,\nmode = H.\n(6.3)"
      },
      "statement": [
        {
          "blockId": "p22-1a1224c4bd",
          "page": 22,
          "quote": "(\n1,\nmode = I,\nx1 + x2 + x3 + x4,\nmode = H.\n(6.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-104",
      "kind": "equation",
      "label": "Equation (6.4)",
      "source": {
        "blockId": "p22-4013f4df08",
        "page": 22,
        "quote": "(ΠF)ij = 5F ′\nij −δijv\n(r = 2),\n(6.4)"
      },
      "statement": [
        {
          "blockId": "p22-4013f4df08",
          "page": 22,
          "quote": "(ΠF)ij = 5F ′\nij −δijv\n(r = 2),\n(6.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-105",
      "kind": "equation",
      "label": "Equation (6.5)",
      "source": {
        "blockId": "p22-8cea1b3984",
        "page": 22,
        "quote": "(ΠF)ijk = 7F ′\nijk −(δijvk + δikvj + δjkvi)\n(r = 3).\n(6.5)"
      },
      "statement": [
        {
          "blockId": "p22-8cea1b3984",
          "page": 22,
          "quote": "(ΠF)ijk = 7F ′\nijk −(δijvk + δikvj + δjkvi)\n(r = 3).\n(6.5)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-106",
      "kind": "equation",
      "label": "Equation (6.6)",
      "source": {
        "blockId": "p22-632c3c3465",
        "page": 22,
        "quote": "(ΠF)ijk = 8F ′\nijk −(2δijvk −δikvj −δjkvi).\n(6.6)"
      },
      "statement": [
        {
          "blockId": "p22-632c3c3465",
          "page": 22,
          "quote": "(ΠF)ijk = 8F ′\nijk −(2δijvk −δikvj −δjkvi).\n(6.6)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-107",
      "kind": "equation",
      "label": "Equation (6.7)",
      "source": {
        "blockId": "p24-18023602ce",
        "page": 24,
        "quote": "h=0\nAhXlAk−1−h −kqlAk,\n(Xl)ij = Xlij,\n(6.7)"
      },
      "statement": [
        {
          "blockId": "p24-18023602ce",
          "page": 24,
          "quote": "h=0\nAhXlAk−1−h −kqlAk,\n(Xl)ij = Xlij,\n(6.7)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-108",
      "kind": "equation",
      "label": "Equation (6.8)",
      "source": {
        "blockId": "p24-003e007e75",
        "page": 24,
        "quote": "Dl tr Ak = k tr(Ak−1Xl) −kql tr Ak.\n(6.8)"
      },
      "statement": [
        {
          "blockId": "p24-003e007e75",
          "page": 24,
          "quote": "Dl tr Ak = k tr(Ak−1Xl) −kql tr Ak.\n(6.8)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-109",
      "kind": "equation",
      "label": "Equation (6.9)",
      "source": {
        "blockId": "p25-26685d701f",
        "page": 25,
        "quote": "∥A∥2\nop ≤θ,\n|Ak|2 ≤θk−1\n(k ≥1),\n(6.9)"
      },
      "statement": [
        {
          "blockId": "p25-26685d701f",
          "page": 25,
          "quote": "∥A∥2\nop ≤θ,\n|Ak|2 ≤θk−1\n(k ≥1),\n(6.9)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-110",
      "kind": "equation",
      "label": "Equation (6.10)",
      "source": {
        "blockId": "p25-34df8deab7",
        "page": 25,
        "quote": "20,\n(6.10)"
      },
      "statement": [
        {
          "blockId": "p25-34df8deab7",
          "page": 25,
          "quote": "20,\n(6.10)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-111",
      "kind": "equation",
      "label": "Equation (6.11)",
      "source": {
        "blockId": "p25-3033ab8bae",
        "page": 25,
        "quote": "7|X|2,\n(X : A)i = XijkAjk.\n(6.11)"
      },
      "statement": [
        {
          "blockId": "p25-3033ab8bae",
          "page": 25,
          "quote": "7|X|2,\n(X : A)i = XijkAjk.\n(6.11)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-112",
      "kind": "equation",
      "label": "Equation (6.12)",
      "source": {
        "blockId": "p25-6744ff863c",
        "page": 25,
        "quote": "b\n.\n(6.12)"
      },
      "statement": [
        {
          "blockId": "p25-6744ff863c",
          "page": 25,
          "quote": "b\n.\n(6.12)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-113",
      "kind": "equation",
      "label": "Equation (6.13)",
      "source": {
        "blockId": "p26-51839e719e",
        "page": 26,
        "quote": "|M(t)| ≤Ltyiyj\n(6.13)"
      },
      "statement": [
        {
          "blockId": "p26-51839e719e",
          "page": 26,
          "quote": "|M(t)| ≤Ltyiyj\n(6.13)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-114",
      "kind": "equation",
      "label": "Equation (6.14)",
      "source": {
        "blockId": "p27-73f0b96db0",
        "page": 27,
        "quote": "i=1\ndiy2\ni −I.\n(6.14)"
      },
      "statement": [
        {
          "blockId": "p27-73f0b96db0",
          "page": 27,
          "quote": "i=1\ndiy2\ni −I.\n(6.14)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-115",
      "kind": "equation",
      "label": "Equation (6.15)",
      "source": {
        "blockId": "p27-67d3cf2ab9",
        "page": 27,
        "quote": "105 .\n(6.15)"
      },
      "statement": [
        {
          "blockId": "p27-67d3cf2ab9",
          "page": 27,
          "quote": "105 .\n(6.15)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-116",
      "kind": "equation",
      "label": "Equation (6.16)",
      "source": {
        "blockId": "p27-e26e4dc531",
        "page": 27,
        "quote": "⌊s⌋⌋by integer arithmetic and increase it by one exactly when n2 < s. Form a symmetric\nmatrix D0 by adding\n|et|bLt\n1 + 1{i̸=j}\n(6.16)"
      },
      "statement": [
        {
          "blockId": "p27-e26e4dc531",
          "page": 27,
          "quote": "⌊s⌋⌋by integer arithmetic and increase it by one exactly when n2 < s. Form a symmetric\nmatrix D0 by adding\n|et|bLt\n1 + 1{i̸=j}\n(6.16)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-117",
      "kind": "equation",
      "label": "Equation (6.17)",
      "source": {
        "blockId": "p27-23b335b01b",
        "page": 27,
        "quote": "to its ij entry and also its ji entry if i ̸= j, starting with zero. The factor 1/2 in an off-diagonal\nentry is required because the associated term in yT D0y is 2(D0)ijyiyj. Lemma 6.4 and the triangle\ninequality therefore give\n|E| ≤yT D0y.\n(6.17)"
      },
      "statement": [
        {
          "blockId": "p27-23b335b01b",
          "page": 27,
          "quote": "to its ij entry and also its ji entry if i ̸= j, starting with zero. The factor 1/2 in an off-diagonal\nentry is required because the associated term in yT D0y is 2(D0)ijyiyj. Lemma 6.4 and the triangle\ninequality therefore give\n|E| ≤yT D0y.\n(6.17)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-118",
      "kind": "equation",
      "label": "Equation (6.18)",
      "source": {
        "blockId": "p28-8ba71d198c",
        "page": 28,
        "quote": "(6.18)"
      },
      "statement": [
        {
          "blockId": "p28-8ba71d198c",
          "page": 28,
          "quote": "(6.18)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-119",
      "kind": "equation",
      "label": "Equation (6.19)",
      "source": {
        "blockId": "p28-3868af73b9",
        "page": 28,
        "quote": "\n\n.\n(6.19)"
      },
      "statement": [
        {
          "blockId": "p28-3868af73b9",
          "page": 28,
          "quote": "\n\n.\n(6.19)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-120",
      "kind": "equation",
      "label": "Equation (6.20)",
      "source": {
        "blockId": "p28-de0f32870a",
        "page": 28,
        "quote": "\n.\n(6.20)"
      },
      "statement": [
        {
          "blockId": "p28-de0f32870a",
          "page": 28,
          "quote": "\n.\n(6.20)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-121",
      "kind": "equation",
      "label": "Equation (6.21)",
      "source": {
        "blockId": "p28-b026237705",
        "page": 28,
        "quote": "= 1.\n(6.21)"
      },
      "statement": [
        {
          "blockId": "p28-b026237705",
          "page": 28,
          "quote": "= 1.\n(6.21)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-122",
      "kind": "equation",
      "label": "Equation (6.22)",
      "source": {
        "blockId": "p29-15d80f220a",
        "page": 29,
        "quote": "α=1\nbα(Uαα + C5α5α + C5αα5),\n(6.22)"
      },
      "statement": [
        {
          "blockId": "p29-15d80f220a",
          "page": 29,
          "quote": "α=1\nbα(Uαα + C5α5α + C5αα5),\n(6.22)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-123",
      "kind": "equation",
      "label": "Equation (6.23)",
      "source": {
        "blockId": "p29-d5abce1feb",
        "page": 29,
        "quote": "!k\n.\n(6.23)"
      },
      "statement": [
        {
          "blockId": "p29-d5abce1feb",
          "page": 29,
          "quote": "!k\n.\n(6.23)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-124",
      "kind": "equation",
      "label": "Equation (6.24)",
      "source": {
        "blockId": "p29-0852f42978",
        "page": 29,
        "quote": "Expanding the two boundary Gram blocks and the Table I expression gives a polynomial represen-\ntative with monomials of degree at most five. A finite way to check the degree assertion is to use\n(6.22): discard positive g powers, substitute the diagonal matrix and normal vector in each of the\n231 rows, and add the displayed terms. All remaining monomials have degree at most five. Each\nboundary Jc has degrees one and two, so its squared terms have degree at most four, or five after\nthe H weight.\nThe following convention gives a small exact list of its coefficients. For an ordinary monomial\nxα of degree j ≤5, replace it by\nxαR⌊(5−j)/2⌋.\n(6.24)"
      },
      "statement": [
        {
          "blockId": "p29-0852f42978",
          "page": 29,
          "quote": "Expanding the two boundary Gram blocks and the Table I expression gives a polynomial represen-\ntative with monomials of degree at most five. A finite way to check the degree assertion is to use\n(6.22): discard positive g powers, substitute the diagonal matrix and normal vector in each of the\n231 rows, and add the displayed terms. All remaining monomials have degree at most five. Each\nboundary Jc has degrees one and two, so its squared terms have degree at most four, or five after\nthe H weight.\nThe following convention gives a small exact list of its coefficients. For an ordinary monomial\nxα of degree j ≤5, replace it by\nxαR⌊(5−j)/2⌋.\n(6.24)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-125",
      "kind": "equation",
      "label": "Equation (6.25)",
      "source": {
        "blockId": "p29-30d50b9011",
        "page": 29,
        "quote": "β\ne(b)\nβ mβ(x)\non R = 1,\n(6.25)"
      },
      "statement": [
        {
          "blockId": "p29-30d50b9011",
          "page": 29,
          "quote": "β\ne(b)\nβ mβ(x)\non R = 1,\n(6.25)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-126",
      "kind": "equation",
      "label": "Equation (6.26)",
      "source": {
        "blockId": "p30-4c979cab5d",
        "page": 30,
        "quote": "(6.26)"
      },
      "statement": [
        {
          "blockId": "p30-4c979cab5d",
          "page": 30,
          "quote": "(6.26)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-127",
      "kind": "equation",
      "label": "Equation (7.1)",
      "source": {
        "blockId": "p30-6b3ec9e1fd",
        "page": 30,
        "quote": "Ωo,+\nw2a−2\no\n|Bo|2 dVo < ∞.\n(7.1)"
      },
      "statement": [
        {
          "blockId": "p30-6b3ec9e1fd",
          "page": 30,
          "quote": "Ωo,+\nw2a−2\no\n|Bo|2 dVo < ∞.\n(7.1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-128",
      "kind": "equation",
      "label": "Equation (7.2)",
      "source": {
        "blockId": "p31-c40865070c",
        "page": 31,
        "quote": "(\nwo(y)2a\n1 + |Bo(y)|/wo(y)\n2,\ny ∈Ωo,+,\n0,\ny ∈Zo,\n(7.2)"
      },
      "statement": [
        {
          "blockId": "p31-c40865070c",
          "page": 31,
          "quote": "(\nwo(y)2a\n1 + |Bo(y)|/wo(y)\n2,\ny ∈Ωo,+,\n0,\ny ∈Zo,\n(7.2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-129",
      "kind": "equation",
      "label": "Equation (7.3)",
      "source": {
        "blockId": "p31-a5342e9d9d",
        "page": 31,
        "quote": "a(M2\no + 1)a∥∆η∥L1(Ωo).\n(7.3)"
      },
      "statement": [
        {
          "blockId": "p31-a5342e9d9d",
          "page": 31,
          "quote": "a(M2\no + 1)a∥∆η∥L1(Ωo).\n(7.3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-130",
      "kind": "equation",
      "label": "Equation (7.4)",
      "source": {
        "blockId": "p32-788c420b9b",
        "page": 32,
        "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
      },
      "statement": [
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-131",
      "kind": "equation",
      "label": "Equation (7.5)",
      "source": {
        "blockId": "p32-864f7ae2ea",
        "page": 32,
        "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
      },
      "statement": [
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-132",
      "kind": "equation",
      "label": "Equation (7.6)",
      "source": {
        "blockId": "p32-013d263a9a",
        "page": 32,
        "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
      },
      "statement": [
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-133",
      "kind": "equation",
      "label": "Equation (7.7)",
      "source": {
        "blockId": "p33-4816d80e83",
        "page": 33,
        "quote": "wo\n.\n(7.7)"
      },
      "statement": [
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-134",
      "kind": "equation",
      "label": "Equation (7.8)",
      "source": {
        "blockId": "p33-3cf50241b6",
        "page": 33,
        "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
      },
      "statement": [
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-135",
      "kind": "equation",
      "label": "Equation (7.9)",
      "source": {
        "blockId": "p33-4ebee114f5",
        "page": 33,
        "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
      },
      "statement": [
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-136",
      "kind": "equation",
      "label": "Equation (7.10)",
      "source": {
        "blockId": "p34-36dc29e37c",
        "page": 34,
        "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
      },
      "statement": [
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-137",
      "kind": "equation",
      "label": "Equation (7.11)",
      "source": {
        "blockId": "p34-2b8f2b4a03",
        "page": 34,
        "quote": "3.\n(7.11)"
      },
      "statement": [
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-138",
      "kind": "equation",
      "label": "Equation (7.12)",
      "source": {
        "blockId": "p34-12feee0731",
        "page": 34,
        "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
      },
      "statement": [
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "external-139",
      "kind": "external",
      "label": "[17] Theorem 1.4",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-140",
      "kind": "external",
      "label": "[4] Theorem 1.1",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-689109c677",
          "page": 49,
          "quote": "[4] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound.\n13 (2011), no. 2, 223–238. doi:10.4171/IFB/255."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-141",
      "kind": "external",
      "label": "[14] Theorem 2",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-17da114e8f",
          "page": 49,
          "quote": "[14] D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm.\nSup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 373–391. Numdam archive."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-142",
      "kind": "external",
      "label": "[6] Theorem 1.1",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-6cc65e0bb3",
          "page": 49,
          "quote": "[6] D. De Silva, F. Ferrari, and S. Salsa, Regularity of higher order in two-phase free boundary\nproblems, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3691–3720. doi:10.1090/tran/7550. Author\nmanuscript: arXiv:1701.07868v2."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-143",
      "kind": "external",
      "label": "[6] Corollary 1.2",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-6cc65e0bb3",
          "page": 49,
          "quote": "[6] D. De Silva, F. Ferrari, and S. Salsa, Regularity of higher order in two-phase free boundary\nproblems, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3691–3720. doi:10.1090/tran/7550. Author\nmanuscript: arXiv:1701.07868v2."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-144",
      "kind": "external",
      "label": "[13] Theorem 1.1",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-cb9f1d7d25",
          "page": 49,
          "quote": "[13] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary\nproblem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. doi:10.1007/s00039-015-0335-6.\nAuthor manuscript: arXiv:1410.7463v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-145",
      "kind": "external",
      "label": "[13] Corollary 1.2",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-cb9f1d7d25",
          "page": 49,
          "quote": "[13] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary\nproblem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. doi:10.1007/s00039-015-0335-6.\nAuthor manuscript: arXiv:1410.7463v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-146",
      "kind": "external",
      "label": "[7] Theorem 1.1",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-650460370e",
          "page": 49,
          "quote": "[7] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math.\n635 (2009), 1–21. doi:10.1515/CRELLE.2009.074."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-147",
      "kind": "external",
      "label": "[11] Theorem 1.4",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-e382e47a48",
          "page": 49,
          "quote": "[11] B. Firester, R. Tsiamis, and Y. Wang, Area-minimizing capillary cones, preprint (2026),\narXiv:2601.18794v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-148",
      "kind": "external",
      "label": "[10] Theorem 1.1",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-3a92df49cd",
          "page": 49,
          "quote": "[10] B. Firester, R. Tsiamis, and Y. Wang, Stability inequalities for one-phase cones, preprint (2026),\narXiv:2601.16966v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-149",
      "kind": "external",
      "label": "[8] Theorem 1.1",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-7ba8a4402f",
          "page": 49,
          "quote": "[8] X. Fern´andez-Real and J. Serra, The stability conjecture for the Bernoulli problem in dimension\nfour, preprint (2026), arXiv:2609.05669v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-150",
      "kind": "external",
      "label": "[9] Theorem 1.3",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "statement": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p49-3f781fe3d6",
          "page": 49,
          "quote": "[9] X. Fern´andez-Real and H. Yu, Linearized equation and generic regularity in the Alt-Caffarelli\nproblem, preprint (2025), arXiv:2510.18330v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-151",
      "kind": "external",
      "label": "[17] Definition 2.11",
      "source": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "statement": [
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-152",
      "kind": "external",
      "label": "[17] Theorem 3.1",
      "source": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "statement": [
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-153",
      "kind": "external",
      "label": "[17] Proposition 4.1",
      "source": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "statement": [
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-154",
      "kind": "external",
      "label": "[17] Proposition 6.2",
      "source": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "statement": [
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-155",
      "kind": "external",
      "label": "[17] Proposition 9.12",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-156",
      "kind": "external",
      "label": "[17] Definition 6.10",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-157",
      "kind": "external",
      "label": "[17] Corollary 8.2",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-158",
      "kind": "external",
      "label": "[17] Theorem 8.1",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-159",
      "kind": "external",
      "label": "[17] Proposition 7.1",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Agent-associated external reference from source context; external statement and proof not retrieved."
    },
    {
      "id": "external-160",
      "kind": "external",
      "label": "[5] Definition 7.1",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p49-cc3b900f63",
          "page": 49,
          "quote": "[5] D. De Silva, F. Ferrari, and S. Salsa, Two-phase problems with distributed sources: regularity\nof the free boundary, Anal. PDE 7 (2014), no. 2, 267–310. doi:10.2140/apde.2014.7.267."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-161",
      "kind": "external",
      "label": "[17] Definition 7.6",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-162",
      "kind": "external",
      "label": "[12] Theorem 4.30",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p49-c80cc7dc40",
          "page": 49,
          "quote": "[12] J. K. Hunter, Notes on Partial Differential Equations, University of California, Davis, revised\nJune 18, 2014. Author’s lecture notes."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-163",
      "kind": "external",
      "label": "[12] Corollary 4.32",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "statement": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p49-c80cc7dc40",
          "page": 49,
          "quote": "[12] J. K. Hunter, Notes on Partial Differential Equations, University of California, Davis, revised\nJune 18, 2014. Author’s lecture notes."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-164",
      "kind": "external",
      "label": "[17] Lemma 10.9",
      "source": {
        "blockId": "p5-37646f2f83",
        "page": 5,
        "quote": "The next two lemmas give the dimension-reduction steps explicitly; compare [17, Lemmas 10.9–10.10]."
      },
      "statement": [
        {
          "blockId": "p5-37646f2f83",
          "page": 5,
          "quote": "The next two lemmas give the dimension-reduction steps explicitly; compare [17, Lemmas 10.9–10.10]."
        },
        {
          "blockId": "p50-e40156af8d",
          "page": 50,
          "quote": "[17] B. Velichkov, Regularity of the One-phase Free Boundaries, Lecture Notes of the Unione\nMatematica Italiana, vol. 28, Springer, Cham, 2023. doi:10.1007/978-3-031-13238-4."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-165",
      "kind": "external",
      "label": "[13] Proposition 2.1",
      "source": {
        "blockId": "p15-5286d7c1ec",
        "page": 15,
        "quote": "The angular constant comes from the sharp radial Hardy threshold, which also enters the cone\ninstability criteria in [13, Propositions 2.1–2.2]. We include the radial calculation to specify the\ncompact supports and the order of limits."
      },
      "statement": [
        {
          "blockId": "p15-5286d7c1ec",
          "page": 15,
          "quote": "The angular constant comes from the sharp radial Hardy threshold, which also enters the cone\ninstability criteria in [13, Propositions 2.1–2.2]. We include the radial calculation to specify the\ncompact supports and the order of limits."
        },
        {
          "blockId": "p49-cb9f1d7d25",
          "page": 49,
          "quote": "[13] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary\nproblem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. doi:10.1007/s00039-015-0335-6.\nAuthor manuscript: arXiv:1410.7463v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-166",
      "kind": "external",
      "label": "[2] Lemma 5.5",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-167",
      "kind": "external",
      "label": "[2] Lemma 7.2",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-168",
      "kind": "external",
      "label": "[2] Proposition 5.2",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-169",
      "kind": "external",
      "label": "[2] Proposition 5.4",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-170",
      "kind": "external",
      "label": "[2] Corollary 3.6",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-171",
      "kind": "external",
      "label": "[2] Lemma 7.1",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-172",
      "kind": "external",
      "label": "[2] Proposition 3.4",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-173",
      "kind": "external",
      "label": "[2] Theorem 7.3",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "external-174",
      "kind": "external",
      "label": "[2] Theorem 1.1",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "statement": [
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p32-07195eea74",
          "page": 32,
          "quote": "wo(y)"
        },
        {
          "blockId": "p32-730e6dfa46",
          "page": 32,
          "quote": "wo(y) ≥2/j,\nsin t ≥2/j"
        },
        {
          "blockId": "p32-0b4390edd5",
          "page": 32,
          "quote": "ξj = 1"
        },
        {
          "blockId": "p32-3ff5f03ffc",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-e03e8d8194",
          "page": 32,
          "quote": "ξj = 0"
        },
        {
          "blockId": "p32-28174604ae",
          "page": 32,
          "quote": "2/j"
        },
        {
          "blockId": "p32-99935fd72e",
          "page": 32,
          "quote": "1/j"
        },
        {
          "blockId": "p32-fdddc3045f",
          "page": 32,
          "quote": "ξj = 0 near the Hessian zero set"
        },
        {
          "blockId": "p32-e1ae1c6ad4",
          "page": 32,
          "quote": "t"
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-c96ebe9994",
          "page": 32,
          "quote": "p+\np−\nt = arcsin(1/j)"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        },
        {
          "blockId": "p32-8503f3e19b",
          "page": 32,
          "quote": "For integers j ≥3 define"
        },
        {
          "blockId": "p32-4009d0672d",
          "page": 32,
          "quote": "αj(y) = ϑ(jwo(y)),\nξj ="
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p32-8dd37cc8e8",
          "page": 32,
          "quote": "Here αj is defined to be zero at Zo, consistently with the displayed formula. It is smooth on Ωo:\nnear any zero of wo it is identically zero, and all its transitions occur where wo is smooth and\npositive. In the suspension, βj vanishes on whole neighborhoods of both tips.\nFigure 1 displays the two exclusions made by the product cutoff in the suspended case. Its\ntransition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate\nabove will control the errors produced in all these bands at once.\nOn Ω+ = {x ∈Ω: w(x) > 0}, use the fields"
        },
        {
          "blockId": "p32-93b0b48beb",
          "page": 32,
          "quote": "h = wap"
        },
        {
          "blockId": "p32-932964097f",
          "page": 32,
          "quote": "f(A, p, g),\nZl = w2a(GlijkXijk + wMl + Pl)"
        },
        {
          "blockId": "p32-c5f33f9c07",
          "page": 32,
          "quote": "constructed in Section 5. By Lemma 5.5, there is a fixed CF ≥1 such that"
        },
        {
          "blockId": "p32-864f7ae2ea",
          "page": 32,
          "quote": "|h| ≤CF wa,\n|∇h| ≤CF wa(1 + w + |X|),\n|Z| ≤CF w2a(1 + w + |X|).\n(7.5)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-0c22fefac6",
          "page": 32,
          "quote": "Lemma 7.2 (Integrated comparison). For each fixed j ≥3,"
        },
        {
          "blockId": "p32-3431e6b64d",
          "page": 32,
          "quote": "Z"
        },
        {
          "blockId": "p32-c0686319f0",
          "page": 32,
          "quote": "ϵ∗"
        },
        {
          "blockId": "p32-013d263a9a",
          "page": 32,
          "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p32-4f4834336d",
          "page": 32,
          "quote": "Ω"
        },
        {
          "blockId": "p33-255f54accb",
          "page": 33,
          "quote": "Proof. The interior comparison of Proposition 5.2 gives\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-8d6e2d16b2",
          "page": 33,
          "quote": "Apply the divergence theorem to the smooth field Fj. In the suspension one may first truncate the\ndomain inside the two neighborhoods where Fj vanishes; the artificial end boundaries have zero\nflux. On the regular side boundary the outward normal is −N. Thus the boundary comparison of\nProposition 5.4, N · Z + Hh2 ≥0, gives\nZ"
        },
        {
          "blockId": "p33-4905da558c",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-8098dfa529",
          "page": 33,
          "quote": "≤\nZ"
        },
        {
          "blockId": "p33-3d41bacdac",
          "page": 33,
          "quote": "∂Ω\nH(ξjh)2 dσ −2\nZ"
        },
        {
          "blockId": "p33-026f65d741",
          "page": 33,
          "quote": "Ω\nξj⟨Z, ∇ξj⟩dV."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p33-b8b38bbf15",
          "page": 33,
          "quote": "Ω"
        },
        {
          "blockId": "p33-fe2147dc4f",
          "page": 33,
          "quote": "Expanding the square and canceling the common terms gives (7.6)."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        },
        {
          "blockId": "p33-637c846acc",
          "page": 33,
          "quote": "7.3\nVanishing of all three errors"
        },
        {
          "blockId": "p33-42ee5abb0a",
          "page": 33,
          "quote": "On Ωo,+, differentiation of αj and |∇wo| ≤|Bo| give"
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-4816d80e83",
          "page": 33,
          "quote": "wo\n.\n(7.7)"
        },
        {
          "blockId": "p33-8e57f38846",
          "page": 33,
          "quote": "To see the constant, wherever ϑ′(jwo) ̸= 0 one has 1 < jwo < 2, so j ≤2/wo; outside this transition\nregion the derivative is zero. The same argument gives"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-3139940803",
          "page": 33,
          "quote": "Equations (7.5) and the preceding bounds give"
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-ff9b62603d",
          "page": 34,
          "quote": "w(t, y) = wo(y)"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-36dc29e37c",
          "page": 34,
          "quote": "s4\nt\n,\ndV = s4\nt dt dVo.\n(7.10)"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-8683bf8b46",
          "page": 34,
          "quote": "√"
        },
        {
          "blockId": "p34-62963b32fc",
          "page": 34,
          "quote": "p"
        },
        {
          "blockId": "p34-ac6855f130",
          "page": 34,
          "quote": "b2o + 3 cos2 t"
        },
        {
          "blockId": "p34-05e199087f",
          "page": 34,
          "quote": "|X| ="
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a158403f8",
          "page": 34,
          "quote": "The warped metric gives the exact cutoff identity"
        },
        {
          "blockId": "p34-1fe44567a1",
          "page": 34,
          "quote": "|∇ξj|2 = α2\nj|β′\nj|2 + s−2\nt β2\nj |∇oαj|2,"
        },
        {
          "blockId": "p34-ef9eec1ebe",
          "page": 34,
          "quote": "so (7.7)–(7.8) imply"
        },
        {
          "blockId": "p34-2b8f2b4a03",
          "page": 34,
          "quote": "3.\n(7.11)"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-69bf8ca4be",
          "page": 34,
          "quote": "Multiplying the bounds for the three errors by the volume factor in (7.10) now gives"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        },
        {
          "blockId": "p34-2c17d7d3df",
          "page": 34,
          "quote": "s2−2a\nt\nDo(y) ≤2s21/25\nt\n\nw29/25\no\n+ w−21/25\no\n|Bo|2\non Ωo,+."
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p34-1efddb603e",
          "page": 34,
          "quote": "Theorem 7.3. The first dimension admitting a nonflat one-homogeneous global minimizer cannot\nbe 5 or 6."
        },
        {
          "blockId": "p34-a04230ec33",
          "page": 34,
          "quote": "Proof. If the first dimension were 5 or 6, the reductions would provide exactly one of the two\nworking domains just considered. In either case, the right side of (7.6) tends to zero. On the original\ncomponent, however,"
        },
        {
          "blockId": "p34-22ad402732",
          "page": 34,
          "quote": "0 < Io :=\nZ"
        },
        {
          "blockId": "p34-43b59a7ccb",
          "page": 34,
          "quote": "Ωo\nw2a\no dVo < ∞."
        },
        {
          "blockId": "p34-81fae47b83",
          "page": 34,
          "quote": "Finiteness follows from boundedness and compactness. Positivity follows from To ̸≡0 and continuity,\nwhich give an open subset of Ωo on which wo > 0. In the nonsuspended case dominated convergence\ngives"
        },
        {
          "blockId": "p34-5217692d2c",
          "page": 34,
          "quote": "Z"
        },
        {
          "blockId": "p34-6099df37b9",
          "page": 34,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-32366896b4",
          "page": 35,
          "quote": "In the suspended case (7.10) gives instead"
        },
        {
          "blockId": "p35-3d16646532",
          "page": 35,
          "quote": "Z"
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-870d7c6300",
          "page": 35,
          "quote": "lim\nj→∞"
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        },
        {
          "blockId": "p35-92873001c2",
          "page": 35,
          "quote": "0\n(sin t)4−2a dt = Io"
        },
        {
          "blockId": "p35-1b0ee7de3a",
          "page": 35,
          "quote": "0\n(sin t)71/25 dt > 0."
        },
        {
          "blockId": "p35-f13a60153b",
          "page": 35,
          "quote": "Here the dominating mass density is the integrable, everywhere defined function (sin t)71/25w29/25\no\n;\nits value is zero wherever wo = 0. These strictly positive limits contradict (7.6), since ϵ∗> 0."
        },
        {
          "blockId": "p35-25b9a7cd3d",
          "page": 35,
          "quote": "The known lower-dimensional classification gives d∗≥5, and the dimension-seven globally\nminimizing example gives d∗≤7, as stated in Section 1. Theorem 7.3 excludes the only two\nremaining possibilities below 7. Hence d∗= 7, completing the proof of Theorem 1.1."
        },
        {
          "blockId": "p35-2871bb21d6",
          "page": 35,
          "quote": "8\nRegularity and sharpness of the dimension bound"
        },
        {
          "blockId": "p35-07f85177a9",
          "page": 35,
          "quote": "We now pass from the classification of homogeneous global minimizers to local free boundaries. This\nstep uses the dimension-reduction theorem of Weiss [18], in the formulation [17, Theorem 1.4 and\nProposition 10.13]."
        },
        {
          "blockId": "p49-75b31f99b3",
          "page": 49,
          "quote": "[2] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lips-\nchitz free boundaries are C1,α, Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. doi:10.4171/RMI/47."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited across source blocks; external statement and proof not retrieved."
    },
    {
      "id": "curated-175",
      "kind": "definition",
      "label": "One-phase Bernoulli energy",
      "source": {
        "blockId": "p1-15b1e52200",
        "page": 1,
        "quote": "For a nonnegative function v ∈H1(B), where B ⊂Rd is a ball, the one-phase Bernoulli energy is"
      },
      "statement": [
        {
          "blockId": "p1-15b1e52200",
          "page": 1,
          "quote": "For a nonnegative function v ∈H1(B), where B ⊂Rd is a ball, the one-phase Bernoulli energy is"
        },
        {
          "blockId": "p1-add8165034",
          "page": 1,
          "quote": "J(v; B) =\nZ"
        },
        {
          "blockId": "p1-7458d4594f",
          "page": 1,
          "quote": "\n|∇v|2 + 1{v>0}\n\ndx.\n(1.1)"
        },
        {
          "blockId": "p1-ca9ee21acf",
          "page": 1,
          "quote": "B"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "curated-176",
      "kind": "definition",
      "label": "Global minimizers, homogeneity and flat solutions",
      "source": {
        "blockId": "p1-1ef6120946",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
      },
      "statement": [
        {
          "blockId": "p1-1ef6120946",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
        },
        {
          "blockId": "p1-ba617d637b",
          "page": 1,
          "quote": "u(x) = (x · e)+,\n|e| = 1."
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "curated-177",
      "kind": "definition",
      "label": "Critical dimension",
      "source": {
        "blockId": "p1-93c4ece4cf",
        "page": 1,
        "quote": "Let d∗be the least dimension containing a nonzero, nonflat, one-homogeneous global minimizer."
      },
      "statement": [
        {
          "blockId": "p1-93c4ece4cf",
          "page": 1,
          "quote": "Let d∗be the least dimension containing a nonzero, nonflat, one-homogeneous global minimizer."
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "curated-178",
      "kind": "definition",
      "label": "Local minimizers, blowups and singular points",
      "source": {
        "blockId": "p1-1398459c75",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
      },
      "statement": [
        {
          "blockId": "p1-1398459c75",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
        }
      ],
      "proof": [],
      "guide": []
    }
  ],
  "edges": [
    {
      "id": "edge-1",
      "from": "equation-47",
      "to": "proposition-3",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p4-656ba12637",
        "page": 4,
        "quote": "rj\n−→U(y)\n(2.1)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-2",
      "from": "equation-48",
      "to": "proposition-3",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p4-e807330307",
        "page": 4,
        "quote": "u = 0,\n∇u = NE\non the free boundary,\n(2.2)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-3",
      "from": "equation-49",
      "to": "lemma-4",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p5-16a211c14c",
        "page": 5,
        "quote": "r\n= ur′(y).\n(2.3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-4",
      "from": "equation-50",
      "to": "lemma-5",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p7-7f67ab3c19",
        "page": 7,
        "quote": "d = d∗∈{5, 6},\n(2.4)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-5",
      "from": "equation-51",
      "to": "proposition-6",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p7-616c575fdc",
        "page": 7,
        "quote": "∆Smogo = −mogo\nin Ωo,\ngo = 0,\n∇Smogo = No\non ∂Ωo,\n(2.5)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-6",
      "from": "equation-52",
      "to": "proposition-6",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p7-5c054ff718",
        "page": 7,
        "quote": "∆Rdu(ry) = r−1\n∆Smogo(y) + mogo(y)\n\n(2.6)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-7",
      "from": "equation-53",
      "to": "lemma-7",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p7-425de8ddc1",
        "page": 7,
        "quote": "Lemma 2.5 (Selection of a nonflat component). There is a component Ωo in Proposition 2.4 such\nthat\nTo := ∇2\nSmogo + go Id ̸≡0\non Ωo.\n(2.7)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-8",
      "from": "equation-54",
      "to": "lemma-7",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p8-81876620ea",
        "page": 8,
        "quote": "D2u(ry)[y, ·] = 0,\nD2u(ry)[v1, v2] = r−1To(y)[v1, v2]\n(v1, v2 ⊥y).\n(2.8)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-9",
      "from": "equation-55",
      "to": "proposition-8",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p8-7722e556c0",
        "page": 8,
        "quote": "eΩ=\n\nΦ(t, y) := (sin t y, cos t) : 0 < t < π, y ∈Ωo\n\t\n,\neg(Φ(t, y)) = sin t go(y).\n(2.9)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-10",
      "from": "equation-58",
      "to": "proposition-9",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p10-ead43e8dd0",
        "page": 10,
        "quote": "∇lBijk −∇iBljk = δljTik + δlkTij −δijTlk −δikTlj,\n(3.3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-11",
      "from": "equation-59",
      "to": "proposition-9",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p10-4f905adc86",
        "page": 10,
        "quote": "∆T = mT,\n∆w2 = 2mw2 + 2|B|2,\n(3.4)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-12",
      "from": "equation-60",
      "to": "proposition-9",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p10-671ef66fb9",
        "page": 10,
        "quote": "∇lpi = Tli −gδli,\n∇lg = pl,\n(3.5)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-13",
      "from": "equation-61",
      "to": "proposition-9",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p10-9cd42a53f7",
        "page": 10,
        "quote": "∇s2 = 2Tp,\n∆s2 = 2w2.\n(3.6)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-14",
      "from": "equation-62",
      "to": "proposition-9",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p10-39b3f272e5",
        "page": 10,
        "quote": "∇lw = w−1TjkBljk,\n|∇w| ≤|B|.\n(3.7)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-15",
      "from": "equation-63",
      "to": "lemma-10",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p11-79aa63b2b7",
        "page": 11,
        "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-16",
      "from": "equation-64",
      "to": "lemma-10",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p11-901a03e7df",
        "page": 11,
        "quote": "Then\nHo > 0,\nTopo = −Hopo,\nwo > 0\non ∂Ωo.\n(3.9)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-17",
      "from": "equation-65",
      "to": "proposition-11",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p11-064ff4fb38",
        "page": 11,
        "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-18",
      "from": "equation-66",
      "to": "proposition-11",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p11-d8ff12d046",
        "page": 11,
        "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-19",
      "from": "equation-67",
      "to": "proposition-11",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p11-ee96e4a03b",
        "page": 11,
        "quote": "Vijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(3.12)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-20",
      "from": "equation-68",
      "to": "proposition-11",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p11-f6ddc57c80",
        "page": 11,
        "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-21",
      "from": "equation-69",
      "to": "proposition-12",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p12-b9af30c473",
        "page": 12,
        "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-22",
      "from": "equation-70",
      "to": "proposition-12",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p12-b9af30c473",
        "page": 12,
        "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-23",
      "from": "equation-71",
      "to": "proposition-12",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p12-fe39066800",
        "page": 12,
        "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-24",
      "from": "equation-72",
      "to": "proposition-12",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p12-8b2672b64e",
        "page": 12,
        "quote": "ew = ρ−1wo,\n| eB|2 = ρ−4\n|Bo|2 + 3c2w2\no\n\n,\neH = ρ−1Ho\non the regular side.\n(3.17)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-25",
      "from": "equation-73",
      "to": "proposition-12",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p12-8a8302c5ce",
        "page": 12,
        "quote": "dV = ρ4 dt dVo,\ndσ = ρ3 dt dσo.\n(3.18)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-26",
      "from": "equation-74",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p13-39ac08043a",
        "page": 13,
        "quote": "∂D\nHEϕ2 dσE,\nHE = −D2u(NE, NE),\n(3.19)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-27",
      "from": "equation-75",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p14-563124ccb1",
        "page": 14,
        "quote": "D\n|∇ϕ|2 dx.\n(3.20)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-28",
      "from": "equation-76",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p14-9f21210952",
        "page": 14,
        "quote": "j(y, 0) = 1,\n∂ξj(y, 0) = div VF (y, 0) = ∆F −2FNENE = 2HE.\n(3.21)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-29",
      "from": "equation-77",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p14-7772931587",
        "page": 14,
        "quote": "and hence\n∂ξ\n\n(1 + |∇F|2)j\n\nξ=0 = 2HE.\n(3.22)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-30",
      "from": "equation-78",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p14-85993fe6a5",
        "page": 14,
        "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-31",
      "from": "equation-79",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p14-0461573878",
        "page": 14,
        "quote": "(1 + |∇(F + εϕ)|2)j = 2 + 2εϕN + 2HEξ + O(ε2 + |εξ| + ξ2).\n(3.24)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-32",
      "from": "equation-80",
      "to": "proposition-13",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p14-b2fb2c5ad6",
        "page": 14,
        "quote": "= 2εϕ0 −ε2HEϕ2\n0 + O(|ε|3).\n(3.25)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-33",
      "from": "equation-81",
      "to": "corollary-14",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p15-9bcb9d1c60",
        "page": 15,
        "quote": "\n|∇v|2 + 4v2\ndV.\n(3.26)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-34",
      "from": "equation-82",
      "to": "corollary-14",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p15-51cc79bd7d",
        "page": 15,
        "quote": "R ψ2 dτ .\n(3.27)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-35",
      "from": "equation-83",
      "to": "corollary-14",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p16-e7c1bb9818",
        "page": 16,
        "quote": "31250000000.\n(4.1)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-36",
      "from": "equation-84",
      "to": "corollary-14",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p16-89211ce53b",
        "page": 16,
        "quote": "DlAij = Xlij −qlAij,\nDlpi = wAli −gzδli,\nDlg = zpl,\nDlδij = 0.\n(4.2)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-37",
      "from": "equation-85",
      "to": "proposition-15",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p16-7cb9687645",
        "page": 16,
        "quote": "+ f|ℓ|2 −(Dlf + 2aqlf)ℓl ≥ϵ∗(z2 + |ℓ|2).\n(4.3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-38",
      "from": "equation-86",
      "to": "proposition-15",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p16-12ac49d162",
        "page": 16,
        "quote": "Then\nS := −λf + Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl ≥δ∗.\n(4.4)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-39",
      "from": "equation-89",
      "to": "lemma-16",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p17-ea70133cef",
        "page": 17,
        "quote": "∇lw = wql,\n∇lAij = Xlij −qlAij,\n∇lpi = wAli −gδli,\n∇lg = pl.\n(5.3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-40",
      "from": "equation-90",
      "to": "lemma-16",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p17-4e99f8fde8",
        "page": 17,
        "quote": "Furthermore,\n1\nwGlijk∇lBijk = 5UjkAjk + Clijk(δljAik + δlkAij).\n(5.4)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-41",
      "from": "equation-93",
      "to": "proposition-17",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p18-9950cf5a0f",
        "page": 18,
        "quote": "div Z ≥4h2 + |∇h|2 + ϵ∗w2a\non Ω+.\n(5.7)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-42",
      "from": "equation-94",
      "to": "proposition-17",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p18-1a4e499ddb",
        "page": 18,
        "quote": "+\n\nDlGlijk + (2a −1)qlGlijk\n\nXijk\n+ w\n\nDlMl + (2a + 1)qlMl\n\n+ DlPl + 2aqlPl.\n(5.8)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-43",
      "from": "equation-95",
      "to": "proposition-17",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p19-8023154c00",
        "page": 19,
        "quote": "4f |F|2,\n(5.9)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-44",
      "from": "equation-96",
      "to": "proposition-17",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p19-7492df17f5",
        "page": 19,
        "quote": "I\n\nz=1 = w−2a\ndiv Z −4h2 −|∇h|2\n+ f|ℓ−ℓ∗|2.\n(5.10)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-45",
      "from": "equation-98",
      "to": "lemma-18",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p19-7966fef82a",
        "page": 19,
        "quote": "w2 = plGlijkVijk = Ujk(S0 + p ⊗p)jk + ClijkplVijk.\n(5.12)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-46",
      "from": "equation-99",
      "to": "proposition-19",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p20-7818ef941d",
        "page": 20,
        "quote": "31250000000,\n(5.13)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-47",
      "from": "equation-100",
      "to": "lemma-20",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p20-451469844c",
        "page": 20,
        "quote": "|h| ≤Cwa,\n|∇h| ≤Cwa(1 + w + |X|),\n|Z| ≤Cw2a(1 + w + |X|).\n(5.14)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-48",
      "from": "equation-109",
      "to": "lemma-23",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p25-26685d701f",
        "page": 25,
        "quote": "∥A∥2\nop ≤θ,\n|Ak|2 ≤θk−1\n(k ≥1),\n(6.9)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-49",
      "from": "equation-110",
      "to": "lemma-23",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p25-34df8deab7",
        "page": 25,
        "quote": "20,\n(6.10)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-50",
      "from": "equation-111",
      "to": "lemma-23",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p25-3033ab8bae",
        "page": 25,
        "quote": "7|X|2,\n(X : A)i = XijkAjk.\n(6.11)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-51",
      "from": "equation-112",
      "to": "lemma-23",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p25-6744ff863c",
        "page": 25,
        "quote": "b\n.\n(6.12)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-52",
      "from": "equation-113",
      "to": "lemma-24",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p26-51839e719e",
        "page": 26,
        "quote": "|M(t)| ≤Ltyiyj\n(6.13)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-53",
      "from": "equation-127",
      "to": "lemma-25",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p30-6b3ec9e1fd",
        "page": 30,
        "quote": "Ωo,+\nw2a−2\no\n|Bo|2 dVo < ∞.\n(7.1)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-54",
      "from": "equation-128",
      "to": "lemma-25",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p31-c40865070c",
        "page": 31,
        "quote": "(\nwo(y)2a\n1 + |Bo(y)|/wo(y)\n2,\ny ∈Ωo,+,\n0,\ny ∈Zo,\n(7.2)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-55",
      "from": "equation-129",
      "to": "lemma-25",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p31-a5342e9d9d",
        "page": 31,
        "quote": "a(M2\no + 1)a∥∆η∥L1(Ωo).\n(7.3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-56",
      "from": "equation-132",
      "to": "lemma-26",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p32-013d263a9a",
        "page": 32,
        "quote": "\nh2|∇ξj|2 + 2ξjh⟨∇h, ∇ξj⟩−2ξj⟨Z, ∇ξj⟩\n\ndV.\n(7.6)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-57",
      "from": "external-28",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-58",
      "from": "external-139",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-59",
      "from": "external-29",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-60",
      "from": "external-140",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-61",
      "from": "external-44",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-62",
      "from": "external-141",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-63",
      "from": "external-142",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-64",
      "from": "external-143",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-65",
      "from": "external-40",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-66",
      "from": "external-30",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-67",
      "from": "external-144",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-68",
      "from": "external-145",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-69",
      "from": "external-146",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-70",
      "from": "external-147",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-71",
      "from": "external-148",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-72",
      "from": "external-149",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-73",
      "from": "external-150",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-74",
      "from": "external-43",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-e66e934670",
        "page": 2,
        "quote": "The obstacle and the proof strategy.\nThe stability inequality compares the Dirichlet energy\nof a test function with a curvature term on the free boundary. The use of the harmonic Hessian\nhas a geometric precedent in Simons’s second-fundamental-form argument for minimal cones [16,\nSection 6.1], a connection developed explicitly by Jerison and Savin. A natural Bernoulli test is\na power of the Hessian norm. Jerison and Savin explain why this unweighted choice does not\ngive a uniform instability criterion in dimensions four through six: the interior and boundary\nestimates impose different restrictions on the power, and their sharp configurations need not agree\n[13, Section 3.2]. Their four-dimensional argument uses a more refined function of the Hessian"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-75",
      "from": "external-40",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-e66e934670",
        "page": 2,
        "quote": "The obstacle and the proof strategy.\nThe stability inequality compares the Dirichlet energy\nof a test function with a curvature term on the free boundary. The use of the harmonic Hessian\nhas a geometric precedent in Simons’s second-fundamental-form argument for minimal cones [16,\nSection 6.1], a connection developed explicitly by Jerison and Savin. A natural Bernoulli test is\na power of the Hessian norm. Jerison and Savin explain why this unweighted choice does not\ngive a uniform instability criterion in dimensions four through six: the interior and boundary\nestimates impose different restrictions on the power, and their sharp configurations need not agree\n[13, Section 3.2]. Their four-dimensional argument uses a more refined function of the Hessian"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-76",
      "from": "external-40",
      "to": "corollary-2",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p3-0c9f90e5b9",
        "page": 3,
        "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-77",
      "from": "external-151",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-78",
      "from": "external-152",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-79",
      "from": "external-153",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-80",
      "from": "external-154",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-81",
      "from": "external-44",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p4-32abdf50fa",
        "page": 4,
        "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-82",
      "from": "external-155",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-83",
      "from": "external-156",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-84",
      "from": "external-157",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-85",
      "from": "external-158",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-86",
      "from": "external-159",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: The Sources and applicability discussion explicitly applies the constant-coefficient theory of [17]; the reference names a result in that source. External source not retrieved or verified."
    },
    {
      "id": "edge-87",
      "from": "external-142",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: This paragraph applies the preceding cited [6, Theorem 1.1 and Corollary 1.2] to obtain higher regularity. External source not retrieved or verified."
    },
    {
      "id": "edge-88",
      "from": "external-143",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "Reviewed external reference: This paragraph applies the preceding cited [6, Theorem 1.1 and Corollary 1.2] to obtain higher regularity. External source not retrieved or verified."
    },
    {
      "id": "edge-89",
      "from": "external-160",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-90",
      "from": "external-161",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-91",
      "from": "external-162",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-92",
      "from": "external-163",
      "to": "proposition-3",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-93",
      "from": "theorem-1",
      "to": "lemma-5",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-d328f4a8f3",
        "page": 7,
        "quote": "same trace and energy b2R2/(R −t) + (R −t). Its derivative at the minimizing parameter t = 0 is\nb2 −1 = 0, so b = 1. This proves flatness in dimension one directly.\nThe lower-dimensional classification of Jerison–Savin [13, Theorem 1.1 and Corollary 1.2] and\nthe dimension-seven minimizing example of De Silva–Jerison [7, Theorem 1.1] give 5 ≤d∗≤7. De\nSilva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors.\nFor the exclusion argument, suppose from now on that"
      },
      "note": ""
    },
    {
      "id": "edge-94",
      "from": "external-144",
      "to": "lemma-5",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-d328f4a8f3",
        "page": 7,
        "quote": "same trace and energy b2R2/(R −t) + (R −t). Its derivative at the minimizing parameter t = 0 is\nb2 −1 = 0, so b = 1. This proves flatness in dimension one directly.\nThe lower-dimensional classification of Jerison–Savin [13, Theorem 1.1 and Corollary 1.2] and\nthe dimension-seven minimizing example of De Silva–Jerison [7, Theorem 1.1] give 5 ≤d∗≤7. De\nSilva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors.\nFor the exclusion argument, suppose from now on that"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-95",
      "from": "external-145",
      "to": "lemma-5",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-d328f4a8f3",
        "page": 7,
        "quote": "same trace and energy b2R2/(R −t) + (R −t). Its derivative at the minimizing parameter t = 0 is\nb2 −1 = 0, so b = 1. This proves flatness in dimension one directly.\nThe lower-dimensional classification of Jerison–Savin [13, Theorem 1.1 and Corollary 1.2] and\nthe dimension-seven minimizing example of De Silva–Jerison [7, Theorem 1.1] give 5 ≤d∗≤7. De\nSilva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors.\nFor the exclusion argument, suppose from now on that"
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-96",
      "from": "proposition-3",
      "to": "proposition-6",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-df0b192be6",
        "page": 7,
        "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
      },
      "note": ""
    },
    {
      "id": "edge-97",
      "from": "lemma-4",
      "to": "proposition-6",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-df0b192be6",
        "page": 7,
        "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
      },
      "note": ""
    },
    {
      "id": "edge-98",
      "from": "lemma-5",
      "to": "proposition-6",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-df0b192be6",
        "page": 7,
        "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
      },
      "note": ""
    },
    {
      "id": "edge-99",
      "from": "equation-52",
      "to": "proposition-6",
      "relation": "equation-reference",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-7608cd04c9",
        "page": 7,
        "quote": "in the positive phase. If go > 0 on the entire sphere, integrating the harmonicity identity over the\nsphere would give mo\nR\ngo = 0, a contradiction. Thus its positive set is proper.\nThe conical free boundary has the radial direction tangent to it at every nonzero point: the\ncurve r 7→rx lies in that free boundary. Consequently its smooth hypersurface is transverse to the\nunit sphere, and their intersection is a smooth hypersurface of Smo. Its regular charts have the\nspherical positive phase exactly on one side. Each point of the sphere therefore has a neighborhood\nwhose intersection with {go > 0} is either connected or empty (using a small ball for points away\nfrom the free boundary). A finite subcover of the sphere shows that there are only finitely many\npositive components. If two component closures met, their common point would be a free-boundary\npoint; the connected positive side of a regular chart there would meet both components, forcing\nthem to coincide. The same charts show that each closure is a smooth compact domain. Each\ncomponent has nonempty boundary because the sphere is connected and the positive set is proper.\nSmoothness of u from its positive side gives smoothness of go on these closures. Equation (2.6)\ngives the interior equation in (2.5). On the boundary, the radial derivative of u is go = 0, so the\nEuclidean gradient is tangent to the sphere. Equation (2.2) then gives the remaining boundary\nconditions."
      },
      "note": ""
    },
    {
      "id": "edge-100",
      "from": "equation-48",
      "to": "proposition-6",
      "relation": "equation-reference",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-7608cd04c9",
        "page": 7,
        "quote": "in the positive phase. If go > 0 on the entire sphere, integrating the harmonicity identity over the\nsphere would give mo\nR\ngo = 0, a contradiction. Thus its positive set is proper.\nThe conical free boundary has the radial direction tangent to it at every nonzero point: the\ncurve r 7→rx lies in that free boundary. Consequently its smooth hypersurface is transverse to the\nunit sphere, and their intersection is a smooth hypersurface of Smo. Its regular charts have the\nspherical positive phase exactly on one side. Each point of the sphere therefore has a neighborhood\nwhose intersection with {go > 0} is either connected or empty (using a small ball for points away\nfrom the free boundary). A finite subcover of the sphere shows that there are only finitely many\npositive components. If two component closures met, their common point would be a free-boundary\npoint; the connected positive side of a regular chart there would meet both components, forcing\nthem to coincide. The same charts show that each closure is a smooth compact domain. Each\ncomponent has nonempty boundary because the sphere is connected and the positive set is proper.\nSmoothness of u from its positive side gives smoothness of go on these closures. Equation (2.6)\ngives the interior equation in (2.5). On the boundary, the radial derivative of u is go = 0, so the\nEuclidean gradient is tangent to the sphere. Equation (2.2) then gives the remaining boundary\nconditions."
      },
      "note": ""
    },
    {
      "id": "edge-101",
      "from": "proposition-6",
      "to": "lemma-7",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-425de8ddc1",
        "page": 7,
        "quote": "Lemma 2.5 (Selection of a nonflat component). There is a component Ωo in Proposition 2.4 such\nthat\nTo := ∇2\nSmogo + go Id ̸≡0\non Ωo.\n(2.7)"
      },
      "note": ""
    },
    {
      "id": "edge-102",
      "from": "proposition-6",
      "to": "lemma-7",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p8-5eb064b940",
        "page": 8,
        "quote": "Suppose that To vanished identically on every positive component.\nThen D2u = 0 on each\ncorresponding connected open cone Di, so u(x) = bi · x + ci there. Homogeneity forces ci = 0, and\npositivity forces bi ̸= 0. Thus Di ⊂Hi := {bi · x > 0}. The set Di is also closed relative to Hi: if\nx ∈Di ∩Hi, continuity gives u(x) = bi · x > 0; a small positive neighborhood of x meets Di, hence\nbelongs to the same component. Since Hi is connected, Di = Hi. The Bernoulli condition at any\nnonzero point of its boundary gives |bi| = 1.\nTwo disjoint open half-spaces through the origin must have opposite unit normals: if their normals\nare e, f with f ̸= −e, the vector e + f lies in both. But opposite half-spaces have spherical closures\nmeeting along the equator e⊥∩Smo, contrary to the pairwise disjoint closures in Proposition 2.4.\nThus there is exactly one positive component, and u = (x · e)+, contrary to nonflatness. This proves\n(2.7)."
      },
      "note": ""
    },
    {
      "id": "edge-103",
      "from": "lemma-7",
      "to": "proposition-8",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p8-05cf6278e3",
        "page": 8,
        "quote": "Proposition 2.6 (Suspension preserves global minimality). If u ≥0 is a global minimizer in Rk,\nthen eu(x, s) = u(x) is a global minimizer in Rk+1. If u is the cone in (2.4) with d = 5, the singular\nfree-boundary set of eu is exactly {0}×R. For the component chosen in Lemma 2.5, the corresponding\npositive component on S5 is"
      },
      "note": ""
    },
    {
      "id": "edge-104",
      "from": "lemma-7",
      "to": "proposition-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-b613bfd01d",
        "page": 9,
        "quote": "Notation for the remaining argument.\nFix the original component Ωo ⊂Smo selected in\nLemma 2.5, where mo ∈{4, 5}. If mo = 5, set Ω= Ωo, g = go, and Σ = ∅. If mo = 4, set Ω= eΩ,\ng = eg, and Σ = {p+, p−}. In either case Ω⊂S5 is connected, and g comes from a nonnegative\none-homogeneous global minimizer in R6. The closure is smooth away from Σ, with positive phase\non one side of its regular boundary\n∂regΩ:= ∂Ω\\ Σ."
      },
      "note": ""
    },
    {
      "id": "edge-105",
      "from": "proposition-9",
      "to": "proposition-12",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-7c1a267d69",
        "page": 12,
        "quote": "Every local identity of Proposition 3.1 holds with m = 5, and Proposition 3.3 holds on the regular\nside. Also es2 < 1 inside and eH, ew > 0 on that side."
      },
      "note": ""
    },
    {
      "id": "edge-106",
      "from": "proposition-11",
      "to": "proposition-12",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-7c1a267d69",
        "page": 12,
        "quote": "Every local identity of Proposition 3.1 holds with m = 5, and Proposition 3.3 holds on the regular\nside. Also es2 < 1 inside and eH, ew > 0 on that side."
      },
      "note": ""
    },
    {
      "id": "edge-107",
      "from": "proposition-9",
      "to": "proposition-12",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-aca19449d5",
        "page": 13,
        "quote": "Adding eg Id proves (3.15). Differentiating that tensor with the displayed connection gives (3.16);\nentries with two or three zero slots vanish. Each term with exactly one zero slot occurs in three\npositions, which proves the factor 3 in (3.17).\nThe inward normal to the side is the horizontal lift of No. Thus the normal-normal entry of\n(3.15) gives eH = ρ−1Ho. The metric on the side is dt2 + ρ2ds2\n∂Ωo, proving (3.18). The trace of (3.15)\nis zero, so ∆eg = −5eg. The interior identities follow locally from Proposition 3.1. The bounds for s2,\nH and w follow from the original compact component and (3.14)–(3.17); no maximum principle at a\ntip is used. The local boundary hypotheses and hence Proposition 3.3 hold on the regular side."
      },
      "note": ""
    },
    {
      "id": "edge-108",
      "from": "proposition-11",
      "to": "proposition-12",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-aca19449d5",
        "page": 13,
        "quote": "Adding eg Id proves (3.15). Differentiating that tensor with the displayed connection gives (3.16);\nentries with two or three zero slots vanish. Each term with exactly one zero slot occurs in three\npositions, which proves the factor 3 in (3.17).\nThe inward normal to the side is the horizontal lift of No. Thus the normal-normal entry of\n(3.15) gives eH = ρ−1Ho. The metric on the side is dt2 + ρ2ds2\n∂Ωo, proving (3.18). The trace of (3.15)\nis zero, so ∆eg = −5eg. The interior identities follow locally from Proposition 3.1. The bounds for s2,\nH and w follow from the original compact component and (3.14)–(3.17); no maximum principle at a\ntip is used. The local boundary hypotheses and hence Proposition 3.3 hold on the regular side."
      },
      "note": ""
    },
    {
      "id": "edge-109",
      "from": "external-165",
      "to": "proposition-13",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-5286d7c1ec",
        "page": 15,
        "quote": "The angular constant comes from the sharp radial Hardy threshold, which also enters the cone\ninstability criteria in [13, Propositions 2.1–2.2]. We include the radial calculation to specify the\ncompact supports and the order of limits."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-110",
      "from": "proposition-13",
      "to": "corollary-14",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p15-aabfeea988",
        "page": 15,
        "quote": "Proof. More generally, consider a link component Ω⊂Sm with angular tests supported in its regular\nclosure, and write D = {ry : r > 0, y ∈Ω}. On its regular side the normal is tangent to Sm, and\n(3.2) gives HE(ry) = r−1H(y). Polar volume has factor rm dr and side area has factor rm−1 dr. For\nζ ∈C∞\nc ((0, ∞)) and an allowed angular test v, put ϕ(ry) = ζ(r)v(y) in Proposition 3.5. It gives\nZ ∞"
      },
      "note": ""
    },
    {
      "id": "edge-111",
      "from": "proposition-9",
      "to": "lemma-16",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p18-17071beaaf",
        "page": 18,
        "quote": "For the skew part, the curvature identity in Proposition 3.1 yields"
      },
      "note": ""
    },
    {
      "id": "edge-112",
      "from": "equation-94",
      "to": "proposition-17",
      "relation": "equation-reference",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p18-842e7465ba",
        "page": 18,
        "quote": "Every Dl in Equation (5.8) is evaluated at z = 1. The three coefficients of q come respectively from\ndifferentiating w2aX, w2a+1, and w2a; in the first expression the quotient X = B/w contributes\n−qlX. Thus their values are exactly"
      },
      "note": ""
    },
    {
      "id": "edge-113",
      "from": "proposition-15",
      "to": "proposition-19",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-cf8d16908c",
        "page": 20,
        "quote": "Proof. Proposition 4.1 gives P = 0 at the boundary, where g = 0. Since X = B/w, Lemma 5.3 gives"
      },
      "note": ""
    },
    {
      "id": "edge-114",
      "from": "lemma-18",
      "to": "proposition-19",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p20-cf8d16908c",
        "page": 20,
        "quote": "Proof. Proposition 4.1 gives P = 0 at the boundary, where g = 0. Since X = B/w, Lemma 5.3 gives"
      },
      "note": ""
    },
    {
      "id": "edge-115",
      "from": "lemma-21",
      "to": "lemma-23",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p26-8d1ed535fc",
        "page": 26,
        "quote": "Expanding the nine terms in |B|2 gives |B|2 = |ζ|2/3+2|Aζ|2/3. The symmetric trace-free projection\nis F = B −J v/7, in the notation of Lemma 6.1. Since |J v|2 = 21|v|2 and ⟨B, J v⟩= 3|v|2,"
      },
      "note": ""
    },
    {
      "id": "edge-116",
      "from": "lemma-22",
      "to": "lemma-24",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p26-e79d4771d9",
        "page": 26,
        "quote": "Lemma 6.4 (Complete graph bound). Every scalar graph retained by Lemma 6.2 satisfies"
      },
      "note": ""
    },
    {
      "id": "edge-117",
      "from": "lemma-23",
      "to": "lemma-24",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p26-93c0366bec",
        "page": 26,
        "quote": "Proof. First separate all scalar trace factors and use Lemma 6.3. A cubic tensor has at most one\nself-edge, since such an edge consumes two of its three incidences. An edge of power k ≥2 reduces\nthat tensor to the vector vi = Xijk(Ak)jk, for which |v| ≤|X| |Ak| ≤|X|θ(k−1)/2. For k = 1 use\n(6.11). These reductions leave at most two cubic vertices and no self-edge on either.\nWe list all possible connected components of the remaining graph. With zero cubic vertices, a\ncomponent is one edge joining two vectors; its contraction is bounded by the operator norm of its\nmatrix times the two vector norms. With one cubic vertex, all its three slots end at vector leaves.\nMove each edge matrix onto its vector, and use tensor Cauchy–Schwarz, obtaining the norm of X\ntimes the three vector norms and the product of the three operator norms.\nIf a component contains two cubic vertices, they are joined by r = 1, 2, or 3 edges; every other\nslot ends at a vector leaf. Contract these leaves first, obtaining rank-r tensors Y, Z with the same\npreceding norm bounds. Up to a permutation of their slots, the remaining contraction is\nD\nY, (Ak1 ⊗· · · ⊗Akr)Z\nE\n."
      },
      "note": ""
    },
    {
      "id": "edge-118",
      "from": "lemma-10",
      "to": "lemma-25",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p31-441dea3d25",
        "page": 31,
        "quote": "Proof. The boundary conclusion of Lemma 3.2 gives wo > 0 at every point of ∂Ωo. Compactness\nand continuity therefore give a boundary neighborhood on which wo has a positive lower bound.\nThe closed set Zo is thus compactly contained in the interior. If it is empty, wo has a positive\nminimum on the whole compact closure, and (7.1) follows immediately from the boundedness of Bo.\nOtherwise choose η ∈C∞\nc (Ωo) with 0 ≤η ≤1, equal to one on an open neighborhood of Zo.\nFor 0 < δ ≤1 put Wδ = w2\no + δ. Although wo need not be smooth at its zeros, Wδ is smooth and\nstrictly positive. The identities in Proposition 3.1 give"
      },
      "note": ""
    },
    {
      "id": "edge-119",
      "from": "proposition-9",
      "to": "lemma-25",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p31-441dea3d25",
        "page": 31,
        "quote": "Proof. The boundary conclusion of Lemma 3.2 gives wo > 0 at every point of ∂Ωo. Compactness\nand continuity therefore give a boundary neighborhood on which wo has a positive lower bound.\nThe closed set Zo is thus compactly contained in the interior. If it is empty, wo has a positive\nminimum on the whole compact closure, and (7.1) follows immediately from the boundedness of Bo.\nOtherwise choose η ∈C∞\nc (Ωo) with 0 ≤η ≤1, equal to one on an open neighborhood of Zo.\nFor 0 < δ ≤1 put Wδ = w2\no + δ. Although wo need not be smooth at its zeros, Wδ is smooth and\nstrictly positive. The identities in Proposition 3.1 give"
      },
      "note": ""
    },
    {
      "id": "edge-120",
      "from": "proposition-3",
      "to": "corollary-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p35-69e0697d40",
        "page": 35,
        "quote": "Proof of Corollary 1.2. Apply the cited dimension-reduction theorem with volume coefficient one\non balls compactly contained in D. Its inputs are uniform flatness regularity, linear nondegeneracy,\ncompactness of blowups, and the fact that all blowup limits are homogeneous global minimizers.\nThese are the constant-coefficient results used in Proposition 2.1; flatness regularity is [17, Theo-\nrem 8.1]. The theorem gives an empty singular set below d∗, local finiteness in dimension d∗, and\nHausdorff dimension at most n −d∗above it. Theorem 1.1 sets d∗= 7. Smoothness at each regular\npoint follows from Proposition 2.1(iii). A countable cover by interior balls gives the assertions on D.\nFor sharpness, let U be the nonflat global minimizer in R7 of De Silva and Jerison [7, Theorem 1.1],\nand set Un(x, y) = U(x) on R7 × Rn−7. Repeated application of Proposition 2.6 shows that Un is\nglobally minimizing. At each point (0, y0), every rescaling of Un is Un itself, by homogeneity and\ntranslation invariance in y. It cannot be flat: a half-space normal would have zero y-component,\nforcing U to be flat. Thus {0} × Rn−7 ⊂Sing(Un). Its dimension is n −7, and the proved upper\nbound gives equality."
      },
      "note": ""
    },
    {
      "id": "edge-121",
      "from": "theorem-1",
      "to": "corollary-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p35-69e0697d40",
        "page": 35,
        "quote": "Proof of Corollary 1.2. Apply the cited dimension-reduction theorem with volume coefficient one\non balls compactly contained in D. Its inputs are uniform flatness regularity, linear nondegeneracy,\ncompactness of blowups, and the fact that all blowup limits are homogeneous global minimizers.\nThese are the constant-coefficient results used in Proposition 2.1; flatness regularity is [17, Theo-\nrem 8.1]. The theorem gives an empty singular set below d∗, local finiteness in dimension d∗, and\nHausdorff dimension at most n −d∗above it. Theorem 1.1 sets d∗= 7. Smoothness at each regular\npoint follows from Proposition 2.1(iii). A countable cover by interior balls gives the assertions on D.\nFor sharpness, let U be the nonflat global minimizer in R7 of De Silva and Jerison [7, Theorem 1.1],\nand set Un(x, y) = U(x) on R7 × Rn−7. Repeated application of Proposition 2.6 shows that Un is\nglobally minimizing. At each point (0, y0), every rescaling of Un is Un itself, by homogeneity and\ntranslation invariance in y. It cannot be flat: a half-space normal would have zero y-component,\nforcing U to be flat. Thus {0} × Rn−7 ⊂Sing(Un). Its dimension is n −7, and the proved upper\nbound gives equality."
      },
      "note": ""
    },
    {
      "id": "edge-122",
      "from": "proposition-8",
      "to": "corollary-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p35-69e0697d40",
        "page": 35,
        "quote": "Proof of Corollary 1.2. Apply the cited dimension-reduction theorem with volume coefficient one\non balls compactly contained in D. Its inputs are uniform flatness regularity, linear nondegeneracy,\ncompactness of blowups, and the fact that all blowup limits are homogeneous global minimizers.\nThese are the constant-coefficient results used in Proposition 2.1; flatness regularity is [17, Theo-\nrem 8.1]. The theorem gives an empty singular set below d∗, local finiteness in dimension d∗, and\nHausdorff dimension at most n −d∗above it. Theorem 1.1 sets d∗= 7. Smoothness at each regular\npoint follows from Proposition 2.1(iii). A countable cover by interior balls gives the assertions on D.\nFor sharpness, let U be the nonflat global minimizer in R7 of De Silva and Jerison [7, Theorem 1.1],\nand set Un(x, y) = U(x) on R7 × Rn−7. Repeated application of Proposition 2.6 shows that Un is\nglobally minimizing. At each point (0, y0), every rescaling of Un is Un itself, by homogeneity and\ntranslation invariance in y. It cannot be flat: a half-space normal would have zero y-component,\nforcing U to be flat. Thus {0} × Rn−7 ⊂Sing(Un). Its dimension is n −7, and the proved upper\nbound gives equality."
      },
      "note": ""
    },
    {
      "id": "edge-123",
      "from": "external-158",
      "to": "corollary-2",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p35-69e0697d40",
        "page": 35,
        "quote": "Proof of Corollary 1.2. Apply the cited dimension-reduction theorem with volume coefficient one\non balls compactly contained in D. Its inputs are uniform flatness regularity, linear nondegeneracy,\ncompactness of blowups, and the fact that all blowup limits are homogeneous global minimizers.\nThese are the constant-coefficient results used in Proposition 2.1; flatness regularity is [17, Theo-\nrem 8.1]. The theorem gives an empty singular set below d∗, local finiteness in dimension d∗, and\nHausdorff dimension at most n −d∗above it. Theorem 1.1 sets d∗= 7. Smoothness at each regular\npoint follows from Proposition 2.1(iii). A countable cover by interior balls gives the assertions on D.\nFor sharpness, let U be the nonflat global minimizer in R7 of De Silva and Jerison [7, Theorem 1.1],\nand set Un(x, y) = U(x) on R7 × Rn−7. Repeated application of Proposition 2.6 shows that Un is\nglobally minimizing. At each point (0, y0), every rescaling of Un is Un itself, by homogeneity and\ntranslation invariance in y. It cannot be flat: a half-space normal would have zero y-component,\nforcing U to be flat. Thus {0} × Rn−7 ⊂Sing(Un). Its dimension is n −7, and the proved upper\nbound gives equality."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-124",
      "from": "curated-175",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-15b1e52200",
        "page": 1,
        "quote": "For a nonnegative function v ∈H1(B), where B ⊂Rd is a ball, the one-phase Bernoulli energy is"
      },
      "note": "Agent-proposed background relation from the cited definition; not proof verification."
    },
    {
      "id": "edge-125",
      "from": "curated-176",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-1ef6120946",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
      },
      "note": "Agent-proposed background relation from the cited definition; not proof verification."
    },
    {
      "id": "edge-126",
      "from": "curated-177",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-93c4ece4cf",
        "page": 1,
        "quote": "Let d∗be the least dimension containing a nonzero, nonflat, one-homogeneous global minimizer."
      },
      "note": "Agent-proposed background relation from the cited definition; not proof verification."
    },
    {
      "id": "edge-127",
      "from": "curated-178",
      "to": "corollary-2",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-1398459c75",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
      },
      "note": "Agent-proposed background relation from the cited definition; not proof verification."
    }
  ],
  "notation": [
    {
      "symbol": "A",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p1-1398459c75",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
      },
      "occurrences": [
        {
          "blockId": "p1-1ef6120946",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
        },
        {
          "blockId": "p1-1398459c75",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
        },
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p2-e66e934670",
          "page": 2,
          "quote": "The obstacle and the proof strategy.\nThe stability inequality compares the Dirichlet energy\nof a test function with a curvature term on the free boundary. The use of the harmonic Hessian\nhas a geometric precedent in Simons’s second-fundamental-form argument for minimal cones [16,\nSection 6.1], a connection developed explicitly by Jerison and Savin. A natural Bernoulli test is\na power of the Hessian norm. Jerison and Savin explain why this unweighted choice does not\ngive a uniform instability criterion in dimensions four through six: the interior and boundary\nestimates impose different restrictions on the power, and their sharp configurations need not agree\n[13, Section 3.2]. Their four-dimensional argument uses a more refined function of the Hessian"
        },
        {
          "blockId": "p3-0c9f90e5b9",
          "page": 3,
          "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
        },
        {
          "blockId": "p3-a5c78f8690",
          "page": 3,
          "quote": "where a ∈(1/2, 1) is fixed, ϵ∗> 0, N points into Ω, and H = −T(N, N). Spherical stability supplies\nthe reverse integrated comparison between the test’s energy and its boundary curvature term. The\nextra positive mass ϵ∗w2a will give the contradiction. The geometry and stability inequality are\ndeveloped in Section 3. We then state the universal tensor inequality and derive the two geometric\ncomparisons; the full exact verification follows in Section 6.\nTwo cancellations make this construction possible. A Laplacian contraction and an antisymmetric\ncurvature contraction eliminate fourth derivatives from div Z. At the boundary, reflection in the\nnormal direction removes the derivative components not determined by the boundary conditions.\nThe remaining inequalities involve only finitely many normalized tensors. We prove them by weighted\nsums of tensor squares and exact rational remainder bounds, with all coefficients and contraction\nconventions supplied in Appendix A.\nFinally, neither the normalized Hessian nor its derivatives are defined at w = 0, and suspension\nintroduces singular tips. These sets cannot be discarded without an estimate. A regularized Bochner\nidentity gives an integrable weight controlling the derivatives near every Hessian zero. The explicit\nsuspension formulas give an integrable power of the distance parameter at both tips. Smooth\nproduct cutoffs therefore remove both obstructions and make all error terms vanish. Section 7\ncompletes the stability contradiction."
        },
        {
          "blockId": "p3-c9b3441728",
          "page": 3,
          "quote": "Organization and conventions.\nSection 2 reduces a hypothetical first nonflat minimizer to a\nsmooth spherical component in dimension six, allowing two suspension tips. Section 3 derives the\ntensor identities and the stability inequality. Section 4 states the universal algebraic estimate, and\nSection 5 derives the pointwise geometric comparison from it. Section 6 gives the complete exact\nverification; Section 7 then removes the cutoffs to obtain the contradiction. Section 8 gives the\nregularity and sharpness consequences. Appendix A gives all coefficient data. All tensor norms and\ncontractions use orthonormal frames. Repeated indices are summed, ∆= tr ∇2, and a boundary\nnormal denoted by N points into the positive phase. The curvature H = −T(N, N) used below is\nnot divided by the dimension of the boundary."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p7-7608cd04c9",
          "page": 7,
          "quote": "in the positive phase. If go > 0 on the entire sphere, integrating the harmonicity identity over the\nsphere would give mo\nR\ngo = 0, a contradiction. Thus its positive set is proper.\nThe conical free boundary has the radial direction tangent to it at every nonzero point: the\ncurve r 7→rx lies in that free boundary. Consequently its smooth hypersurface is transverse to the\nunit sphere, and their intersection is a smooth hypersurface of Smo. Its regular charts have the\nspherical positive phase exactly on one side. Each point of the sphere therefore has a neighborhood\nwhose intersection with {go > 0} is either connected or empty (using a small ball for points away\nfrom the free boundary). A finite subcover of the sphere shows that there are only finitely many\npositive components. If two component closures met, their common point would be a free-boundary\npoint; the connected positive side of a regular chart there would meet both components, forcing\nthem to coincide. The same charts show that each closure is a smooth compact domain. Each\ncomponent has nonempty boundary because the sphere is connected and the positive set is proper.\nSmoothness of u from its positive side gives smoothness of go on these closures. Equation (2.6)\ngives the interior equation in (2.5). On the boundary, the radial derivative of u is go = 0, so the\nEuclidean gradient is tangent to the sphere. Equation (2.2) then gives the remaining boundary\nconditions."
        },
        {
          "blockId": "p11-a3924f6d0b",
          "page": 11,
          "quote": "A = T/w,\nλ = TNN/w = −H/w,\nS0 = λA −A2,"
        },
        {
          "blockId": "p11-f6ddc57c80",
          "page": 11,
          "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
        },
        {
          "blockId": "p12-4584ce2868",
          "page": 12,
          "quote": "The zero trace of B then gives BNNN = w2.\nNormalization gives S0p = 0 and tr S0 = −|A|2 = −1. Consequently Vijj = pi tr S0 + 2(S0p)i +\npi = 0. In an orthonormal frame with first vector p = N, one has VNαβ = (S0)αβ and VNNN = 1.\nThese are exactly the normalized identities (3.11), and VNjk = S0,jk + pjpk. Finally, T has only its\nnormal-normal and tangential-tangential blocks, so the odd-normal formulas account for the entire\ncontraction TjkBNjk. Thus"
        },
        {
          "blockId": "p12-a1dcbd3ddc",
          "page": 12,
          "quote": "∂Nw = w−1TjkBNjk = w2Ajk(S0 + p ⊗p)jk = w2(2λ −tr A3),"
        },
        {
          "blockId": "p16-e2bbbf27a2",
          "page": 16,
          "quote": "4\nA universal algebraic certificate"
        },
        {
          "blockId": "p16-eb9722a177",
          "page": 16,
          "quote": "The comparison needed for the stability inequality is a statement about finite-dimensional tensors.\nWe state it with free algebraic data before deriving its geometric consequences. All tensor spaces\nhere are over R5, with the Euclidean inner product and full tensor norm; repeated indices range\nfrom 1 to 5 and are summed. A symmetric tensor is called trace-free if contraction of any two slots\nis zero. We write STFr(R5) for the space of symmetric trace-free tensors of rank r. Fix"
        },
        {
          "blockId": "p16-6e69ee52c4",
          "page": 16,
          "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
        },
        {
          "blockId": "p16-4f79bc733f",
          "page": 16,
          "quote": "where p, ℓ∈R5 and g, w, z ∈R are otherwise free. In particular, w and z need not be positive. On\ntensor polynomials in A, p, g, δ, define Dl by the product rule and"
        },
        {
          "blockId": "p16-bae9fc97ae",
          "page": 16,
          "quote": "Proposition 4.1 (Universal inequalities). There are rational tensor polynomials f, Ujk, Klijk, Ml, Pl\nin A, p, g, δ, with the following properties.\nThey are equivariant under orthogonal changes of\ncoordinates, with f invariant. Their coefficients are specified by Table I in Appendix A, using the\ndecoding in Section 6.1. Put"
        },
        {
          "blockId": "p16-102340751e",
          "page": 16,
          "quote": "In particular, f ≥ϵ∗whenever g2 + |p|2 ≤1.\nFor every symmetric trace-free unit matrix A and vector p satisfying"
        },
        {
          "blockId": "p16-cc99c86d04",
          "page": 16,
          "quote": "define\nS0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk."
        }
      ],
      "occurrenceCount": 72
    },
    {
      "symbol": "Clijk",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p17-19ab409168",
        "page": 17,
        "quote": "All assertions up to the side boundary use derivatives from the positive side. In the suspended case\nthe two tips are excluded throughout Section 5. We apply the universal inequalities of Proposition 4.1,\nwhose proof follows in Section 6. For its coefficient polynomials f, U, K, M, P, with their stated slot\norders, set\nClijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk.\n(5.1)"
      },
      "occurrences": [
        {
          "blockId": "p16-b848250caa",
          "page": 16,
          "quote": "Clijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk."
        },
        {
          "blockId": "p16-af17922025",
          "page": 16,
          "quote": "I := z2 [5UjkAjk + Clijk(δljAik + δlkAij) −4f]"
        },
        {
          "blockId": "p17-19ab409168",
          "page": 17,
          "quote": "All assertions up to the side boundary use derivatives from the positive side. In the suspended case\nthe two tips are excluded throughout Section 5. We apply the universal inequalities of Proposition 4.1,\nwhose proof follows in Section 6. For its coefficient polynomials f, U, K, M, P, with their stated slot\norders, set\nClijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk.\n(5.1)"
        },
        {
          "blockId": "p17-5b77bc6948",
          "page": 17,
          "quote": "Thus Ciljk = −Clijk. No other slot symmetry of U or K will be needed. The orthogonal equivariance\nasserted in Proposition 4.1 ensures that all the fields defined below are independent of the orthonormal\nframe."
        },
        {
          "blockId": "p17-4e99f8fde8",
          "page": 17,
          "quote": "Furthermore,\n1\nwGlijk∇lBijk = 5UjkAjk + Clijk(δljAik + δlkAij).\n(5.4)"
        },
        {
          "blockId": "p18-e9ec2e0d65",
          "page": 18,
          "quote": "2Clijk(δljAik + δlkAij −δijAlk −δikAlj)"
        },
        {
          "blockId": "p18-a658af75df",
          "page": 18,
          "quote": "= Clijk(δljAik + δlkAij)."
        },
        {
          "blockId": "p18-3f9b6dca97",
          "page": 18,
          "quote": "The last equality follows by interchanging the summed indices l, i in the last two terms and using\nCiljk = −Clijk. Equivalently, Clijk∇lBijk equals Klijk(∇lBijk −∇iBljk); thus the factor 1/2 belongs\nto the formula with C, not to the formula with K.\nThis calculation eliminates every undetermined fourth derivative of g. Indeed, if two possible\ntensors ∇lBijk have the same Laplacian contraction and the same skew difference in l, i, their\ndifference has zero contraction with δliUjk and with Clijk. No constraint on the remaining fourth\nderivative entries is required."
        },
        {
          "blockId": "p18-4ac9acb3fe",
          "page": 18,
          "quote": "w−2a div Z = 5UjkAjk + Clijk(δljAik + δlkAij)"
        }
      ],
      "occurrenceCount": 9,
      "latex": "C_{lijk}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "D",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p1-1398459c75",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
      },
      "occurrences": [
        {
          "blockId": "p1-1398459c75",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
        },
        {
          "blockId": "p1-63eea2c5f4",
          "page": 1,
          "quote": "Corollary 1.2 (Sharp singular-set bound). Let n ≥2, let D ⊂Rn be open, and let u ∈H1\nloc(D) be\na nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is\nsmooth outside Sing(u). This set is empty when n ≤6, is locally finite when n = 7, and satisfies"
        },
        {
          "blockId": "p4-5aee8e3401",
          "page": 4,
          "quote": "Proposition 2.1 (Regularity and blowups of minimizers). Let d ≥2 and let u ≥0 be a local\nminimizer of the one-phase energy with volume coefficient 1 in an open set D ⊂Rd."
        },
        {
          "blockId": "p4-cd834ef0a2",
          "page": 4,
          "quote": "(ii) If x0 ∈D ∩∂{u > 0}, every sequence rj ↓0 has a subsequence for which"
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p8-81876620ea",
          "page": 8,
          "quote": "D2u(ry)[y, ·] = 0,\nD2u(ry)[v1, v2] = r−1To(y)[v1, v2]\n(v1, v2 ⊥y).\n(2.8)"
        },
        {
          "blockId": "p8-5eb064b940",
          "page": 8,
          "quote": "Suppose that To vanished identically on every positive component.\nThen D2u = 0 on each\ncorresponding connected open cone Di, so u(x) = bi · x + ci there. Homogeneity forces ci = 0, and\npositivity forces bi ̸= 0. Thus Di ⊂Hi := {bi · x > 0}. The set Di is also closed relative to Hi: if\nx ∈Di ∩Hi, continuity gives u(x) = bi · x > 0; a small positive neighborhood of x meets Di, hence\nbelongs to the same component. Since Hi is connected, Di = Hi. The Bernoulli condition at any\nnonzero point of its boundary gives |bi| = 1.\nTwo disjoint open half-spaces through the origin must have opposite unit normals: if their normals\nare e, f with f ̸= −e, the vector e + f lies in both. But opposite half-spaces have spherical closures\nmeeting along the equator e⊥∩Smo, contrary to the pairwise disjoint closures in Proposition 2.4.\nThus there is exactly one positive component, and u = (x · e)+, contrary to nonflatness. This proves\n(2.7)."
        },
        {
          "blockId": "p10-ea839c2676",
          "page": 10,
          "quote": "D2u(ry) = r−1\n0\n0\n0\nT(y)"
        },
        {
          "blockId": "p10-0dca6cc3ad",
          "page": 10,
          "quote": "Indeed, Euler’s identity gives D2u y = 0, and differentiation of ∇Rm+1u = gy + p along a tangent\nvector gives the tangential block ∇2g + g Id at radius one. Homogeneity then gives the factor r−1.\nThe same gradient formula identifies s2 with |∇Rm+1u|2 on the unit sphere."
        },
        {
          "blockId": "p13-2b3a07bdc3",
          "page": 13,
          "quote": "Proposition 3.5 (Euclidean stability on a component). Let D be the selected positivity component\nof a global minimizing cone in Rm+1. Let ϕ be any real smooth function up to the regular boundary\nof D, with compact support in that regular closure away from the vertex and from any other singular\npoints. Then\nZ"
        },
        {
          "blockId": "p13-4c29be1598",
          "page": 13,
          "quote": "D\n|∇ϕ|2 dx ≥\nZ"
        },
        {
          "blockId": "p13-39ac08043a",
          "page": 13,
          "quote": "∂D\nHEϕ2 dσE,\nHE = −D2u(NE, NE),\n(3.19)"
        },
        {
          "blockId": "p13-a694bf1a3d",
          "page": 13,
          "quote": "Proof. Every regular free-boundary point has a neighborhood where the positive phase is the single\nside belonging to D. Compactness of the boundary portion meeting supp ϕ gives finitely many such\nneighborhoods and a collar separated from every other positivity component and every singular\npoint. Choose the collar over a slightly larger boundary portion, so that the extension of ϕ can\nvanish near its lateral edge. Extend ϕ smoothly across this boundary with compact support in these\nneighborhoods. Also extend the positive side of u smoothly to a function F on the collar, with its\ninterior jets unchanged. Since u = 0 and ∇u = NE on the boundary, shrinking the collar makes\nF negative on its exterior side and positive on its interior side. Such an extension need only be\nsmooth to finite order sufficient for the expansions below; smooth boundary regularity supplies this,\nand a partition of unity joins the local extensions while preserving the interior function and its jets.\nFor a real parameter ε of either sufficiently small sign, use (F + εϕ)+ in the collar, u + εϕ on the\nrest of the affected portion of D, and u elsewhere. These definitions agree: F = u on the interior of\nthe collar, and on the compact interior support outside a smaller collar u has a positive minimum."
        },
        {
          "blockId": "p14-66c1ab50cc",
          "page": 14,
          "quote": "Near the exterior and lateral edges of the modification the function agrees with u. The resulting\ncompetitor uε is nonnegative, belongs to H1\nloc, and has compactly supported difference from u in a\ncontaining ball. It is therefore admissible for the original global minimizer. This construction uses\nno separate minimizing assertion for D.\nWe compute the change in energy in such a ball. First integrate the perturbed density over the\nfixed original domain D, and then account for the signed boundary layer where its positivity set\nchanges. The fixed-domain contribution is"
        },
        {
          "blockId": "p14-72f4405762",
          "page": 14,
          "quote": "D\n⟨∇u, ∇ϕ⟩dx + ε2\nZ"
        },
        {
          "blockId": "p14-194d0e1184",
          "page": 14,
          "quote": "D\n|∇ϕ|2 dx"
        },
        {
          "blockId": "p14-9b869eba9c",
          "page": 14,
          "quote": "∂D\nϕ dσE + ε2\nZ"
        },
        {
          "blockId": "p14-563124ccb1",
          "page": 14,
          "quote": "D\n|∇ϕ|2 dx.\n(3.20)"
        },
        {
          "blockId": "p14-443810c310",
          "page": 14,
          "quote": "The equality ∆F = 0 on the original boundary follows from matching the interior second derivatives;\nharmonicity outside D is unnecessary. Moreover,"
        }
      ],
      "occurrenceCount": 34
    },
    {
      "symbol": "Fl",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p19-fb5f4cbc43",
        "page": 19,
        "quote": "Put\nFl = Dlf + 2aqlf,\nℓ∗= F"
      },
      "occurrences": [
        {
          "blockId": "p19-fb5f4cbc43",
          "page": 19,
          "quote": "Put\nFl = Dlf + 2aqlf,\nℓ∗= F"
        },
        {
          "blockId": "p19-4886bf6683",
          "page": 19,
          "quote": "2√f Fl,\n|∇h|2 = w2a"
        }
      ],
      "occurrenceCount": 2,
      "latex": "F_{l}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "Ho",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p11-79aa63b2b7",
        "page": 11,
        "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
      },
      "occurrences": [
        {
          "blockId": "p11-79aa63b2b7",
          "page": 11,
          "quote": "Lemma 3.2 (Strict gradient bound on the original component). On Ωo one has s2\no < 1. At every\nboundary point, put\nHo = −To(No, No).\n(3.8)"
        },
        {
          "blockId": "p11-901a03e7df",
          "page": 11,
          "quote": "Then\nHo > 0,\nTopo = −Hopo,\nwo > 0\non ∂Ωo.\n(3.9)"
        },
        {
          "blockId": "p11-48cf81081f",
          "page": 11,
          "quote": "Proof. The boundary conditions give s2\no = 1 on ∂Ωo. By (3.6), s2\no is subharmonic. The maximum\nprinciple on the compact closure gives s2\no ≤1. If equality held at an interior point, the strong\nmaximum principle would make s2\no constant on the connected component, and then ∆s2\no = 2w2\no\nwould imply To ≡0. Thus s2\no < 1 inside.\nThe smooth boundary has the interior sphere property at each point. The Hopf boundary\nlemma therefore gives ∂Nos2\no < 0. Since po = No there, (3.6) identifies this derivative as 2To(No, No),\nproving Ho > 0. For a tangent vector τ to ∂Ωo, tangential differentiation of s2\no = 1 similarly gives\nTo(τ, No) = 0. This proves the eigenvector assertion and wo ≥Ho > 0. Continuity and compactness\nof the boundary give a positive minimum of wo and a neighborhood of the boundary disjoint from\nits zero set."
        },
        {
          "blockId": "p12-8b2672b64e",
          "page": 12,
          "quote": "ew = ρ−1wo,\n| eB|2 = ρ−4\n|Bo|2 + 3c2w2\no\n\n,\neH = ρ−1Ho\non the regular side.\n(3.17)"
        },
        {
          "blockId": "p13-aca19449d5",
          "page": 13,
          "quote": "Adding eg Id proves (3.15). Differentiating that tensor with the displayed connection gives (3.16);\nentries with two or three zero slots vanish. Each term with exactly one zero slot occurs in three\npositions, which proves the factor 3 in (3.17).\nThe inward normal to the side is the horizontal lift of No. Thus the normal-normal entry of\n(3.15) gives eH = ρ−1Ho. The metric on the side is dt2 + ρ2ds2\n∂Ωo, proving (3.18). The trace of (3.15)\nis zero, so ∆eg = −5eg. The interior identities follow locally from Proposition 3.1. The bounds for s2,\nH and w follow from the original compact component and (3.14)–(3.17); no maximum principle at a\ntip is used. The local boundary hypotheses and hence Proposition 3.3 hold on the regular side."
        }
      ],
      "occurrenceCount": 5,
      "latex": "H_{o}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "Kn",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p33-d88ba6541b",
        "page": 33,
        "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
      },
      "occurrences": [
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        }
      ],
      "occurrenceCount": 4,
      "latex": "K_{n}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "L",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-474378faca",
        "page": 6,
        "quote": "Let L be a Lipschitz constant for u on a fixed neighborhood of x0. Since u(x0) = 0, for |y| ≤M\nand small r,"
      },
      "occurrences": [
        {
          "blockId": "p4-3a8780641d",
          "page": 4,
          "quote": "locally uniformly in Rd. The limit is a nonzero, nonnegative, one-homogeneous global minimizer.\nAlong a further subsequence the convergence is strong in H1\nloc, and the positivity indicators\nconverge in L1\nloc."
        },
        {
          "blockId": "p6-474378faca",
          "page": 6,
          "quote": "Let L be a Lipschitz constant for u on a fixed neighborhood of x0. Since u(x0) = 0, for |y| ≤M\nand small r,"
        },
        {
          "blockId": "p6-909e57afa6",
          "page": 6,
          "quote": "|ur′(y) −ur(y)| ≤L|r −r′|M"
        },
        {
          "blockId": "p6-9d760e4b18",
          "page": 6,
          "quote": "Let ψ be the zero extension of q −v. For L > 1, choose ηL : R →[0, 1] equal to 1 on [−L, L], equal\nto 0 outside [−L −1, L + 1], and linear on each intervening interval. Define"
        },
        {
          "blockId": "p6-2f894589b6",
          "page": 6,
          "quote": "VL(y, s) = U(y, s) + ηL(s)ψ(y)."
        },
        {
          "blockId": "p6-0789ceed61",
          "page": 6,
          "quote": "This function is nonnegative: on B × R it is the convex combination (1 −ηL)v + ηLq. Its difference\nfrom U belongs to H1 and has compact support in a sufficiently large ball BL ⊂Rk+1, so it belongs\nto H1\n0(BL).\nThe energy difference on B × [−L, L] is −2Lδ. On either transition cylinder, convexity gives"
        },
        {
          "blockId": "p6-40fb45e6d6",
          "page": 6,
          "quote": "|∇yVL|2 ≤(1 −ηL)|∇v|2 + ηL|∇q|2,\n|∂sVL|2 ≤|q −v|2."
        },
        {
          "blockId": "p6-0c9d12b210",
          "page": 6,
          "quote": "independently of L. Outside these three cylinders the energy is unchanged. Thus"
        },
        {
          "blockId": "p6-1a6a7a5e80",
          "page": 6,
          "quote": "J(VL; BL) −J(U; BL) ≤−2Lδ + C < 0"
        },
        {
          "blockId": "p6-1035b8716b",
          "page": 6,
          "quote": "for large L, contradicting minimality of U."
        },
        {
          "blockId": "p26-63ab49ec44",
          "page": 26,
          "quote": "and let i, j ∈{1, 2, 3, 4} be its two degree-one types, counted with multiplicity and ordered so i ≤j.\nLet h be its g exponent and b its number of p endpoints. Define L2\nt as a product of the following\nrational factors:"
        },
        {
          "blockId": "p27-491e7509c6",
          "page": 27,
          "quote": "n2\nt ≥1010L2\nt ,\nbLt = nt"
        },
        {
          "blockId": "p27-6bbdfda212",
          "page": 27,
          "quote": "There is no floating-point square root in this prescription. For a rational s = 1010L2\nt , compute\nn = ⌊\np"
        },
        {
          "blockId": "p27-23be23f54d",
          "page": 27,
          "quote": "For clarity, the exact finite calculation needed here is the following. Decode all rows of Tables I\nand II; expand the nineteen Gram blocks by unordered row pairs and the finite projections (6.4)–\n(6.6); subtract each Table I contribution to (4.3); and add the four contractions with coefficients\ndi. Use the contraction, derivative, and key ordering rules above after each operation. Combine all\nsigned coefficients at equal keys. For each nonzero key compute its rational L2\nt , perform (6.15), and"
        },
        {
          "blockId": "p30-c89a582f54",
          "page": 30,
          "quote": "The tensor To is smooth on the compact closure; its norm wo is continuous there and smooth where\nit is positive. The working domain Ω⊂S5 is Ωo when mo = 5, and its suspension when mo = 4.\nOn the regular part of this working domain we use the unadorned notation of Sections 3 and 5. In\nparticular, Z denotes the flux vector field, whereas Zo denotes the original Hessian zero set. Set\nMo = ∥wo∥L∞(Ωo). We fix"
        },
        {
          "blockId": "p31-08a0ccbfd4",
          "page": 31,
          "quote": "belongs to L1(Ωo)."
        },
        {
          "blockId": "p31-a5342e9d9d",
          "page": 31,
          "quote": "a(M2\no + 1)a∥∆η∥L1(Ωo).\n(7.3)"
        },
        {
          "blockId": "p31-da05d9263f",
          "page": 31,
          "quote": "Ωo,+\n(1 −η)w2a−2\no\n|Bo|2 dVo ≤b2a−2\nη\n∥Bo∥2\nL2(Ωo)."
        },
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        }
      ],
      "occurrenceCount": 19
    },
    {
      "symbol": "Mo",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p30-c89a582f54",
        "page": 30,
        "quote": "The tensor To is smooth on the compact closure; its norm wo is continuous there and smooth where\nit is positive. The working domain Ω⊂S5 is Ωo when mo = 5, and its suspension when mo = 4.\nOn the regular part of this working domain we use the unadorned notation of Sections 3 and 5. In\nparticular, Z denotes the flux vector field, whereas Zo denotes the original Hessian zero set. Set\nMo = ∥wo∥L∞(Ωo). We fix"
      },
      "occurrences": [
        {
          "blockId": "p30-c89a582f54",
          "page": 30,
          "quote": "The tensor To is smooth on the compact closure; its norm wo is continuous there and smooth where\nit is positive. The working domain Ω⊂S5 is Ωo when mo = 5, and its suspension when mo = 4.\nOn the regular part of this working domain we use the unadorned notation of Sections 3 and 5. In\nparticular, Z denotes the flux vector field, whereas Zo denotes the original Hessian zero set. Set\nMo = ∥wo∥L∞(Ωo). We fix"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        },
        {
          "blockId": "p34-30f6e2c51a",
          "page": 34,
          "quote": "This inequality is initially read on Ωo,+, and both sides are assigned zero on Zo. Its constant depends\nonly on the fixed certificate, a, ϑ, and Mo, and is independent of j, t, and y. It also shows explicitly\nwhy the original weighted |Bo|2 estimate suffices: by (7.2),"
        }
      ],
      "occurrenceCount": 4,
      "latex": "M_{o}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "R",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-19acdad6af",
        "page": 5,
        "quote": "Proof. Write ur = ux0,r. For fixed t ∈R and small r > 0, set r′ = r/(1 + rt) > 0. Homogeneity\ngives the exact identity"
      },
      "occurrences": [
        {
          "blockId": "p1-4296a3c860",
          "page": 1,
          "quote": "Theorem 1.1. Every nonzero one-homogeneous global minimizer of (1.1) in Rd, 1 ≤d ≤6, is flat.\nThere is a nonflat one-homogeneous global minimizer in R7. Consequently,"
        },
        {
          "blockId": "p5-16fb8ef603",
          "page": 5,
          "quote": "U(y + tx0) = U(y)\n(y ∈Rd, t ∈R)."
        },
        {
          "blockId": "p5-19acdad6af",
          "page": 5,
          "quote": "Proof. Write ur = ux0,r. For fixed t ∈R and small r > 0, set r′ = r/(1 + rt) > 0. Homogeneity\ngives the exact identity"
        },
        {
          "blockId": "p6-35c36a36fe",
          "page": 6,
          "quote": "Lemma 2.3 (Minimality of a translation-invariant factor). Suppose U(y, s) = v(y) is a global\nminimizer in Rk × R, where v ≥0 belongs to H1\nloc(Rk). Then v is a global minimizer in Rk."
        },
        {
          "blockId": "p6-9d760e4b18",
          "page": 6,
          "quote": "Let ψ be the zero extension of q −v. For L > 1, choose ηL : R →[0, 1] equal to 1 on [−L, L], equal\nto 0 outside [−L −1, L + 1], and linear on each intervening interval. Define"
        },
        {
          "blockId": "p6-0789ceed61",
          "page": 6,
          "quote": "This function is nonnegative: on B × R it is the convex combination (1 −ηL)v + ηLq. Its difference\nfrom U belongs to H1 and has compact support in a sufficiently large ball BL ⊂Rk+1, so it belongs\nto H1\n0(BL).\nThe energy difference on B × [−L, L] is −2Lδ. On either transition cylinder, convexity gives"
        },
        {
          "blockId": "p6-5aff418e48",
          "page": 6,
          "quote": "In dimension one, every nonnegative one-homogeneous Sobolev function has the form u(x) =\nb+x+ + b−(−x)+ with b+, b−≥0. If both slopes are positive, a nonnegative smooth bump ϕ\nsupported in (−R, R) with ϕ(0) > 0 preserves positivity volume and gives"
        },
        {
          "blockId": "p6-7aa4a96b63",
          "page": 6,
          "quote": "J(u + τϕ; (−R, R)) −J(u; (−R, R)) = −2τ(b+ + b−)ϕ(0) + τ 2\nZ R"
        },
        {
          "blockId": "p6-83bb31912e",
          "page": 6,
          "quote": "−R\n|ϕ′|2 < 0"
        },
        {
          "blockId": "p6-05b288455c",
          "page": 6,
          "quote": "for small τ > 0. Thus a nonzero minimizing cone has just one positive slope. After reflection write\nu(x) = bx+, b > 0. The competitors vt(x) = bR(x −t)+/(R −t) on (−R, R), |t| < R, have the"
        },
        {
          "blockId": "p7-d328f4a8f3",
          "page": 7,
          "quote": "same trace and energy b2R2/(R −t) + (R −t). Its derivative at the minimizing parameter t = 0 is\nb2 −1 = 0, so b = 1. This proves flatness in dimension one directly.\nThe lower-dimensional classification of Jerison–Savin [13, Theorem 1.1 and Corollary 1.2] and\nthe dimension-seven minimizing example of De Silva–Jerison [7, Theorem 1.1] give 5 ≤d∗≤7. De\nSilva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors.\nFor the exclusion argument, suppose from now on that"
        },
        {
          "blockId": "p7-df0b192be6",
          "page": 7,
          "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
        },
        {
          "blockId": "p7-7608cd04c9",
          "page": 7,
          "quote": "in the positive phase. If go > 0 on the entire sphere, integrating the harmonicity identity over the\nsphere would give mo\nR\ngo = 0, a contradiction. Thus its positive set is proper.\nThe conical free boundary has the radial direction tangent to it at every nonzero point: the\ncurve r 7→rx lies in that free boundary. Consequently its smooth hypersurface is transverse to the\nunit sphere, and their intersection is a smooth hypersurface of Smo. Its regular charts have the\nspherical positive phase exactly on one side. Each point of the sphere therefore has a neighborhood\nwhose intersection with {go > 0} is either connected or empty (using a small ball for points away\nfrom the free boundary). A finite subcover of the sphere shows that there are only finitely many\npositive components. If two component closures met, their common point would be a free-boundary\npoint; the connected positive side of a regular chart there would meet both components, forcing\nthem to coincide. The same charts show that each closure is a smooth compact domain. Each\ncomponent has nonempty boundary because the sphere is connected and the positive set is proper.\nSmoothness of u from its positive side gives smoothness of go on these closures. Equation (2.6)\ngives the interior equation in (2.5). On the boundary, the radial derivative of u is go = 0, so the\nEuclidean gradient is tangent to the sphere. Equation (2.2) then gives the remaining boundary\nconditions."
        },
        {
          "blockId": "p8-05cf6278e3",
          "page": 8,
          "quote": "Proposition 2.6 (Suspension preserves global minimality). If u ≥0 is a global minimizer in Rk,\nthen eu(x, s) = u(x) is a global minimizer in Rk+1. If u is the cone in (2.4) with d = 5, the singular\nfree-boundary set of eu is exactly {0}×R. For the component chosen in Lemma 2.5, the corresponding\npositive component on S5 is"
        },
        {
          "blockId": "p8-fdfcec216b",
          "page": 8,
          "quote": "Proof. Fix a ball B = BR((xc, sc)) ⊂Rk+1 and a nonnegative V ∈H1(B) with V −eu ∈H1\n0(B).\nExtend that difference by zero to F ∈H1(Rk+1). For |s −sc| < R, let"
        },
        {
          "blockId": "p8-c3e3c2d6dd",
          "page": 8,
          "quote": "R2−(s−sc)2(xc)"
        },
        {
          "blockId": "p8-6ae0fb328c",
          "page": 8,
          "quote": "J(eu; B) =\nZ sc+R"
        },
        {
          "blockId": "p8-f86a1e4e3c",
          "page": 8,
          "quote": "sc−R\nJ(u; Bs) ds"
        },
        {
          "blockId": "p8-cf7551d8db",
          "page": 8,
          "quote": "≤\nZ sc+R"
        },
        {
          "blockId": "p8-1e92aa4b2f",
          "page": 8,
          "quote": "sc−R\nJ(Vs; Bs) ds ≤J(V ; B)."
        }
      ],
      "occurrenceCount": 49
    },
    {
      "symbol": "S0",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-cc99c86d04",
        "page": 16,
        "quote": "define\nS0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk."
      },
      "occurrences": [
        {
          "blockId": "p11-a3924f6d0b",
          "page": 11,
          "quote": "A = T/w,\nλ = TNN/w = −H/w,\nS0 = λA −A2,"
        },
        {
          "blockId": "p11-ee96e4a03b",
          "page": 11,
          "quote": "Vijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(3.12)"
        },
        {
          "blockId": "p11-7580df6f09",
          "page": 11,
          "quote": "Then |p| = 1, Ap = λp, λ < 0, S0p = 0, tr S0 = −1, and V is trace-free. The tensors B/w2 and V\nagree in every component with an odd number of normal slots. Moreover,"
        },
        {
          "blockId": "p11-f6ddc57c80",
          "page": 11,
          "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
        },
        {
          "blockId": "p12-4584ce2868",
          "page": 12,
          "quote": "The zero trace of B then gives BNNN = w2.\nNormalization gives S0p = 0 and tr S0 = −|A|2 = −1. Consequently Vijj = pi tr S0 + 2(S0p)i +\npi = 0. In an orthonormal frame with first vector p = N, one has VNαβ = (S0)αβ and VNNN = 1.\nThese are exactly the normalized identities (3.11), and VNjk = S0,jk + pjpk. Finally, T has only its\nnormal-normal and tangential-tangential blocks, so the odd-normal formulas account for the entire\ncontraction TjkBNjk. Thus"
        },
        {
          "blockId": "p12-a1dcbd3ddc",
          "page": 12,
          "quote": "∂Nw = w−1TjkBNjk = w2Ajk(S0 + p ⊗p)jk = w2(2λ −tr A3),"
        },
        {
          "blockId": "p16-cc99c86d04",
          "page": 16,
          "quote": "define\nS0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk."
        },
        {
          "blockId": "p16-12ac49d162",
          "page": 16,
          "quote": "Then\nS := −λf + Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl ≥δ∗.\n(4.4)"
        },
        {
          "blockId": "p19-d3b87440c1",
          "page": 19,
          "quote": "S0 = λA −A2,\nVijk = pi(S0)jk + pj(S0)ki + pk(S0)ij + pipjpk.\n(5.11)"
        },
        {
          "blockId": "p19-7966fef82a",
          "page": 19,
          "quote": "w2 = plGlijkVijk = Ujk(S0 + p ⊗p)jk + ClijkplVijk.\n(5.12)"
        },
        {
          "blockId": "p20-b941ebff60",
          "page": 20,
          "quote": "w2\n= λAαβ −(A2)αβ = (S0)αβ,\nBNNN"
        },
        {
          "blockId": "p20-e5286ce9bf",
          "page": 20,
          "quote": "piVijk = (S0)jk + pjpk."
        },
        {
          "blockId": "p20-eea01575b0",
          "page": 20,
          "quote": "N · Z = w2a+1 (Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl) ,"
        },
        {
          "blockId": "p29-2c0a52f30e",
          "page": 29,
          "quote": "No sign or distinctness condition is imposed on the xi; the normal eigenvalue condition is exactly\nP\ni xi > 0. In these coordinates (S0)55 = 0 and (S0)αα = bα := λxα −x2\nα. The only nonzero entries\nof V are V555 = 1 and the three permutations of V5αα = bα. Since C55jk = 0, a direct component\nformula for the boundary polynomial is therefore"
        }
      ],
      "occurrenceCount": 14,
      "latex": "S_{0}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "T",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-0c9f90e5b9",
        "page": 3,
        "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
      },
      "occurrences": [
        {
          "blockId": "p3-0c9f90e5b9",
          "page": 3,
          "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
        },
        {
          "blockId": "p3-a5c78f8690",
          "page": 3,
          "quote": "where a ∈(1/2, 1) is fixed, ϵ∗> 0, N points into Ω, and H = −T(N, N). Spherical stability supplies\nthe reverse integrated comparison between the test’s energy and its boundary curvature term. The\nextra positive mass ϵ∗w2a will give the contradiction. The geometry and stability inequality are\ndeveloped in Section 3. We then state the universal tensor inequality and derive the two geometric\ncomparisons; the full exact verification follows in Section 6.\nTwo cancellations make this construction possible. A Laplacian contraction and an antisymmetric\ncurvature contraction eliminate fourth derivatives from div Z. At the boundary, reflection in the\nnormal direction removes the derivative components not determined by the boundary conditions.\nThe remaining inequalities involve only finitely many normalized tensors. We prove them by weighted\nsums of tensor squares and exact rational remainder bounds, with all coefficients and contraction\nconventions supplied in Appendix A.\nFinally, neither the normalized Hessian nor its derivatives are defined at w = 0, and suspension\nintroduces singular tips. These sets cannot be discarded without an estimate. A regularized Bochner\nidentity gives an integrable weight controlling the derivatives near every Hessian zero. The explicit\nsuspension formulas give an integrable power of the distance parameter at both tips. Smooth\nproduct cutoffs therefore remove both obstructions and make all error terms vanish. Section 7\ncompletes the stability contradiction."
        },
        {
          "blockId": "p3-c9b3441728",
          "page": 3,
          "quote": "Organization and conventions.\nSection 2 reduces a hypothetical first nonflat minimizer to a\nsmooth spherical component in dimension six, allowing two suspension tips. Section 3 derives the\ntensor identities and the stability inequality. Section 4 states the universal algebraic estimate, and\nSection 5 derives the pointwise geometric comparison from it. Section 6 gives the complete exact\nverification; Section 7 then removes the cutoffs to obtain the contradiction. Section 8 gives the\nregularity and sharpness consequences. Appendix A gives all coefficient data. All tensor norms and\ncontractions use orthonormal frames. Repeated indices are summed, ∆= tr ∇2, and a boundary\nnormal denoted by N points into the positive phase. The curvature H = −T(N, N) used below is\nnot divided by the dimension of the boundary."
        },
        {
          "blockId": "p10-ae1fb87ac5",
          "page": 10,
          "quote": "p = ∇g,\nT = ∇2g + g Id,\nBijk = ∇iTjk,\nw = |T|,\ns2 = g2 + |p|2.\n(3.1)"
        },
        {
          "blockId": "p10-ea839c2676",
          "page": 10,
          "quote": "D2u(ry) = r−1\n0\n0\n0\nT(y)"
        },
        {
          "blockId": "p10-ce852626e7",
          "page": 10,
          "quote": "Proposition 3.1 (Interior identities). The tensor T is symmetric and trace-free, and B is fully\nsymmetric and trace-free in every pair of slots. In particular div T = 0. The following identities\nhold:"
        },
        {
          "blockId": "p10-4f905adc86",
          "page": 10,
          "quote": "∆T = mT,\n∆w2 = 2mw2 + 2|B|2,\n(3.4)"
        },
        {
          "blockId": "p10-12ceae48c3",
          "page": 10,
          "quote": "Proof. The trace equation is tr T = ∆g + mg = 0. On the unit sphere the commutation rule for the\ncovector p gives\n∇i(∇2g)jk −∇j(∇2g)ik = δikpj −δjkpi."
        },
        {
          "blockId": "p10-3fb8cef4c5",
          "page": 10,
          "quote": "Adding the derivatives of g Id cancels the right side. Thus T is Codazzi: ∇iTjk = ∇jTik. Together\nwith symmetry in j, k, this proves full symmetry of B. Differentiating tr T = 0 shows Bijj = 0; full\nsymmetry gives all other traces and hence div T = 0.\nFor clarity, the same curvature convention on a covariant two-tensor is"
        },
        {
          "blockId": "p10-b7b15cdbf0",
          "page": 10,
          "quote": "([∇l, ∇i]T)jk = δljTik + δlkTij −δijTlk −δikTlj."
        },
        {
          "blockId": "p10-d0dac569b1",
          "page": 10,
          "quote": "mTjk + Tjk −Tjk −δjk tr T = mTjk."
        },
        {
          "blockId": "p10-685261eec2",
          "page": 10,
          "quote": "The product rule for |T|2 now proves (3.4). The definitions give (3.5) and"
        },
        {
          "blockId": "p10-df9c6a96db",
          "page": 10,
          "quote": "Taking the divergence and using div T = 0 and tr T = 0 gives ∆s2 = 2Tli(Tli −gδli) = 2w2. Finally,\ndifferentiating w2 and applying Cauchy–Schwarz proves both gradient estimates."
        },
        {
          "blockId": "p11-fdd2a71ec4",
          "page": 11,
          "quote": "Proposition 3.3 (Boundary jets). Suppose g = 0, p = N, and T(N, N) = −H < 0 on a smooth\nboundary portion, where N is inward. Let Greek indices be tangent to this boundary and put"
        },
        {
          "blockId": "p11-c696783e4a",
          "page": 11,
          "quote": "Then TNα = 0, Tαβ = hαβ, and H = tr h; in particular H is a sum of principal curvatures with the\nspecified convention, without division by m −1. If ∇∂is the boundary connection, all components\nof B are described, up to permutation of slots, by"
        },
        {
          "blockId": "p11-d8ff12d046",
          "page": 11,
          "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
        },
        {
          "blockId": "p11-a243f0a817",
          "page": 11,
          "quote": "The components with an odd number of normal slots are thus determined by the pointwise tensor T;\nthe other components involve tangential derivatives of curvature. The tensor V below encodes this\nodd-normal part after normalization. Here w > 0. Define the normalized tensors and eigenvalue"
        },
        {
          "blockId": "p11-a3924f6d0b",
          "page": 11,
          "quote": "A = T/w,\nλ = TNN/w = −H/w,\nS0 = λA −A2,"
        },
        {
          "blockId": "p12-7bd615b1c8",
          "page": 12,
          "quote": "Proof. On the boundary, tangential differentiation of |p|2 = 1 gives TNα = 0, and ∇αp = Teα\nbecause g = 0. Since p = N along the boundary, this proves Tαβ = hαβ. The trace equation for\nT gives tr h = −TNN = H. In differentiating three tangential entries of T, the normal connection\nterms pair with T(N, eα) = 0. This yields the first formula in (3.10). Likewise,"
        },
        {
          "blockId": "p12-8755606744",
          "page": 12,
          "quote": "∂αT(N, N) = BαNN + 2T(∇αN, N) = BαNN,"
        }
      ],
      "occurrenceCount": 31
    },
    {
      "symbol": "U",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-7b73106fd2",
        "page": 5,
        "quote": "Lemma 2.2 (Translation invariance of an off-origin blowup). Let u be a one-homogeneous global\nminimizer, let 0 ̸= x0 ∈∂{u > 0}, and let U be any limit in (2.1). Then"
      },
      "occurrences": [
        {
          "blockId": "p4-656ba12637",
          "page": 4,
          "quote": "rj\n−→U(y)\n(2.1)"
        },
        {
          "blockId": "p5-7b73106fd2",
          "page": 5,
          "quote": "Lemma 2.2 (Translation invariance of an off-origin blowup). Let u be a one-homogeneous global\nminimizer, let 0 ̸= x0 ∈∂{u > 0}, and let U be any limit in (2.1). Then"
        },
        {
          "blockId": "p5-16fb8ef603",
          "page": 5,
          "quote": "U(y + tx0) = U(y)\n(y ∈Rd, t ∈R)."
        },
        {
          "blockId": "p6-35c36a36fe",
          "page": 6,
          "quote": "Lemma 2.3 (Minimality of a translation-invariant factor). Suppose U(y, s) = v(y) is a global\nminimizer in Rk × R, where v ≥0 belongs to H1\nloc(Rk). Then v is a global minimizer in Rk."
        },
        {
          "blockId": "p6-2f894589b6",
          "page": 6,
          "quote": "VL(y, s) = U(y, s) + ηL(s)ψ(y)."
        },
        {
          "blockId": "p6-0789ceed61",
          "page": 6,
          "quote": "This function is nonnegative: on B × R it is the convex combination (1 −ηL)v + ηLq. Its difference\nfrom U belongs to H1 and has compact support in a sufficiently large ball BL ⊂Rk+1, so it belongs\nto H1\n0(BL).\nThe energy difference on B × [−L, L] is −2Lδ. On either transition cylinder, convexity gives"
        },
        {
          "blockId": "p6-1a6a7a5e80",
          "page": 6,
          "quote": "J(VL; BL) −J(U; BL) ≤−2Lδ + C < 0"
        },
        {
          "blockId": "p6-1035b8716b",
          "page": 6,
          "quote": "for large L, contradicting minimality of U."
        },
        {
          "blockId": "p7-df0b192be6",
          "page": 7,
          "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
        },
        {
          "blockId": "p16-5719369278",
          "page": 16,
          "quote": "Moreover, U and K are even in p, and P = 0 when g = 0."
        },
        {
          "blockId": "p17-19ab409168",
          "page": 17,
          "quote": "All assertions up to the side boundary use derivatives from the positive side. In the suspended case\nthe two tips are excluded throughout Section 5. We apply the universal inequalities of Proposition 4.1,\nwhose proof follows in Section 6. For its coefficient polynomials f, U, K, M, P, with their stated slot\norders, set\nClijk = Klijk −Kiljk,\nGlijk = δliUjk + Clijk.\n(5.1)"
        },
        {
          "blockId": "p17-5b77bc6948",
          "page": 17,
          "quote": "Thus Ciljk = −Clijk. No other slot symmetry of U or K will be needed. The orthogonal equivariance\nasserted in Proposition 4.1 ensures that all the fields defined below are independent of the orthonormal\nframe."
        },
        {
          "blockId": "p17-2efa1d3509",
          "page": 17,
          "quote": "on the admissible data.\nHere Dl is applied only to coefficient polynomials in A, p, g, δ. Differentiation of the factors w\nand X, which are not coefficient variables, is performed separately. In particular, no rule for DlX\nor Dlq is being assumed.\nFor the U part of (5.4),"
        },
        {
          "blockId": "p19-1b7b8d3435",
          "page": 19,
          "quote": "Proof. Let R = Id −2p ⊗p be reflection in p⊥. Since A is symmetric and Ap = λp, it preserves p⊥,\nand hence RART = A. Also Rp = −p. Orthogonal equivariance and the even parity in p of U, K,\nand therefore of G, give"
        },
        {
          "blockId": "p19-99832b79b9",
          "page": 19,
          "quote": "Choose any orthonormal basis with its last vector p. A component of a three-tensor with k normal\nslots is multiplied by (−1)k under R⊗3. It follows that F has zero components whenever the number\nof normal slots is even. This assertion concerns the full unsymmetrized tensor F; no permutation of\nits slots, and no extra symmetry of U or K, has been used."
        },
        {
          "blockId": "p21-75940db39f",
          "page": 21,
          "quote": "We return to the free algebraic data of Section 4 to prove Proposition 4.1. All tensors are again over\nR5. The scalars w, z and vector ℓare free, and X is an arbitrary symmetric trace-free three-tensor.\nTable I specifies the polynomials f, U, K, M, P, and Table II specifies nonnegative comparison forms.\nFor the interior inequality, we compare I with a positive diagonal quadratic form in (w, z, ℓ, X)\nplus projected tensor squares, and bound the remaining error term by term. For the boundary\ninequality, the analogous comparison uses scalar squares in the four tangential eigenvalues and a\nuniform bound on eleven remainder coefficients. We give the contraction rules and rational bounds\nin full."
        },
        {
          "blockId": "p21-5111bc0d6f",
          "page": 21,
          "quote": "For F ∈{f, U, K, M, P}, let rF be respectively 0, 2, 4, 1, 1, with slot orders respectively empty, (j, k),\n(l, i, j, k), (l), (l). If a row of Table I consists of F, s, c, its contribution is cMrF (s)/Q; thus each\npolynomial is given by"
        },
        {
          "blockId": "p21-040b98ffcc",
          "page": 21,
          "quote": "Table I contains 34, 57, 34, 74, 32 rows for f, U, K, M, P, respectively. Every free slot occurs exactly\nonce in every row, and these rows have no X, ℓ, w, z factors. Every P row has a positive g exponent.\nBecause an edge has two endpoints and U, K have even rank, their numbers of p endpoints are\neven. This proves the parity assertions of Proposition 4.1 directly from the definitions. The same\ndefinitions show that all five polynomials are equivariant under orthogonal changes of coordinates,\nwith f invariant."
        },
        {
          "blockId": "p25-225c20d6c9",
          "page": 25,
          "quote": "The linear contribution of each Table I row is obtained by substitution into (4.3), forming\nC = K −K(12) for a K row. One check on the slot order and coefficients is the single row U = A\nwith coefficient one. Its interior contribution is"
        },
        {
          "blockId": "p29-588804483d",
          "page": 29,
          "quote": "S = −λf + U55 + M5 +"
        }
      ],
      "occurrenceCount": 26
    },
    {
      "symbol": "X",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-6e69ee52c4",
        "page": 16,
        "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
      },
      "occurrences": [
        {
          "blockId": "p12-390df1acf6",
          "page": 12,
          "quote": "This proves the first formula in (3.11). Taking its tangential trace gives\nX"
        },
        {
          "blockId": "p12-ecab27bc42",
          "page": 12,
          "quote": "α\nBNαα = TNN\nX"
        },
        {
          "blockId": "p12-80f71c8f3c",
          "page": 12,
          "quote": "α\nTαα −\nX"
        },
        {
          "blockId": "p12-b912b80307",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −T 2\nNN −\nX"
        },
        {
          "blockId": "p16-6e69ee52c4",
          "page": 16,
          "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
        },
        {
          "blockId": "p16-194cc9dd41",
          "page": 16,
          "quote": "This is a first derivative with values in the enlarged polynomial algebra containing X, w, z; no\nderivative of these new variables is used."
        },
        {
          "blockId": "p17-0498dc48f3",
          "page": 17,
          "quote": "w,\nX = B"
        },
        {
          "blockId": "p17-fad04e79c7",
          "page": 17,
          "quote": "The identities of Proposition 3.1 show that A ∈STF2(R5), |A| = 1, and X ∈STF3(R5). Moreover,\ng2 + |p|2 ≤1, so these are admissible data for the interior certificate. The homogenizing variable in\nthat certificate is set to z = 1 in every geometric application."
        },
        {
          "blockId": "p17-2efa1d3509",
          "page": 17,
          "quote": "on the admissible data.\nHere Dl is applied only to coefficient polynomials in A, p, g, δ. Differentiation of the factors w\nand X, which are not coefficient variables, is performed separately. In particular, no rule for DlX\nor Dlq is being assumed.\nFor the U part of (5.4),"
        },
        {
          "blockId": "p18-55233111ba",
          "page": 18,
          "quote": "on the entire normalized coefficient domain. The verification in Section 6.5 obtains this bound\nusing the free algebraic data X = w = z = 0. Since f is independent of w, this permitted algebraic\nsubstitution does not impose w = 0 on the geometric data: the fields below are defined on Ω+,\nwhere w > 0.\nDefine on Ω+\nh = wap"
        },
        {
          "blockId": "p18-842e7465ba",
          "page": 18,
          "quote": "Every Dl in Equation (5.8) is evaluated at z = 1. The three coefficients of q come respectively from\ndifferentiating w2aX, w2a+1, and w2a; in the first expression the quotient X = B/w contributes\n−qlX. Thus their values are exactly"
        },
        {
          "blockId": "p20-cf8d16908c",
          "page": 20,
          "quote": "Proof. Proposition 4.1 gives P = 0 at the boundary, where g = 0. Since X = B/w, Lemma 5.3 gives"
        },
        {
          "blockId": "p20-451469844c",
          "page": 20,
          "quote": "|h| ≤Cwa,\n|∇h| ≤Cwa(1 + w + |X|),\n|Z| ≤Cw2a(1 + w + |X|).\n(5.14)"
        },
        {
          "blockId": "p20-f7d8142cce",
          "page": 20,
          "quote": "The constant is independent of w, |X|, and all subsequent cutoff parameters."
        },
        {
          "blockId": "p20-cba39cc8cd",
          "page": 20,
          "quote": "is compact. The coefficient polynomials and their first derivatives with respect to their coefficient\nvariables are therefore uniformly bounded there. Also |q| ≤|X| by Cauchy–Schwarz. The rules (5.3)\nconsequently imply\n|Df| + |q| |f| ≤C(1 + w + |X|)."
        },
        {
          "blockId": "p21-75940db39f",
          "page": 21,
          "quote": "We return to the free algebraic data of Section 4 to prove Proposition 4.1. All tensors are again over\nR5. The scalars w, z and vector ℓare free, and X is an arbitrary symmetric trace-free three-tensor.\nTable I specifies the polynomials f, U, K, M, P, and Table II specifies nonnegative comparison forms.\nFor the interior inequality, we compare I with a positive diagonal quadratic form in (w, z, ℓ, X)\nplus projected tensor squares, and bound the remaining error term by term. For the boundary\ninequality, the analogous comparison uses scalar squares in the four tangential eigenvalues and a\nuniform bound on eleven remainder coefficients. We give the contraction rules and rational bounds\nin full."
        },
        {
          "blockId": "p21-e2ab8e7e5f",
          "page": 21,
          "quote": "Each later triple uvk specifies an edge carrying the matrix Ak, with A0 = δ. An endpoint labeled\ni ∈{0, . . . , r −1} is the ith free slot, starting with slot zero. An endpoint labeled 5 carries one copy\nof p, and one labeled 6 carries one copy of ℓ. Three incidences of label 7 are contracted against the\nthree slots of one X; three incidences of label 8, if present, use a second copy of X. Each occurrence\nof a vector endpoint is a separate vector factor. Matrix entries, vector entries, and tensor entries\nare multiplied, and all nonfree slots are summed. Edges are unoriented because A is symmetric.\nPermuting the three incidences at an X does not change its value because X is symmetric. There\nare no implicit multiplicities or factorials in this convention.\nWrite Mr(s) for the tensor encoded by the word s. For example,"
        },
        {
          "blockId": "p21-d2e315f7be",
          "page": 21,
          "quote": "X"
        },
        {
          "blockId": "p21-040b98ffcc",
          "page": 21,
          "quote": "Table I contains 34, 57, 34, 74, 32 rows for f, U, K, M, P, respectively. Every free slot occurs exactly\nonce in every row, and these rows have no X, ℓ, w, z factors. Every P row has a positive g exponent.\nBecause an edge has two endpoints and U, K have even rank, their numbers of p endpoints are\neven. This proves the parity assertions of Proposition 4.1 directly from the definitions. The same\ndefinitions show that all five polynomials are equivariant under orthogonal changes of coordinates,\nwith f invariant."
        },
        {
          "blockId": "p22-6f28dc0c8d",
          "page": 22,
          "quote": "N\nX"
        }
      ],
      "occurrenceCount": 69
    },
    {
      "symbol": "bo",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p33-d88ba6541b",
        "page": 33,
        "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
      },
      "occurrences": [
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        }
      ],
      "occurrenceCount": 5,
      "latex": "b_{o}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "g",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-0c9f90e5b9",
        "page": 3,
        "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
      },
      "occurrences": [
        {
          "blockId": "p3-0c9f90e5b9",
          "page": 3,
          "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
        },
        {
          "blockId": "p9-b613bfd01d",
          "page": 9,
          "quote": "Notation for the remaining argument.\nFix the original component Ωo ⊂Smo selected in\nLemma 2.5, where mo ∈{4, 5}. If mo = 5, set Ω= Ωo, g = go, and Σ = ∅. If mo = 4, set Ω= eΩ,\ng = eg, and Σ = {p+, p−}. In either case Ω⊂S5 is connected, and g comes from a nonnegative\none-homogeneous global minimizer in R6. The closure is smooth away from Σ, with positive phase\non one side of its regular boundary\n∂regΩ:= ∂Ω\\ Σ."
        },
        {
          "blockId": "p10-bb1981e605",
          "page": 10,
          "quote": "For a smooth solution of ∆g = −mg on a domain in Sm, set"
        },
        {
          "blockId": "p10-ae1fb87ac5",
          "page": 10,
          "quote": "p = ∇g,\nT = ∇2g + g Id,\nBijk = ∇iTjk,\nw = |T|,\ns2 = g2 + |p|2.\n(3.1)"
        },
        {
          "blockId": "p10-0dca6cc3ad",
          "page": 10,
          "quote": "Indeed, Euler’s identity gives D2u y = 0, and differentiation of ∇Rm+1u = gy + p along a tangent\nvector gives the tangential block ∇2g + g Id at radius one. Homogeneity then gives the factor r−1.\nThe same gradient formula identifies s2 with |∇Rm+1u|2 on the unit sphere."
        },
        {
          "blockId": "p10-671ef66fb9",
          "page": 10,
          "quote": "∇lpi = Tli −gδli,\n∇lg = pl,\n(3.5)"
        },
        {
          "blockId": "p10-12ceae48c3",
          "page": 10,
          "quote": "Proof. The trace equation is tr T = ∆g + mg = 0. On the unit sphere the commutation rule for the\ncovector p gives\n∇i(∇2g)jk −∇j(∇2g)ik = δikpj −δjkpi."
        },
        {
          "blockId": "p10-3fb8cef4c5",
          "page": 10,
          "quote": "Adding the derivatives of g Id cancels the right side. Thus T is Codazzi: ∇iTjk = ∇jTik. Together\nwith symmetry in j, k, this proves full symmetry of B. Differentiating tr T = 0 shows Bijj = 0; full\nsymmetry gives all other traces and hence div T = 0.\nFor clarity, the same curvature convention on a covariant two-tensor is"
        },
        {
          "blockId": "p10-3dd350e865",
          "page": 10,
          "quote": "∇ls2 = 2gpl + 2pi(Tli −gδli) = 2Tlipi."
        },
        {
          "blockId": "p10-df9c6a96db",
          "page": 10,
          "quote": "Taking the divergence and using div T = 0 and tr T = 0 gives ∆s2 = 2Tli(Tli −gδli) = 2w2. Finally,\ndifferentiating w2 and applying Cauchy–Schwarz proves both gradient estimates."
        },
        {
          "blockId": "p11-fdd2a71ec4",
          "page": 11,
          "quote": "Proposition 3.3 (Boundary jets). Suppose g = 0, p = N, and T(N, N) = −H < 0 on a smooth\nboundary portion, where N is inward. Let Greek indices be tangent to this boundary and put"
        },
        {
          "blockId": "p12-7bd615b1c8",
          "page": 12,
          "quote": "Proof. On the boundary, tangential differentiation of |p|2 = 1 gives TNα = 0, and ∇αp = Teα\nbecause g = 0. Since p = N along the boundary, this proves Tαβ = hαβ. The trace equation for\nT gives tr h = −TNN = H. In differentiating three tangential entries of T, the normal connection\nterms pair with T(N, eα) = 0. This yields the first formula in (3.10). Likewise,"
        },
        {
          "blockId": "p13-6249f48ba3",
          "page": 13,
          "quote": "We now use the unadorned notation Ω, g, p, T, B, w, H for the working component in S5 defined\nat the end of Section 2. It is either Ωo when mo = 5, or the above suspension when mo = 4. All\nboundary integrals mean integrals over the regular boundary, with the two tips omitted in the\nsuspended case. The original data retain the subscript o when needed."
        },
        {
          "blockId": "p16-6e69ee52c4",
          "page": 16,
          "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
        },
        {
          "blockId": "p16-4f79bc733f",
          "page": 16,
          "quote": "where p, ℓ∈R5 and g, w, z ∈R are otherwise free. In particular, w and z need not be positive. On\ntensor polynomials in A, p, g, δ, define Dl by the product rule and"
        },
        {
          "blockId": "p16-bae9fc97ae",
          "page": 16,
          "quote": "Proposition 4.1 (Universal inequalities). There are rational tensor polynomials f, Ujk, Klijk, Ml, Pl\nin A, p, g, δ, with the following properties.\nThey are equivariant under orthogonal changes of\ncoordinates, with f invariant. Their coefficients are specified by Table I in Appendix A, using the\ndecoding in Section 6.1. Put"
        },
        {
          "blockId": "p16-102340751e",
          "page": 16,
          "quote": "In particular, f ≥ϵ∗whenever g2 + |p|2 ≤1.\nFor every symmetric trace-free unit matrix A and vector p satisfying"
        },
        {
          "blockId": "p16-57f3a15ba4",
          "page": 16,
          "quote": "g = 0,\n|p| = 1,\nAp = λp,\nλ < 0,"
        },
        {
          "blockId": "p16-5719369278",
          "page": 16,
          "quote": "Moreover, U and K are even in p, and P = 0 when g = 0."
        },
        {
          "blockId": "p17-fad04e79c7",
          "page": 17,
          "quote": "The identities of Proposition 3.1 show that A ∈STF2(R5), |A| = 1, and X ∈STF3(R5). Moreover,\ng2 + |p|2 ≤1, so these are admissible data for the interior certificate. The homogenizing variable in\nthat certificate is set to z = 1 in every geometric application."
        }
      ],
      "occurrenceCount": 47
    },
    {
      "symbol": "h",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-0c9f90e5b9",
        "page": 3,
        "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
      },
      "occurrences": [
        {
          "blockId": "p3-0c9f90e5b9",
          "page": 3,
          "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
        },
        {
          "blockId": "p3-5ad1b21119",
          "page": 3,
          "quote": "div Z ≥|∇h|2 + 4h2 + ϵ∗w2a,\nN · Z + Hh2 ≥0,"
        },
        {
          "blockId": "p11-4388b4b044",
          "page": 11,
          "quote": "hαβ = ⟨∇αN, eβ⟩."
        },
        {
          "blockId": "p11-c696783e4a",
          "page": 11,
          "quote": "Then TNα = 0, Tαβ = hαβ, and H = tr h; in particular H is a sum of principal curvatures with the\nspecified convention, without division by m −1. If ∇∂is the boundary connection, all components\nof B are described, up to permutation of slots, by"
        },
        {
          "blockId": "p11-064ff4fb38",
          "page": 11,
          "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
        },
        {
          "blockId": "p12-7bd615b1c8",
          "page": 12,
          "quote": "Proof. On the boundary, tangential differentiation of |p|2 = 1 gives TNα = 0, and ∇αp = Teα\nbecause g = 0. Since p = N along the boundary, this proves Tαβ = hαβ. The trace equation for\nT gives tr h = −TNN = H. In differentiating three tangential entries of T, the normal connection\nterms pair with T(N, eα) = 0. This yields the first formula in (3.10). Likewise,"
        },
        {
          "blockId": "p12-e7c20dd423",
          "page": 12,
          "quote": "which proves the second formula by symmetry of B.\nFor a tangent field eα, differentiate T(N, eα) = 0 along eβ. Since ⟨N, ∇βeα⟩= −hβα, this gives"
        },
        {
          "blockId": "p18-55233111ba",
          "page": 18,
          "quote": "on the entire normalized coefficient domain. The verification in Section 6.5 obtains this bound\nusing the free algebraic data X = w = z = 0. Since f is independent of w, this permitted algebraic\nsubstitution does not impose w = 0 on the geometric data: the fields below are defined on Ω+,\nwhere w > 0.\nDefine on Ω+\nh = wap"
        },
        {
          "blockId": "p18-dba8c2d892",
          "page": 18,
          "quote": "Proposition 5.2. The scalar h and vector field Z satisfy"
        },
        {
          "blockId": "p18-9950cf5a0f",
          "page": 18,
          "quote": "div Z ≥4h2 + |∇h|2 + ϵ∗w2a\non Ω+.\n(5.7)"
        },
        {
          "blockId": "p19-d28ab51ebc",
          "page": 19,
          "quote": "Differentiating h and completing the square give, respectively,"
        },
        {
          "blockId": "p19-4886bf6683",
          "page": 19,
          "quote": "2√f Fl,\n|∇h|2 = w2a"
        },
        {
          "blockId": "p19-7492df17f5",
          "page": 19,
          "quote": "I\n\nz=1 = w−2a\ndiv Z −4h2 −|∇h|2\n+ f|ℓ−ℓ∗|2.\n(5.10)"
        },
        {
          "blockId": "p19-6d735e7032",
          "page": 19,
          "quote": "div Z −4h2 −|∇h|2 ≥ϵ∗w2a(1 + |ℓ∗|2),"
        },
        {
          "blockId": "p20-451469844c",
          "page": 20,
          "quote": "|h| ≤Cwa,\n|∇h| ≤Cwa(1 + w + |X|),\n|Z| ≤Cw2a(1 + w + |X|).\n(5.14)"
        },
        {
          "blockId": "p20-c1ad42d75d",
          "page": 20,
          "quote": "Use (5.5) and (5.9) for the estimates on h and ∇h, and use (5.6) and the boundedness of G, M, P\nfor the estimate on Z."
        },
        {
          "blockId": "p24-18023602ce",
          "page": 24,
          "quote": "h=0\nAhXlAk−1−h −kqlAk,\n(Xl)ij = Xlij,\n(6.7)"
        },
        {
          "blockId": "p24-8f3e8fb1e8",
          "page": 24,
          "quote": "For k = 0 the sums and derivatives are zero. These formulas follow by differentiating each matrix\nfactor using (4.2); cyclic invariance of trace makes all k insertion positions on a trace equal. In\ngraph form, inserting Xl on an edge of power k splits it into two paths of lengths h, k −1 −h\nending at a new X, whose last slot is the derivative index l. The negative terms from all A factors\ncombine into minus the total A degree times ql times the original graph. Each p endpoint is in turn\nreplaced by wAl· −gzδl·, and a factor gh gives hzgh−1pl. If an edge has two p endpoints, these are\ntwo product-rule terms, including their multiplicity when they coincide.\nThese operations are also sound if traces were simplified first, since"
        },
        {
          "blockId": "p25-5f06dbaca2",
          "page": 25,
          "quote": "If h, b > 0 and g2 + |p|2 ≤1, then"
        },
        {
          "blockId": "p25-138422542a",
          "page": 25,
          "quote": "|g|2h|p|2b ≤\n\nh\nh + b"
        }
      ],
      "occurrenceCount": 31
    },
    {
      "symbol": "mo",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p7-36823df07f",
        "page": 7,
        "quote": "Proposition 2.4 (Smooth original link). Every point of ∂{u > 0} \\ {0} is regular. Put mo = d −1\nand go = u|Smo. The set {go > 0} is nonempty and proper, has finitely many connected components,\nand their closures are pairwise disjoint compact smooth manifolds with boundary. On each such\ncomponent Ωo, the restriction of go is smooth up to the boundary from within Ωo and satisfies"
      },
      "occurrences": [
        {
          "blockId": "p7-36823df07f",
          "page": 7,
          "quote": "Proposition 2.4 (Smooth original link). Every point of ∂{u > 0} \\ {0} is regular. Put mo = d −1\nand go = u|Smo. The set {go > 0} is nonempty and proper, has finitely many connected components,\nand their closures are pairwise disjoint compact smooth manifolds with boundary. On each such\ncomponent Ωo, the restriction of go is smooth up to the boundary from within Ωo and satisfies"
        },
        {
          "blockId": "p7-7608cd04c9",
          "page": 7,
          "quote": "in the positive phase. If go > 0 on the entire sphere, integrating the harmonicity identity over the\nsphere would give mo\nR\ngo = 0, a contradiction. Thus its positive set is proper.\nThe conical free boundary has the radial direction tangent to it at every nonzero point: the\ncurve r 7→rx lies in that free boundary. Consequently its smooth hypersurface is transverse to the\nunit sphere, and their intersection is a smooth hypersurface of Smo. Its regular charts have the\nspherical positive phase exactly on one side. Each point of the sphere therefore has a neighborhood\nwhose intersection with {go > 0} is either connected or empty (using a small ball for points away\nfrom the free boundary). A finite subcover of the sphere shows that there are only finitely many\npositive components. If two component closures met, their common point would be a free-boundary\npoint; the connected positive side of a regular chart there would meet both components, forcing\nthem to coincide. The same charts show that each closure is a smooth compact domain. Each\ncomponent has nonempty boundary because the sphere is connected and the positive set is proper.\nSmoothness of u from its positive side gives smoothness of go on these closures. Equation (2.6)\ngives the interior equation in (2.5). On the boundary, the radial derivative of u is go = 0, so the\nEuclidean gradient is tangent to the sphere. Equation (2.2) then gives the remaining boundary\nconditions."
        },
        {
          "blockId": "p9-b613bfd01d",
          "page": 9,
          "quote": "Notation for the remaining argument.\nFix the original component Ωo ⊂Smo selected in\nLemma 2.5, where mo ∈{4, 5}. If mo = 5, set Ω= Ωo, g = go, and Σ = ∅. If mo = 4, set Ω= eΩ,\ng = eg, and Σ = {p+, p−}. In either case Ω⊂S5 is connected, and g comes from a nonnegative\none-homogeneous global minimizer in R6. The closure is smooth away from Σ, with positive phase\non one side of its regular boundary\n∂regΩ:= ∂Ω\\ Σ."
        },
        {
          "blockId": "p9-b2dcf5b6ec",
          "page": 9,
          "quote": "Let Ωo ⊂Smo, mo ∈{4, 5}, be the original component supplied by Proposition 2.4 and Lemma 2.5.\nIts closure is a compact smooth manifold with nonempty boundary. The function go is smooth up\nto that boundary, is positive in Ωo, and satisfies"
        },
        {
          "blockId": "p12-6e66e62020",
          "page": 12,
          "quote": "Proposition 3.4 (Suspended data). If mo = 4, write ρ = sin t, c = cos t and parameterize the\nregular part of the suspended component by"
        },
        {
          "blockId": "p13-6249f48ba3",
          "page": 13,
          "quote": "We now use the unadorned notation Ω, g, p, T, B, w, H for the working component in S5 defined\nat the end of Section 2. It is either Ωo when mo = 5, or the above suspension when mo = 4. All\nboundary integrals mean integrals over the regular boundary, with the two tips omitted in the\nsuspended case. The original data retain the subscript o when needed."
        },
        {
          "blockId": "p15-91cc1d2aa7",
          "page": 15,
          "quote": "R(−βψ + k−1ψ′)2 dτ, and\nR\nψψ′ = 0 by compact support.\nDivide the separated stability inequality by its positive radial factor and let k →∞. In tangent\ndimension m = 5, one has β2 = 4, proving (3.26). The same computation on an original link with\nmo = 4 would give 9/4; the working link always has m = 5.\nThe order of limits is fixed: for each angular test and each finite k, first take ε →0 in the second\nvariation, and then take k →∞. Angular cutoffs used later are removed only after (3.26) has been\nobtained for each fixed smooth test. The collar and the allowable size of ε may depend on these\nfixed supports."
        },
        {
          "blockId": "p30-5af4fd6cfe",
          "page": 30,
          "quote": "We now justify the integration of the pointwise comparison. Retain the original smooth component\nΩo ⊂Smo, with mo ∈{4, 5}, from Proposition 2.4 and Lemma 2.5. Its tensor To is not identically\nzero. Write"
        },
        {
          "blockId": "p30-c89a582f54",
          "page": 30,
          "quote": "The tensor To is smooth on the compact closure; its norm wo is continuous there and smooth where\nit is positive. The working domain Ω⊂S5 is Ωo when mo = 5, and its suspension when mo = 4.\nOn the regular part of this working domain we use the unadorned notation of Sections 3 and 5. In\nparticular, Z denotes the flux vector field, whereas Zo denotes the original Hessian zero set. Set\nMo = ∥wo∥L∞(Ωo). We fix"
        },
        {
          "blockId": "p32-788c420b9b",
          "page": 32,
          "quote": "(\nαj,\nmo = 5,\nαj(y)βj(t),\nβj(t) = ϑ(j sin t),\nmo = 4.\n(7.4)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        }
      ],
      "occurrenceCount": 12,
      "latex": "m_{o}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "nt",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p27-45ed2819cd",
        "page": 27,
        "quote": "The squared norms in Equation (6.14) are polynomial contractions. Expand and collect E =\nP\nt etM(t) by Section 6.3, discarding zero coefficients only after their signed contributions have\nbeen summed. All coefficients are rational. The denominator Q2 suffices: Gram terms have that\ndenominator, and the denominators of a, di, and all Table I coefficients divide it after multiplication\nby Q2.\nFor each retained key let nt be the least nonnegative integer satisfying"
      },
      "occurrences": [
        {
          "blockId": "p27-45ed2819cd",
          "page": 27,
          "quote": "The squared norms in Equation (6.14) are polynomial contractions. Expand and collect E =\nP\nt etM(t) by Section 6.3, discarding zero coefficients only after their signed contributions have\nbeen summed. All coefficients are rational. The denominator Q2 suffices: Gram terms have that\ndenominator, and the denominators of a, di, and all Table I coefficients divide it after multiplication\nby Q2.\nFor each retained key let nt be the least nonnegative integer satisfying"
        },
        {
          "blockId": "p27-491e7509c6",
          "page": 27,
          "quote": "n2\nt ≥1010L2\nt ,\nbLt = nt"
        }
      ],
      "occurrenceCount": 2,
      "latex": "n_{t}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "ql",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-6e69ee52c4",
        "page": 16,
        "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
      },
      "occurrences": [
        {
          "blockId": "p16-6e69ee52c4",
          "page": 16,
          "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
        },
        {
          "blockId": "p17-064052db40",
          "page": 17,
          "quote": "w ,\nql = XljkAjk.\n(5.2)"
        },
        {
          "blockId": "p17-391bd90f4b",
          "page": 17,
          "quote": "Dl(tr A) = 0,\nDl(|A|2) = 2ql −2ql = 0"
        },
        {
          "blockId": "p21-7fe613ffd3",
          "page": 21,
          "quote": "M0(200550) = g2|p|2,\nM4(000500520131)lijk = plpjAik,\nM1(000771700)l = ql."
        },
        {
          "blockId": "p24-8f3e8fb1e8",
          "page": 24,
          "quote": "For k = 0 the sums and derivatives are zero. These formulas follow by differentiating each matrix\nfactor using (4.2); cyclic invariance of trace makes all k insertion positions on a trace equal. In\ngraph form, inserting Xl on an edge of power k splits it into two paths of lengths h, k −1 −h\nending at a new X, whose last slot is the derivative index l. The negative terms from all A factors\ncombine into minus the total A degree times ql times the original graph. Each p endpoint is in turn\nreplaced by wAl· −gzδl·, and a factor gh gives hzgh−1pl. If an edge has two p endpoints, these are\ntwo product-rule terms, including their multiplicity when they coincide.\nThese operations are also sound if traces were simplified first, since"
        },
        {
          "blockId": "p24-684730476e",
          "page": 24,
          "quote": "Dl tr A = 0,\nDl tr A2 = 2XlijAij −2ql|A|2 = 0."
        },
        {
          "blockId": "p24-508d7d7d45",
          "page": 24,
          "quote": "Proof. The incidence assertions hold for each table row by its decoding. A disjoint union preserves\nthem. A permutation only renames free slots. Contracting two free slots removes their unique\nincidences and joins the remaining ends, or closes a trace; it does not change the number of slots of\nany cubic tensor. The derivative insertion described above replaces one edge by two incidences of a\nnew X and supplies its third incidence as a new free slot. Trace differentiation does the same with\na self-edge. The ql term supplies two incidences on its self-edge and the third at l. Derivatives of p\nand g supply a free derivative slot without introducing a cubic tensor. These statements prove the\nincidence assertions by induction over the operations.\nTable I has degree zero and its derivative has degree one in the indicated variables. The formula\nfor I multiplies such derivatives by one further factor, or a degree-zero coefficient by two factors.\nEach Table II row has degree one; projection, contraction, and multiplication by χ preserve that\ndegree. Squaring produces degree two. In particular no operation used here needs three copies of X.\nTermination is immediate from the finite tables, finite permutation sums, and the finite product\nrule; every contraction reduces the number of free slots by two."
        }
      ],
      "occurrenceCount": 7,
      "latex": "q_{l}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "rF",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p21-5111bc0d6f",
        "page": 21,
        "quote": "For F ∈{f, U, K, M, P}, let rF be respectively 0, 2, 4, 1, 1, with slot orders respectively empty, (j, k),\n(l, i, j, k), (l), (l). If a row of Table I consists of F, s, c, its contribution is cMrF (s)/Q; thus each\npolynomial is given by"
      },
      "occurrences": [
        {
          "blockId": "p21-5111bc0d6f",
          "page": 21,
          "quote": "For F ∈{f, U, K, M, P}, let rF be respectively 0, 2, 4, 1, 1, with slot orders respectively empty, (j, k),\n(l, i, j, k), (l), (l). If a row of Table I consists of F, s, c, its contribution is cMrF (s)/Q; thus each\npolynomial is given by"
        }
      ],
      "occurrenceCount": 1,
      "latex": "r_{F}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "s",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p2-dcac2de92c",
        "page": 2,
        "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
      },
      "occurrences": [
        {
          "blockId": "p2-dcac2de92c",
          "page": 2,
          "quote": "The upper bound is the standard dimension-reduction consequence of the critical dimension,\nand sharpness follows by taking Cartesian products of the De Silva–Jerison cone with Euclidean\nspace. We verify these deductions in Section 8.\nThe variational theory of the one-phase Bernoulli problem begins with Alt and Caffarelli’s\nexistence and regularity theory [1]. Weiss’s monotonicity formula and dimension reduction [18]\nconnect the singularities of general minimizers to homogeneous global minimizers; see also [17,\nTheorem 1.4 and Chapters 9–10]. Thus the first dimension admitting a nonflat minimizing cone\ndetermines both the threshold for singular free boundaries and the codimension bound furnished by\nthis theory. Our main result answers positively the critical-dimension question whether d∗= 7.\nA second strand of the theory passes from weak free boundaries to classical ones. Caffarelli’s\nHarnack-inequality approach established regularity for Lipschitz free boundaries [2]. De Silva\ndeveloped a different improvement-of-flatness argument allowing a nonzero right-hand side [4,\nTheorems 1.1–1.2]; this is the approach presented in the regularity chapters of [17]. Higher regularity\nhas classical antecedents in Kinderlehrer and Nirenberg [14, Theorem 2]. The version used here, for\nLipschitz viscosity solutions, is due to De Silva, Ferrari and Salsa [6, Theorem 1.1 and Corollary 1.2].\nThese results make a classification of homogeneous minimizing blowups applicable to general local\nminimizers.\nAs recalled by Jerison and Savin [13, Introduction], the dimension-three rigidity result for stable\nhomogeneous solutions with smooth cross-section is due to Caffarelli, Jerison and Kenig [3]. Jerison\nand Savin proved that stable homogeneous solutions with smooth cross-section are flat in dimensions\nat most four [13, Theorem 1.1]; their regularity consequence for minimizers [13, Corollary 1.2]\ngives d∗≥5. Their argument constructs functions of the Hessian satisfying inequalities for the\nlinearized free-boundary problem. In the other direction, De Silva and Jerison constructed a nonflat\nhomogeneous global minimizer in dimension seven [7, Theorem 1.1], proving d∗≤7. The minimizing\nproperty in that construction is with respect to all Sobolev competitors with the prescribed boundary\ntrace. Firester, Tsiamis and Wang also construct further symmetric minimizing one-phase cones,\nincluding a second example in dimension seven [11, Theorem 1.4].\nRecent work further describes cones with symmetry. Firester, Tsiamis and Wang [10, Theo-\nrem 1.1] prove that their cohomogeneity-one cones with O(n −k) × O(k) symmetry, 1 ≤k ≤n −2,\nare unstable for n ≤6 and strictly stable for n ≥7. The exclusion proved here applies to globally\nminimizing cones without a symmetry assumption.\nCone classification also differs from rigidity of arbitrary entire stable solutions. Fern´andez-Real\nand Serra prove that every global classical stable solution in dimension four is one-dimensional [8,\nTheorem 1.1]. Their result removes homogeneity in that dimension.\nFor an admissible increasing family of nonnegative boundary data, Fern´andez-Real and Yu prove\nregularity through dimension six for almost every parameter [9, Assumption 1 and Theorem 1.3].\nThis is a generic-data result: the increase is required at a fixed linear rate on the positive set of the\nsmaller boundary datum."
        },
        {
          "blockId": "p2-e66e934670",
          "page": 2,
          "quote": "The obstacle and the proof strategy.\nThe stability inequality compares the Dirichlet energy\nof a test function with a curvature term on the free boundary. The use of the harmonic Hessian\nhas a geometric precedent in Simons’s second-fundamental-form argument for minimal cones [16,\nSection 6.1], a connection developed explicitly by Jerison and Savin. A natural Bernoulli test is\na power of the Hessian norm. Jerison and Savin explain why this unweighted choice does not\ngive a uniform instability criterion in dimensions four through six: the interior and boundary\nestimates impose different restrictions on the power, and their sharp configurations need not agree\n[13, Section 3.2]. Their four-dimensional argument uses a more refined function of the Hessian"
        },
        {
          "blockId": "p3-0c9f90e5b9",
          "page": 3,
          "quote": "eigenvalues [13, Section 4].\nOur proof uses the normalized shape of the Hessian to construct both a scalar test and a separate\nvector field whose divergence controls the test’s energy. The two constructions are chosen together,\nso the interior and boundary estimates retain the same tensor data. The proof has three stages.\nFirst, suppose the first nonflat minimizing cone occurs in dimension five or six.\nBlowups\naway from its vertex reduce the dimension and are therefore flat. Regularity theory makes each\ncomponent of its spherical positive set smooth, and at least one component has nonzero Hessian. A\nfive-dimensional cone can be made independent of one extra coordinate without losing minimality.\nIts spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a\ndomain Ω⊂S5 with a smooth regular boundary and, possibly, these two tips (Section 2).\nSecond, let g be the restriction of the resulting cone to the sphere, and set T = ∇2g + g Id and\nw = |T|. Where w > 0, we construct a positive scalar h from w, the normalized tensor T/w, g, and\n∇g, and a vector field Z from the same data and ∇T/w. The target inequalities are"
        },
        {
          "blockId": "p3-a5c78f8690",
          "page": 3,
          "quote": "where a ∈(1/2, 1) is fixed, ϵ∗> 0, N points into Ω, and H = −T(N, N). Spherical stability supplies\nthe reverse integrated comparison between the test’s energy and its boundary curvature term. The\nextra positive mass ϵ∗w2a will give the contradiction. The geometry and stability inequality are\ndeveloped in Section 3. We then state the universal tensor inequality and derive the two geometric\ncomparisons; the full exact verification follows in Section 6.\nTwo cancellations make this construction possible. A Laplacian contraction and an antisymmetric\ncurvature contraction eliminate fourth derivatives from div Z. At the boundary, reflection in the\nnormal direction removes the derivative components not determined by the boundary conditions.\nThe remaining inequalities involve only finitely many normalized tensors. We prove them by weighted\nsums of tensor squares and exact rational remainder bounds, with all coefficients and contraction\nconventions supplied in Appendix A.\nFinally, neither the normalized Hessian nor its derivatives are defined at w = 0, and suspension\nintroduces singular tips. These sets cannot be discarded without an estimate. A regularized Bochner\nidentity gives an integrable weight controlling the derivatives near every Hessian zero. The explicit\nsuspension formulas give an integrable power of the distance parameter at both tips. Smooth\nproduct cutoffs therefore remove both obstructions and make all error terms vanish. Section 7\ncompletes the stability contradiction."
        },
        {
          "blockId": "p6-35c36a36fe",
          "page": 6,
          "quote": "Lemma 2.3 (Minimality of a translation-invariant factor). Suppose U(y, s) = v(y) is a global\nminimizer in Rk × R, where v ≥0 belongs to H1\nloc(Rk). Then v is a global minimizer in Rk."
        },
        {
          "blockId": "p6-2f894589b6",
          "page": 6,
          "quote": "VL(y, s) = U(y, s) + ηL(s)ψ(y)."
        },
        {
          "blockId": "p7-d328f4a8f3",
          "page": 7,
          "quote": "same trace and energy b2R2/(R −t) + (R −t). Its derivative at the minimizing parameter t = 0 is\nb2 −1 = 0, so b = 1. This proves flatness in dimension one directly.\nThe lower-dimensional classification of Jerison–Savin [13, Theorem 1.1 and Corollary 1.2] and\nthe dimension-seven minimizing example of De Silva–Jerison [7, Theorem 1.1] give 5 ≤d∗≤7. De\nSilva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors.\nFor the exclusion argument, suppose from now on that"
        },
        {
          "blockId": "p7-df0b192be6",
          "page": 7,
          "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
        },
        {
          "blockId": "p8-ffe8311d4a",
          "page": 8,
          "quote": "Differentiating this formula, or using Euler’s identity for the radial entries, gives"
        },
        {
          "blockId": "p8-05cf6278e3",
          "page": 8,
          "quote": "Proposition 2.6 (Suspension preserves global minimality). If u ≥0 is a global minimizer in Rk,\nthen eu(x, s) = u(x) is a global minimizer in Rk+1. If u is the cone in (2.4) with d = 5, the singular\nfree-boundary set of eu is exactly {0}×R. For the component chosen in Lemma 2.5, the corresponding\npositive component on S5 is"
        },
        {
          "blockId": "p8-fdfcec216b",
          "page": 8,
          "quote": "Proof. Fix a ball B = BR((xc, sc)) ⊂Rk+1 and a nonnegative V ∈H1(B) with V −eu ∈H1\n0(B).\nExtend that difference by zero to F ∈H1(Rk+1). For |s −sc| < R, let"
        },
        {
          "blockId": "p8-c3e3c2d6dd",
          "page": 8,
          "quote": "R2−(s−sc)2(xc)"
        },
        {
          "blockId": "p8-26bd66ccb6",
          "page": 8,
          "quote": "be the horizontal slice. For almost every such s, F(·, s)|Bs ∈H1\n0(Bs). To see the zero-boundary\nassertion without a trace regularity assumption on V , approximate F in H1(Rk+1) by the zero\nextensions of functions in C∞\nc (B). After passing to a subsequence, Fubini’s Theorem gives convergence\nof their slices to F(·, s) in H1(Rk) for almost every s. Each approximating slice lies in C∞\nc (Bs), so\nthe assertion follows by closure.\nFor almost every s, Vs = V (·, s) is therefore an admissible nonnegative competitor for u in Bs.\nIntegrating its minimizing inequality and using the nonnegativity of the additional derivative energy,\nwe obtain"
        },
        {
          "blockId": "p9-6271b34b61",
          "page": 9,
          "quote": "shows that all its off-axis points are regular: their positive phase is the product of a smooth one-sided\nregular chart for u and an interval. At any axis point z0 = (0, s0), every rescaling is exactly"
        },
        {
          "blockId": "p9-a95f4394b0",
          "page": 9,
          "quote": "eu(z0 + r(x, s))"
        },
        {
          "blockId": "p9-251ecacf7c",
          "page": 9,
          "quote": "r\n= eu(x, s)."
        },
        {
          "blockId": "p9-6c6e1e3c68",
          "page": 9,
          "quote": "All spherical boundary values below are taken from the positive side. Tests on Ωwill be smooth and\nsupported away from Σ. Every compact portion of the selected Euclidean cone’s closure that avoids\nthe origin and, in the suspended case, the axis is separated from all other positive components. This\nfollows from the disjoint compact original closures, and in the suspension from the same fact on any\nclosed subinterval of (0, π). Consequently the local variations on this component can be made as\nvariations of the global minimizer. No minimizing property of the selected component extended by\nzero is needed."
        },
        {
          "blockId": "p10-ae1fb87ac5",
          "page": 10,
          "quote": "p = ∇g,\nT = ∇2g + g Id,\nBijk = ∇iTjk,\nw = |T|,\ns2 = g2 + |p|2.\n(3.1)"
        },
        {
          "blockId": "p10-4d5809125b",
          "page": 10,
          "quote": "The notation s2 denotes the displayed smooth function, including at points where it vanishes. If\nu(ry) = rg(y) is the homogeneous extension, then in the radial and tangential orthonormal frame"
        },
        {
          "blockId": "p10-0dca6cc3ad",
          "page": 10,
          "quote": "Indeed, Euler’s identity gives D2u y = 0, and differentiation of ∇Rm+1u = gy + p along a tangent\nvector gives the tangential block ∇2g + g Id at radius one. Homogeneity then gives the factor r−1.\nThe same gradient formula identifies s2 with |∇Rm+1u|2 on the unit sphere."
        }
      ],
      "occurrenceCount": 42
    },
    {
      "symbol": "st",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p34-a6c4f669ee",
        "page": 34,
        "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
      },
      "occurrences": [
        {
          "blockId": "p22-0d90457bbb",
          "page": 22,
          "quote": "t=1\nctcMr(st)."
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-1c9ed1e5dd",
          "page": 34,
          "quote": "st\n,\n|B(t, y)|2 = |Bo(y)|2 + 3 cos2(t)wo(y)2"
        },
        {
          "blockId": "p34-56a963e041",
          "page": 34,
          "quote": "st\n≤"
        },
        {
          "blockId": "p34-be5ed89957",
          "page": 34,
          "quote": "3\nst\n(1 + bo)."
        },
        {
          "blockId": "p34-2a8be806b3",
          "page": 34,
          "quote": "st\n(1 + bo),\n1 + w + |X| ≤Ks"
        },
        {
          "blockId": "p34-819d186303",
          "page": 34,
          "quote": "st\n(1 + bo),\nKs = 1 + Mo +\n√"
        }
      ],
      "occurrenceCount": 7,
      "latex": "s_{t}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "u",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p1-1398459c75",
        "page": 1,
        "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
      },
      "occurrences": [
        {
          "blockId": "p1-1ef6120946",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
        },
        {
          "blockId": "p1-ba617d637b",
          "page": 1,
          "quote": "u(x) = (x · e)+,\n|e| = 1."
        },
        {
          "blockId": "p1-1398459c75",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(D) is a local minimizer in an open set D ⊂Rn if it satisfies the same\ncomparison on every ball compactly contained in D. At an interior free-boundary point x0, a blowup\nis a locally uniform limit of u(x0 + rx)/r along radii tending to zero. A point is regular if one such\nlimit is a flat solution, and singular otherwise. Write Sing(u) for the set of singular points in D."
        },
        {
          "blockId": "p1-63eea2c5f4",
          "page": 1,
          "quote": "Corollary 1.2 (Sharp singular-set bound). Let n ≥2, let D ⊂Rn be open, and let u ∈H1\nloc(D) be\na nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is\nsmooth outside Sing(u). This set is empty when n ≤6, is locally finite when n = 7, and satisfies"
        },
        {
          "blockId": "p1-43352643db",
          "page": 1,
          "quote": "dimH Sing(u) ≤n −7\n(n ≥7)."
        },
        {
          "blockId": "p4-67ac06f5ce",
          "page": 4,
          "quote": "The ballwise convention also gives comparison on every bounded open set O ⋐Rd: extend an H1\n0(O)\ncompetitor difference by zero into a ball containing O, apply minimality there, and cancel the energy\noutside O. Conversely, comparison on bounded open sets includes comparison on balls. It is even\nenough to have comparison against arbitrary admissible competitors in every origin-centered ball,\nby the same extension argument. There is no restriction to homogeneous competitors.\nThe convention that competitors are nonnegative makes no difference to the standard one-phase\nresults. If v −u ∈H1\n0(O) and u ≥0, then v+ −u ∈H1\n0(O) and\nZ"
        },
        {
          "blockId": "p4-97da25c832",
          "page": 4,
          "quote": "For completeness, the assertion about H1\n0(O) follows by approximating v −u in H1 by compactly\nsupported smooth functions and applying the continuity of positive-part truncation in H1; the\ntruncated differences in the approximation have compact support in O."
        },
        {
          "blockId": "p4-5aee8e3401",
          "page": 4,
          "quote": "Proposition 2.1 (Regularity and blowups of minimizers). Let d ≥2 and let u ≥0 be a local\nminimizer of the one-phase energy with volume coefficient 1 in an open set D ⊂Rd."
        },
        {
          "blockId": "p4-87d718bcb3",
          "page": 4,
          "quote": "(i) There is a locally Lipschitz representative of u, and it is harmonic in {u > 0}."
        },
        {
          "blockId": "p4-cd834ef0a2",
          "page": 4,
          "quote": "(ii) If x0 ∈D ∩∂{u > 0}, every sequence rj ↓0 has a subsequence for which"
        },
        {
          "blockId": "p4-6b51763d0e",
          "page": 4,
          "quote": "ux0,rj(y) := u(x0 + rjy)"
        },
        {
          "blockId": "p4-d4e305255d",
          "page": 4,
          "quote": "(iii) If one such limit equals (y · e)+ with |e| = 1, then near x0 the free boundary is a smooth\nhypersurface and {u > 0} is exactly one of its sides. The function u is smooth up to that\nhypersurface from the positive side, and"
        },
        {
          "blockId": "p4-e807330307",
          "page": 4,
          "quote": "u = 0,\n∇u = NE\non the free boundary,\n(2.2)"
        },
        {
          "blockId": "p4-377923367a",
          "page": 4,
          "quote": "where NE is the Euclidean unit normal pointing into {u > 0}."
        },
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p5-9ef6667a7a",
          "page": 5,
          "quote": "As in Section 1, a free-boundary point is regular if it has a half-space blowup and singular\notherwise. Proposition 2.1 explains the geometric meaning of regularity. Conversely, a smooth\none-sided free boundary with (2.2) has a half-space blowup by its first-order Taylor expansion.\nWe henceforth use the continuous representative of a minimizer. If it is one-homogeneous almost\neverywhere, it is one-homogeneous everywhere: for each fixed r > 0, both sides of u(rx) = ru(x)\nare continuous and agree almost everywhere."
        },
        {
          "blockId": "p5-7b73106fd2",
          "page": 5,
          "quote": "Lemma 2.2 (Translation invariance of an off-origin blowup). Let u be a one-homogeneous global\nminimizer, let 0 ̸= x0 ∈∂{u > 0}, and let U be any limit in (2.1). Then"
        },
        {
          "blockId": "p5-a73e49f286",
          "page": 5,
          "quote": "ur(y + tx0) = u\n\n(1 + rt)(x0 + r′y)\n"
        },
        {
          "blockId": "p6-474378faca",
          "page": 6,
          "quote": "Let L be a Lipschitz constant for u on a fixed neighborhood of x0. Since u(x0) = 0, for |y| ≤M\nand small r,"
        }
      ],
      "occurrenceCount": 44
    },
    {
      "symbol": "ur",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-19acdad6af",
        "page": 5,
        "quote": "Proof. Write ur = ux0,r. For fixed t ∈R and small r > 0, set r′ = r/(1 + rt) > 0. Homogeneity\ngives the exact identity"
      },
      "occurrences": [
        {
          "blockId": "p5-19acdad6af",
          "page": 5,
          "quote": "Proof. Write ur = ux0,r. For fixed t ∈R and small r > 0, set r′ = r/(1 + rt) > 0. Homogeneity\ngives the exact identity"
        },
        {
          "blockId": "p5-a73e49f286",
          "page": 5,
          "quote": "ur(y + tx0) = u\n\n(1 + rt)(x0 + r′y)\n"
        },
        {
          "blockId": "p5-16a211c14c",
          "page": 5,
          "quote": "r\n= ur′(y).\n(2.3)"
        },
        {
          "blockId": "p6-909e57afa6",
          "page": 6,
          "quote": "|ur′(y) −ur(y)| ≤L|r −r′|M"
        }
      ],
      "occurrenceCount": 4,
      "latex": "u_{r}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "v",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p4-67ac06f5ce",
        "page": 4,
        "quote": "The ballwise convention also gives comparison on every bounded open set O ⋐Rd: extend an H1\n0(O)\ncompetitor difference by zero into a ball containing O, apply minimality there, and cancel the energy\noutside O. Conversely, comparison on bounded open sets includes comparison on balls. It is even\nenough to have comparison against arbitrary admissible competitors in every origin-centered ball,\nby the same extension argument. There is no restriction to homogeneous competitors.\nThe convention that competitors are nonnegative makes no difference to the standard one-phase\nresults. If v −u ∈H1\n0(O) and u ≥0, then v+ −u ∈H1\n0(O) and\nZ"
      },
      "occurrences": [
        {
          "blockId": "p1-15b1e52200",
          "page": 1,
          "quote": "For a nonnegative function v ∈H1(B), where B ⊂Rd is a ball, the one-phase Bernoulli energy is"
        },
        {
          "blockId": "p1-add8165034",
          "page": 1,
          "quote": "J(v; B) =\nZ"
        },
        {
          "blockId": "p1-7458d4594f",
          "page": 1,
          "quote": "\n|∇v|2 + 1{v>0}\n\ndx.\n(1.1)"
        },
        {
          "blockId": "p1-1ef6120946",
          "page": 1,
          "quote": "A nonnegative u ∈H1\nloc(Rd) is a global minimizer if J(u; B) ≤J(v; B) for every ball B and every\nnonnegative v ∈H1(B) with v −u ∈H1\n0(B). The competitors in these comparisons are arbitrary\nSobolev functions with the prescribed trace. A function is one-homogeneous if u(rx) = ru(x) for\nevery r > 0, almost everywhere in x. The nonzero flat solutions are"
        },
        {
          "blockId": "p4-67ac06f5ce",
          "page": 4,
          "quote": "The ballwise convention also gives comparison on every bounded open set O ⋐Rd: extend an H1\n0(O)\ncompetitor difference by zero into a ball containing O, apply minimality there, and cancel the energy\noutside O. Conversely, comparison on bounded open sets includes comparison on balls. It is even\nenough to have comparison against arbitrary admissible competitors in every origin-centered ball,\nby the same extension argument. There is no restriction to homogeneous competitors.\nThe convention that competitors are nonnegative makes no difference to the standard one-phase\nresults. If v −u ∈H1\n0(O) and u ≥0, then v+ −u ∈H1\n0(O) and\nZ"
        },
        {
          "blockId": "p4-83de7802d7",
          "page": 4,
          "quote": "\n|∇v|2 + 1{v>0}\n\n."
        },
        {
          "blockId": "p4-6ded8d5ea2",
          "page": 4,
          "quote": "\n|∇v+|2 + 1{v+>0}\n\n≤\nZ"
        },
        {
          "blockId": "p4-97da25c832",
          "page": 4,
          "quote": "For completeness, the assertion about H1\n0(O) follows by approximating v −u in H1 by compactly\nsupported smooth functions and applying the continuity of positive-part truncation in H1; the\ntruncated differences in the approximation have compact support in O."
        },
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p6-35c36a36fe",
          "page": 6,
          "quote": "Lemma 2.3 (Minimality of a translation-invariant factor). Suppose U(y, s) = v(y) is a global\nminimizer in Rk × R, where v ≥0 belongs to H1\nloc(Rk). Then v is a global minimizer in Rk."
        },
        {
          "blockId": "p6-5a86e5b19a",
          "page": 6,
          "quote": "Proof. Suppose that a ball B ⊂Rk admits a nonnegative q ∈H1(B) with q −v ∈H1\n0(B) and"
        },
        {
          "blockId": "p6-e4883ffbed",
          "page": 6,
          "quote": "J(v; B) −J(q; B) = δ > 0."
        },
        {
          "blockId": "p6-9d760e4b18",
          "page": 6,
          "quote": "Let ψ be the zero extension of q −v. For L > 1, choose ηL : R →[0, 1] equal to 1 on [−L, L], equal\nto 0 outside [−L −1, L + 1], and linear on each intervening interval. Define"
        },
        {
          "blockId": "p6-0789ceed61",
          "page": 6,
          "quote": "This function is nonnegative: on B × R it is the convex combination (1 −ηL)v + ηLq. Its difference\nfrom U belongs to H1 and has compact support in a sufficiently large ball BL ⊂Rk+1, so it belongs\nto H1\n0(BL).\nThe energy difference on B × [−L, L] is −2Lδ. On either transition cylinder, convexity gives"
        },
        {
          "blockId": "p6-40fb45e6d6",
          "page": 6,
          "quote": "|∇yVL|2 ≤(1 −ηL)|∇v|2 + ηL|∇q|2,\n|∂sVL|2 ≤|q −v|2."
        },
        {
          "blockId": "p6-a280a08341",
          "page": 6,
          "quote": "\n|∇q|2 + |q −v|2 + 1\n\n,"
        },
        {
          "blockId": "p7-df0b192be6",
          "page": 7,
          "quote": "Proof. At a nonzero free-boundary point x0, choose a blowup U from Proposition 2.1. By Lemma 2.2,\nin orthogonal coordinates Rd = x⊥\n0 × R it has the form U(y, s) = v(y). The factor v is locally\nLipschitz, nonnegative, nonzero and one-homogeneous.\nLemma 2.3 shows that it is a global\nminimizer. Since its dimension is d −1 < d∗, it is flat. Therefore U itself is a half-space solution,\nand Proposition 2.1 makes x0 regular.\nThe spherical positive set is nonempty by nontriviality. The polar-coordinate formula for the\nEuclidean Laplacian of u(ry) = rgo(y) gives"
        },
        {
          "blockId": "p8-81876620ea",
          "page": 8,
          "quote": "D2u(ry)[y, ·] = 0,\nD2u(ry)[v1, v2] = r−1To(y)[v1, v2]\n(v1, v2 ⊥y).\n(2.8)"
        },
        {
          "blockId": "p15-e7b7ce4464",
          "page": 15,
          "quote": "Corollary 3.6 (Spherical stability). For every real function v smooth up to the regular boundary of\nthe working component Ω⊂S5, with compact support in its closure away from the tips if present,\nZ"
        },
        {
          "blockId": "p15-9bcb9d1c60",
          "page": 15,
          "quote": "\n|∇v|2 + 4v2\ndV.\n(3.26)"
        }
      ],
      "occurrenceCount": 37
    },
    {
      "symbol": "vj",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p32-0405f3618c",
        "page": 32,
        "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
      },
      "occurrences": [
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p33-fbe7c4bfd3",
          "page": 33,
          "quote": "Corollary 3.6 applies to vj and bounds the boundary integral by\nZ"
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        }
      ],
      "occurrenceCount": 3,
      "latex": "v_{j}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "µ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p25-4e4810cb21",
        "page": 25,
        "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
      },
      "occurrences": [
        {
          "blockId": "p25-4e4810cb21",
          "page": 25,
          "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "Γ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p22-4ec0cc74d9",
        "page": 22,
        "quote": "Here Π is specified below by r and kind. There are nineteen such blocks with a total of 529 monomial\nrows. Every row has degree one in (w, z, ℓ, X), assigning degree one to each component of these\nvariables and degree zero to A, p, g. Hence Γ has degree two in these variables.\nThe two remaining blocks have rank −1 and fourteen rows each. In these blocks the four digits\nof a row word are an exponent β = (β1, . . . , β4), encoding xβ for x ∈R4. Both coefficient matrices\nhave five columns. Set"
      },
      "occurrences": [
        {
          "blockId": "p22-dbfc2410c7",
          "page": 22,
          "quote": "Its contribution to Γ is"
        },
        {
          "blockId": "p22-4ec0cc74d9",
          "page": 22,
          "quote": "Here Π is specified below by r and kind. There are nineteen such blocks with a total of 529 monomial\nrows. Every row has degree one in (w, z, ℓ, X), assigning degree one to each component of these\nvariables and degree zero to A, p, g. Hence Γ has degree two in these variables.\nThe two remaining blocks have rank −1 and fourteen rows each. In these blocks the four digits\nof a row word are an exponent β = (β1, . . . , β4), encoding xβ for x ∈R4. Both coefficient matrices\nhave five columns. Set"
        },
        {
          "blockId": "p23-0a22271950",
          "page": 23,
          "quote": "Lemma 6.1. All these operators are self-adjoint and positive semidefinite on their entire tensor\nspaces. In particular, Γ ≥0 whenever χ ≥0."
        },
        {
          "blockId": "p24-a934f8e471",
          "page": 24,
          "quote": "Lemma 6.2 (Closure of the graph calculation). The preceding rules evaluate all terms of I and Γ\nby a finite sum of scalar graphs. In every intermediate graph a free slot has exactly one incidence,\neach cubic tensor has exactly three incidences, and there are at most two cubic tensors. Every\nresulting scalar graph has degree two in (w, z, ℓ, X)."
        },
        {
          "blockId": "p27-02eb384e55",
          "page": 27,
          "quote": "4, 1\n\n,\nE = Γ +"
        },
        {
          "blockId": "p28-9c043f5e74",
          "page": 28,
          "quote": "Use Γ ≥0 in (6.14) and then (6.17). It follows that I is at least the sum of y2\ni times the four\npositive slacks in (6.20). In particular the coefficients of z2 and |ℓ|2 are at least ϵ∗, proving (4.3).\nTo obtain the stated positivity of f, keep A, p, g fixed, set X = w = z = 0, and choose |ℓ| = 1. Then\nq = 0, every D term is zero, and I = f."
        },
        {
          "blockId": "p36-ab20d66874",
          "page": 36,
          "quote": "In Table I the first field specifies the polynomial, the second its monomial, and the third its\ncoefficient numerator. The free-slot orders are Ujk, Klijk, Ml, and Pl, while f has rank zero. The\nnumbers of rows for (f, U, M, P, K) are (34, 57, 74, 32, 34), for a total of 231.\nEach line of Table II beginning with @ starts a separate block and gives its rank, projection\nlabel, and multiplier mode. For each coefficient column, sum the monomials with the coefficients\nin that column divided by Q to obtain a tensor Jc. For nonnegative rank the block contributes\nρ P\nc⟨Jc, ΠJc⟩to Γ, where the modes I and B mean ρ = 1 and ρ = χ = 1 −|p|2 −g2, respectively.\nThe labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not\ninstruct any further normalization of the coefficients. Identical headers in different positions still\ndefine distinct blocks.\nFor rank −1, the four code digits instead give the four individual exponents of x1, x2, x3, x4. The\nblock contributes to Λ the average over all 24 permutations of ρ P\nc J2\nc . Here I means ρ = 1, while H\nmeans ρ = P\ni xi = −λ. The averaging and boundary normalization R = P\ni x2\ni + (P\ni xi)2 = 1 are\nthose of Section 6.2. Table II contains 21 blocks: 19 interior blocks with 529 rows and two boundary\nblocks with 14 rows each, hence 557 coefficient rows in total. Each block retains its own number of\ncoefficient columns."
        }
      ],
      "occurrenceCount": 7
    },
    {
      "symbol": "Λ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p29-703434ff94",
        "page": 29,
        "quote": "Expand (6.24) and sum the coefficients of all exponent vectors whose nondecreasing rearrangement\nis β. Their sum is the coefficient of mβ after averaging: indeed Av(xα) = mβ for every member α\nof that orbit, including when exponents repeat. Thus sorting exponents and summing coefficients\nintroduces no extra factor of 24. Since S and the constant are symmetric and Λ was defined by\naveraging, the resulting averaged polynomial still equals Eb on R = 1.\nPrecisely, using the row-pair expansion of Section 6.3 for the two rank-−1 blocks, subtracting\nthe boundary contributions from Table I with denominator Q, and adding the homogenized constant\n1/4, one obtains\nEb(x) =\nX"
      },
      "occurrences": [
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p22-5b933e18eb",
          "page": 22,
          "quote": "Λ(x) =\nX"
        },
        {
          "blockId": "p22-433f847b38",
          "page": 22,
          "quote": "Consequently Λ ≥0 when P xi ≥0."
        },
        {
          "blockId": "p29-4d4cefc91e",
          "page": 29,
          "quote": "Eb = Λ + 1"
        },
        {
          "blockId": "p29-703434ff94",
          "page": 29,
          "quote": "Expand (6.24) and sum the coefficients of all exponent vectors whose nondecreasing rearrangement\nis β. Their sum is the coefficient of mβ after averaging: indeed Av(xα) = mβ for every member α\nof that orbit, including when exponents repeat. Thus sorting exponents and summing coefficients\nintroduces no extra factor of 24. Since S and the constant are symmetric and Λ was defined by\naveraging, the resulting averaged polynomial still equals Eb on R = 1.\nPrecisely, using the row-pair expansion of Section 6.3 for the two rank-−1 blocks, subtracting\nthe boundary contributions from Table I with denominator Q, and adding the homogenized constant\n1/4, one obtains\nEb(x) =\nX"
        },
        {
          "blockId": "p30-d103829520",
          "page": 30,
          "quote": "S = Λ + 1"
        },
        {
          "blockId": "p36-ab20d66874",
          "page": 36,
          "quote": "In Table I the first field specifies the polynomial, the second its monomial, and the third its\ncoefficient numerator. The free-slot orders are Ujk, Klijk, Ml, and Pl, while f has rank zero. The\nnumbers of rows for (f, U, M, P, K) are (34, 57, 74, 32, 34), for a total of 231.\nEach line of Table II beginning with @ starts a separate block and gives its rank, projection\nlabel, and multiplier mode. For each coefficient column, sum the monomials with the coefficients\nin that column divided by Q to obtain a tensor Jc. For nonnegative rank the block contributes\nρ P\nc⟨Jc, ΠJc⟩to Γ, where the modes I and B mean ρ = 1 and ρ = χ = 1 −|p|2 −g2, respectively.\nThe labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not\ninstruct any further normalization of the coefficients. Identical headers in different positions still\ndefine distinct blocks.\nFor rank −1, the four code digits instead give the four individual exponents of x1, x2, x3, x4. The\nblock contributes to Λ the average over all 24 permutations of ρ P\nc J2\nc . Here I means ρ = 1, while H\nmeans ρ = P\ni xi = −λ. The averaging and boundary normalization R = P\ni x2\ni + (P\ni xi)2 = 1 are\nthose of Section 6.2. Table II contains 21 blocks: 19 interior blocks with 529 rows and two boundary\nblocks with 14 rows each, hence 557 coefficient rows in total. Each block retains its own number of\ncoefficient columns."
        }
      ],
      "occurrenceCount": 7
    },
    {
      "symbol": "Π",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p22-4ec0cc74d9",
        "page": 22,
        "quote": "Here Π is specified below by r and kind. There are nineteen such blocks with a total of 529 monomial\nrows. Every row has degree one in (w, z, ℓ, X), assigning degree one to each component of these\nvariables and degree zero to A, p, g. Hence Γ has degree two in these variables.\nThe two remaining blocks have rank −1 and fourteen rows each. In these blocks the four digits\nof a row word are an exponent β = (β1, . . . , β4), encoding xβ for x ∈R4. Both coefficient matrices\nhave five columns. Set"
      },
      "occurrences": [
        {
          "blockId": "p22-4ec0cc74d9",
          "page": 22,
          "quote": "Here Π is specified below by r and kind. There are nineteen such blocks with a total of 529 monomial\nrows. Every row has degree one in (w, z, ℓ, X), assigning degree one to each component of these\nvariables and degree zero to A, p, g. Hence Γ has degree two in these variables.\nThe two remaining blocks have rank −1 and fourteen rows each. In these blocks the four digits\nof a row word are an exponent β = (β1, . . . , β4), encoding xβ for x ∈R4. Both coefficient matrices\nhave five columns. Set"
        },
        {
          "blockId": "p22-070d569830",
          "page": 22,
          "quote": "For an arbitrary rank-r tensor F, let F π denote slot permutation by π. We use the averaged\nsymmetrization Sym F = (r!)−1 P\nπ∈Sr F π. The operator Trab contracts slots a, b with the Euclidean\nmetric, leaving the remaining slots in their original order. The label none means Π = Id. The label\nanti means\nΠF =\nX"
        },
        {
          "blockId": "p25-616b2bfb28",
          "page": 25,
          "quote": "and its boundary contribution is 2λ −tr A3. To expand a Gram block, sum over unordered pairs\nof rows s, t: the coefficient is P\nc cscctc/Q2, doubled when s ̸= t, and the graph polynomial is the\ncontraction of row t with Π applied to row s, multiplied by ρ. This uses self-adjointness of Π and\naccounts for every term of (6.2) exactly once."
        }
      ],
      "occurrenceCount": 3
    },
    {
      "symbol": "Σ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p9-b613bfd01d",
        "page": 9,
        "quote": "Notation for the remaining argument.\nFix the original component Ωo ⊂Smo selected in\nLemma 2.5, where mo ∈{4, 5}. If mo = 5, set Ω= Ωo, g = go, and Σ = ∅. If mo = 4, set Ω= eΩ,\ng = eg, and Σ = {p+, p−}. In either case Ω⊂S5 is connected, and g comes from a nonnegative\none-homogeneous global minimizer in R6. The closure is smooth away from Σ, with positive phase\non one side of its regular boundary\n∂regΩ:= ∂Ω\\ Σ."
      },
      "occurrences": [
        {
          "blockId": "p9-b613bfd01d",
          "page": 9,
          "quote": "Notation for the remaining argument.\nFix the original component Ωo ⊂Smo selected in\nLemma 2.5, where mo ∈{4, 5}. If mo = 5, set Ω= Ωo, g = go, and Σ = ∅. If mo = 4, set Ω= eΩ,\ng = eg, and Σ = {p+, p−}. In either case Ω⊂S5 is connected, and g comes from a nonnegative\none-homogeneous global minimizer in R6. The closure is smooth away from Σ, with positive phase\non one side of its regular boundary\n∂regΩ:= ∂Ω\\ Σ."
        },
        {
          "blockId": "p9-6c6e1e3c68",
          "page": 9,
          "quote": "All spherical boundary values below are taken from the positive side. Tests on Ωwill be smooth and\nsupported away from Σ. Every compact portion of the selected Euclidean cone’s closure that avoids\nthe origin and, in the suspended case, the axis is separated from all other positive components. This\nfollows from the disjoint compact original closures, and in the suspension from the same fact on any\nclosed subinterval of (0, π). Consequently the local variations on this component can be made as\nvariations of the global minimizer. No minimizing property of the selected component extended by\nzero is needed."
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "Φ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p32-d5aa0817e2",
        "page": 32,
        "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
      },
      "occurrences": [
        {
          "blockId": "p8-7722e556c0",
          "page": 8,
          "quote": "eΩ=\n\nΦ(t, y) := (sin t y, cos t) : 0 < t < π, y ∈Ωo\n\t\n,\neg(Φ(t, y)) = sin t go(y).\n(2.9)"
        },
        {
          "blockId": "p8-9d96cf4b31",
          "page": 8,
          "quote": "Its boundary consists of the smooth side Φ((0, π) × ∂Ωo) and the two singular tips p+ = (0, 1) and\np−= (0, −1)."
        },
        {
          "blockId": "p9-6632f13d2e",
          "page": 9,
          "quote": "This is nonflat, so no axis point is regular. The axis meets S5 only in p+ and p−.\nAway from these two points, Φ is a smooth product coordinate map from (0, π) × S4 to\nS5 \\ {p+, p−}. Positivity in these coordinates is precisely go(y) > 0, so connected components are\nas in (2.9). Since Ωo is compact, its suspended closure is obtained by adding the two endpoints of\nthe interval, with each endpoint collapsed to its tip. Its remaining boundary is exactly the stated\nsmooth side."
        },
        {
          "blockId": "p32-d5aa0817e2",
          "page": 32,
          "quote": "Figure 1: The product cutoff on a suspended component. The horizontal variable is the suspension\nangle in Φ(t, y) = (sin t y, cos t); the two end slices collapse to the tips p+ and p−. The vertical\nvariable records the value of wo(y), not a coordinate on Ωo. Gray regions have ξj = 0, and the blue\nregion has ξj = 1; the intervening bands contain the smooth transitions. The scales are schematic,\nand no shape or regularity of the Hessian zero set is represented."
        }
      ],
      "occurrenceCount": 4
    },
    {
      "symbol": "α",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "occurrences": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p11-d8ff12d046",
          "page": 11,
          "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
        },
        {
          "blockId": "p12-e7c20dd423",
          "page": 12,
          "quote": "which proves the second formula by symmetry of B.\nFor a tangent field eα, differentiate T(N, eα) = 0 along eβ. Since ⟨N, ∇βeα⟩= −hβα, this gives"
        },
        {
          "blockId": "p12-84d5476660",
          "page": 12,
          "quote": "0 = BβNα + (T 2)βα −TNNTβα."
        },
        {
          "blockId": "p12-ecab27bc42",
          "page": 12,
          "quote": "α\nBNαα = TNN\nX"
        },
        {
          "blockId": "p12-80f71c8f3c",
          "page": 12,
          "quote": "α\nTαα −\nX"
        },
        {
          "blockId": "p12-b900895217",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −w2."
        },
        {
          "blockId": "p12-b912b80307",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −T 2\nNN −\nX"
        },
        {
          "blockId": "p12-4584ce2868",
          "page": 12,
          "quote": "The zero trace of B then gives BNNN = w2.\nNormalization gives S0p = 0 and tr S0 = −|A|2 = −1. Consequently Vijj = pi tr S0 + 2(S0p)i +\npi = 0. In an orthonormal frame with first vector p = N, one has VNαβ = (S0)αβ and VNNN = 1.\nThese are exactly the normalized identities (3.11), and VNjk = S0,jk + pjpk. Finally, T has only its\nnormal-normal and tangential-tangential blocks, so the odd-normal formulas account for the entire\ncontraction TjkBNjk. Thus"
        },
        {
          "blockId": "p12-b9af30c473",
          "page": 12,
          "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
        },
        {
          "blockId": "p12-fe39066800",
          "page": 12,
          "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
        },
        {
          "blockId": "p13-1921e32005",
          "page": 13,
          "quote": "∇E0E0 = ∇E0Eα = 0,\n∇EαE0 = (c/ρ)Eα,\n∇EαEβ = −(c/ρ)δαβE0."
        },
        {
          "blockId": "p13-13d5fe94da",
          "page": 13,
          "quote": "(∇2eg)00 = −ρgo,\n(∇2eg)0α = 0,\n(∇2eg)αβ = ρ−1(∇2\nogo)αβ + (c2/ρ)goδαβ."
        },
        {
          "blockId": "p20-044918f5ba",
          "page": 20,
          "quote": "For tangent indices α, β, the boundary jet identities give"
        },
        {
          "blockId": "p20-b941ebff60",
          "page": 20,
          "quote": "w2\n= λAαβ −(A2)αβ = (S0)αβ,\nBNNN"
        },
        {
          "blockId": "p25-2eaaddba0c",
          "page": 25,
          "quote": "Proof. Let the eigenvalues of A be α1, . . . , α5. For each i,"
        },
        {
          "blockId": "p25-435108ca7b",
          "page": 25,
          "quote": "α2\ni ="
        },
        {
          "blockId": "p25-338a35ec9b",
          "page": 25,
          "quote": "j̸=i\nα2\nj = 4(1 −α2\ni )."
        },
        {
          "blockId": "p25-4e4810cb21",
          "page": 25,
          "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
        },
        {
          "blockId": "p29-2c0a52f30e",
          "page": 29,
          "quote": "No sign or distinctness condition is imposed on the xi; the normal eigenvalue condition is exactly\nP\ni xi > 0. In these coordinates (S0)55 = 0 and (S0)αα = bα := λxα −x2\nα. The only nonzero entries\nof V are V555 = 1 and the three permutations of V5αα = bα. Since C55jk = 0, a direct component\nformula for the boundary polynomial is therefore"
        }
      ],
      "occurrenceCount": 23
    },
    {
      "symbol": "β",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p15-294693052d",
        "page": 15,
        "quote": "For each fixed ζ and v, the Euclidean support lies in a finite annulus and is separated from the\nsingular set and other components, so the preceding second variation applies.\nSet β = (m −1)/2. For fζ(r) = rβζ(r), integration of the cross term gives the radial Hardy\nidentity\nZ ∞"
      },
      "occurrences": [
        {
          "blockId": "p11-4388b4b044",
          "page": 11,
          "quote": "hαβ = ⟨∇αN, eβ⟩."
        },
        {
          "blockId": "p11-c696783e4a",
          "page": 11,
          "quote": "Then TNα = 0, Tαβ = hαβ, and H = tr h; in particular H is a sum of principal curvatures with the\nspecified convention, without division by m −1. If ∇∂is the boundary connection, all components\nof B are described, up to permutation of slots, by"
        },
        {
          "blockId": "p11-064ff4fb38",
          "page": 11,
          "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
        },
        {
          "blockId": "p11-d8ff12d046",
          "page": 11,
          "quote": "BNαβ = TNNTαβ −(T 2)αβ,\nBNNN = w2.\n(3.11)"
        },
        {
          "blockId": "p12-7bd615b1c8",
          "page": 12,
          "quote": "Proof. On the boundary, tangential differentiation of |p|2 = 1 gives TNα = 0, and ∇αp = Teα\nbecause g = 0. Since p = N along the boundary, this proves Tαβ = hαβ. The trace equation for\nT gives tr h = −TNN = H. In differentiating three tangential entries of T, the normal connection\nterms pair with T(N, eα) = 0. This yields the first formula in (3.10). Likewise,"
        },
        {
          "blockId": "p12-84d5476660",
          "page": 12,
          "quote": "0 = BβNα + (T 2)βα −TNNTβα."
        },
        {
          "blockId": "p12-b900895217",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −w2."
        },
        {
          "blockId": "p12-b912b80307",
          "page": 12,
          "quote": "α,β\nT 2\nαβ = −T 2\nNN −\nX"
        },
        {
          "blockId": "p12-4584ce2868",
          "page": 12,
          "quote": "The zero trace of B then gives BNNN = w2.\nNormalization gives S0p = 0 and tr S0 = −|A|2 = −1. Consequently Vijj = pi tr S0 + 2(S0p)i +\npi = 0. In an orthonormal frame with first vector p = N, one has VNαβ = (S0)αβ and VNNN = 1.\nThese are exactly the normalized identities (3.11), and VNjk = S0,jk + pjpk. Finally, T has only its\nnormal-normal and tangential-tangential blocks, so the odd-normal formulas account for the entire\ncontraction TjkBNjk. Thus"
        },
        {
          "blockId": "p12-b9af30c473",
          "page": 12,
          "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
        },
        {
          "blockId": "p12-fe39066800",
          "page": 12,
          "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
        },
        {
          "blockId": "p13-13d5fe94da",
          "page": 13,
          "quote": "(∇2eg)00 = −ρgo,\n(∇2eg)0α = 0,\n(∇2eg)αβ = ρ−1(∇2\nogo)αβ + (c2/ρ)goδαβ."
        },
        {
          "blockId": "p15-294693052d",
          "page": 15,
          "quote": "For each fixed ζ and v, the Euclidean support lies in a finite annulus and is separated from the\nsingular set and other components, so the preceding second variation applies.\nSet β = (m −1)/2. For fζ(r) = rβζ(r), integration of the cross term gives the radial Hardy\nidentity\nZ ∞"
        },
        {
          "blockId": "p15-22db447160",
          "page": 15,
          "quote": "0\nrm|ζ′|2 dr −β2\nZ ∞"
        },
        {
          "blockId": "p15-12704680e3",
          "page": 15,
          "quote": "To approach its constant, choose nonzero ψ ∈C∞\nc (R) and set ζk(r) = r−βψ((log r)/k).\nThe\nsubstitution τ = (log r)/k yields\nR ∞\n0 rm|ζ′\nk|2 dr\nR ∞\n0 rm−2ζ2\nk dr = β2 + 1"
        },
        {
          "blockId": "p15-91cc1d2aa7",
          "page": 15,
          "quote": "R(−βψ + k−1ψ′)2 dτ, and\nR\nψψ′ = 0 by compact support.\nDivide the separated stability inequality by its positive radial factor and let k →∞. In tangent\ndimension m = 5, one has β2 = 4, proving (3.26). The same computation on an original link with\nmo = 4 would give 9/4; the working link always has m = 5.\nThe order of limits is fixed: for each angular test and each finite k, first take ε →0 in the second\nvariation, and then take k →∞. Angular cutoffs used later are removed only after (3.26) has been\nobtained for each fixed smooth test. The collar and the allowable size of ε may depend on these\nfixed supports."
        },
        {
          "blockId": "p20-044918f5ba",
          "page": 20,
          "quote": "For tangent indices α, β, the boundary jet identities give"
        },
        {
          "blockId": "p20-9e3441a439",
          "page": 20,
          "quote": "BNαβ"
        },
        {
          "blockId": "p20-b941ebff60",
          "page": 20,
          "quote": "w2\n= λAαβ −(A2)αβ = (S0)αβ,\nBNNN"
        },
        {
          "blockId": "p22-4ec0cc74d9",
          "page": 22,
          "quote": "Here Π is specified below by r and kind. There are nineteen such blocks with a total of 529 monomial\nrows. Every row has degree one in (w, z, ℓ, X), assigning degree one to each component of these\nvariables and degree zero to A, p, g. Hence Γ has degree two in these variables.\nThe two remaining blocks have rank −1 and fourteen rows each. In these blocks the four digits\nof a row word are an exponent β = (β1, . . . , β4), encoding xβ for x ∈R4. Both coefficient matrices\nhave five columns. Set"
        }
      ],
      "occurrenceCount": 27
    },
    {
      "symbol": "γ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "occurrences": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p11-064ff4fb38",
          "page": 11,
          "quote": "Bαβγ = (∇∂\nαh)βγ,\nBNNα = −∂αH,\n(3.10)"
        },
        {
          "blockId": "p12-fe39066800",
          "page": 12,
          "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
        }
      ],
      "occurrenceCount": 3
    },
    {
      "symbol": "δ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-4f79bc733f",
        "page": 16,
        "quote": "where p, ℓ∈R5 and g, w, z ∈R are otherwise free. In particular, w and z need not be positive. On\ntensor polynomials in A, p, g, δ, define Dl by the product rule and"
      },
      "occurrences": [
        {
          "blockId": "p6-e4883ffbed",
          "page": 6,
          "quote": "J(v; B) −J(q; B) = δ > 0."
        },
        {
          "blockId": "p13-1921e32005",
          "page": 13,
          "quote": "∇E0E0 = ∇E0Eα = 0,\n∇EαE0 = (c/ρ)Eα,\n∇EαEβ = −(c/ρ)δαβE0."
        },
        {
          "blockId": "p16-0026491c88",
          "page": 16,
          "quote": "50,\nQ = 250000,\nϵ∗=\n438907\n12000000,\nδ∗= 6984137359"
        },
        {
          "blockId": "p16-4f79bc733f",
          "page": 16,
          "quote": "where p, ℓ∈R5 and g, w, z ∈R are otherwise free. In particular, w and z need not be positive. On\ntensor polynomials in A, p, g, δ, define Dl by the product rule and"
        },
        {
          "blockId": "p16-bae9fc97ae",
          "page": 16,
          "quote": "Proposition 4.1 (Universal inequalities). There are rational tensor polynomials f, Ujk, Klijk, Ml, Pl\nin A, p, g, δ, with the following properties.\nThey are equivariant under orthogonal changes of\ncoordinates, with f invariant. Their coefficients are specified by Table I in Appendix A, using the\ndecoding in Section 6.1. Put"
        },
        {
          "blockId": "p16-12ac49d162",
          "page": 16,
          "quote": "Then\nS := −λf + Ujk(S0 + p ⊗p)jk + ClijkplVijk + plMl ≥δ∗.\n(4.4)"
        },
        {
          "blockId": "p17-e955da3b7c",
          "page": 17,
          "quote": "Lemma 5.1. At the data (5.2), the derivation Dl of (4.2), evaluated at z = 1, equals covariant\ndifferentiation on every coefficient polynomial in A, p, g, δ. In particular,"
        },
        {
          "blockId": "p17-2efa1d3509",
          "page": 17,
          "quote": "on the admissible data.\nHere Dl is applied only to coefficient polynomials in A, p, g, δ. Differentiation of the factors w\nand X, which are not coefficient variables, is performed separately. In particular, no rule for DlX\nor Dlq is being assumed.\nFor the U part of (5.4),"
        },
        {
          "blockId": "p20-c8cb318614",
          "page": 20,
          "quote": "N · Z + Hh2 = w2a+1S ≥δ∗w2a+1 > 0,\nδ∗= 6984137359"
        },
        {
          "blockId": "p21-e2ab8e7e5f",
          "page": 21,
          "quote": "Each later triple uvk specifies an edge carrying the matrix Ak, with A0 = δ. An endpoint labeled\ni ∈{0, . . . , r −1} is the ith free slot, starting with slot zero. An endpoint labeled 5 carries one copy\nof p, and one labeled 6 carries one copy of ℓ. Three incidences of label 7 are contracted against the\nthree slots of one X; three incidences of label 8, if present, use a second copy of X. Each occurrence\nof a vector endpoint is a separate vector factor. Matrix entries, vector entries, and tensor entries\nare multiplied, and all nonfree slots are summed. Edges are unoriented because A is symmetric.\nPermuting the three incidences at an X does not change its value because X is symmetric. There\nare no implicit multiplicities or factorials in this convention.\nWrite Mr(s) for the tensor encoded by the word s. For example,"
        },
        {
          "blockId": "p23-2c341b9f8d",
          "page": 23,
          "quote": "Proof. Each slot permutation is orthogonal, with adjoint its inverse. Averaging over all permutations,\nwith or without their signs, gives the orthogonal projection onto symmetric or alternating tensors.\nThus anti is r! times an orthogonal projection. For a symmetric matrix S, S −δ tr S/5 is its\northogonal trace-free projection. Since F ′ = 2 Sym F in (6.4), that operator is ten times the\nsymmetric trace-free projection.\nFor rank three define\n(J v)ijk = δijvk + δikvj + δjkvi."
        },
        {
          "blockId": "p30-0559e72ebf",
          "page": 30,
          "quote": "31250000000 = δ∗."
        },
        {
          "blockId": "p31-441dea3d25",
          "page": 31,
          "quote": "Proof. The boundary conclusion of Lemma 3.2 gives wo > 0 at every point of ∂Ωo. Compactness\nand continuity therefore give a boundary neighborhood on which wo has a positive lower bound.\nThe closed set Zo is thus compactly contained in the interior. If it is empty, wo has a positive\nminimum on the whole compact closure, and (7.1) follows immediately from the boundedness of Bo.\nOtherwise choose η ∈C∞\nc (Ωo) with 0 ≤η ≤1, equal to one on an open neighborhood of Zo.\nFor 0 < δ ≤1 put Wδ = w2\no + δ. Although wo need not be smooth at its zeros, Wδ is smooth and\nstrictly positive. The identities in Proposition 3.1 give"
        },
        {
          "blockId": "p31-a9e55efa94",
          "page": 31,
          "quote": "Ωo ηW a−1\nδ\n|Bo|2 dVo. Applying the chain rule to W a\nδ and integrating twice by parts against\nthe compactly supported η yields"
        },
        {
          "blockId": "p31-cfddc1d416",
          "page": 31,
          "quote": "Ωo\nηW a−1\nδ\n∆Wδ dVo"
        },
        {
          "blockId": "p31-31447e09ef",
          "page": 31,
          "quote": "Ωo\n(∆η)W a\nδ dVo + (1 −a)\nZ"
        },
        {
          "blockId": "p31-c544ea662e",
          "page": 31,
          "quote": "Ωo\nηW a−2\nδ\n|∇Wδ|2 dVo"
        },
        {
          "blockId": "p31-8d32359266",
          "page": 31,
          "quote": "Ωo\n(∆η)W a\nδ dVo + 4(1 −a)Iδ."
        },
        {
          "blockId": "p31-fd4079e419",
          "page": 31,
          "quote": "As δ ↓0, the nonnegative integrands restricted to Ωo,+ increase to ηw2a−2\no\n|Bo|2 because a < 1.\nMonotone convergence and (7.3) control that integral. The compact set supp(1 −η) ⊂Ωo is disjoint\nfrom Zo; writing bη > 0 for the minimum of wo there, the remaining integral is bounded by\nZ"
        },
        {
          "blockId": "p31-6e1b1f928c",
          "page": 31,
          "quote": "Thus all constants in this argument depend only on the fixed original component and its smooth\ndata, on a, and on the chosen interior cutoff η, and are independent of δ. Finally, on Ωo,+,"
        }
      ],
      "occurrenceCount": 20
    },
    {
      "symbol": "ε",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p5-9d1a526b83",
        "page": 5,
        "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
      },
      "occurrences": [
        {
          "blockId": "p5-9d1a526b83",
          "page": 5,
          "quote": "property of the limit; Proposition 9.12 gives its one-homogeneity. All these statements can be\napplied in a bounded ball compactly contained in D before taking the rescalings. In particular, the\nassertion about minimality is the minimizing compactness result, including the volume term.\nDefinition 6.10 and Corollary 8.2, with the one-sided graph conclusion in Theorem 8.1, give a\nC1,α regular neighborhood from a single half-space blowup. Proposition 7.1 supplies the viscosity\nBernoulli condition.\nTo justify all subsequent boundary derivatives, use [6, Theorem 1.1 and Corollary 1.2] with\nf+ = f−= 0 and G(b) = 1 + b. We check the two-phase viscosity condition of [5, Definition 7.1]\nwhen u ≥0. Let v be a comparison touching u at a free-boundary point x0, with a C2 interface\nand C2 branches having nonnegative normal slopes a and b on its positive and negative sides. If\nv touches from below, then v+ ≤u in a neighborhood. For a > 0, extend its positive branch to\na signed C2 function ϕ, negative just across the interface; then ϕ ≤u and touches at x0. The\none-phase condition gives a ≤1 ≤G(b); the case a = 0 is immediate. If v touches from above, the\nneighborhood inequality v ≥u ≥0 forces v = u = 0 throughout its nonpositive side, so b = 0. A\nC2 extension ϕ of its positive branch satisfies ϕ+ ≥u on both sides and touches at x0. Thus the\none-phase condition gives a ≥1 = G(0). These uses of the one-phase condition are justified by [17,\nDefinition 7.6 and Proposition 7.1]: to meet its smooth-test convention, replace each C2 extension\nby its quadratic Taylor polynomial minus, respectively plus, ε|x −x0|2. For any ε > 0, these are\nsmooth lower and upper supports in a sufficiently small neighborhood, with the same contact value\nand gradient.\nThe interior equations hold because u is harmonic where positive and identically zero on the\ninterior of its zero set. The function G is smooth, strictly increasing and unbounded, with G(0) = 1;\nthe cited results allow the negative phase to vanish. The preceding C1,α one-sided graph satisfies\nthe slab flatness hypothesis after rotation and sufficiently small dilation, while the local Lipschitz\nbound is preserved. Theorem 1.1 gives C2,γ regularity, and Corollary 1.2 gives smoothness. Local\nharmonic boundary regularity with zero Dirichlet data [12, Theorems 4.30–4.31 and Corollary 4.32]\nthen gives smoothness of u from the positive side. These estimates apply after flattening a smooth\nboundary patch and localizing away from its artificial boundary; iteration and Sobolev embedding\ngive every required derivative. The Bernoulli condition and nonnegativity give (2.2)."
        },
        {
          "blockId": "p13-a694bf1a3d",
          "page": 13,
          "quote": "Proof. Every regular free-boundary point has a neighborhood where the positive phase is the single\nside belonging to D. Compactness of the boundary portion meeting supp ϕ gives finitely many such\nneighborhoods and a collar separated from every other positivity component and every singular\npoint. Choose the collar over a slightly larger boundary portion, so that the extension of ϕ can\nvanish near its lateral edge. Extend ϕ smoothly across this boundary with compact support in these\nneighborhoods. Also extend the positive side of u smoothly to a function F on the collar, with its\ninterior jets unchanged. Since u = 0 and ∇u = NE on the boundary, shrinking the collar makes\nF negative on its exterior side and positive on its interior side. Such an extension need only be\nsmooth to finite order sufficient for the expansions below; smooth boundary regularity supplies this,\nand a partition of unity joins the local extensions while preserving the interior function and its jets.\nFor a real parameter ε of either sufficiently small sign, use (F + εϕ)+ in the collar, u + εϕ on the\nrest of the affected portion of D, and u elsewhere. These definitions agree: F = u on the interior of\nthe collar, and on the compact interior support outside a smaller collar u has a positive minimum."
        },
        {
          "blockId": "p14-5471f5971d",
          "page": 14,
          "quote": "2ε\nZ"
        },
        {
          "blockId": "p14-72f4405762",
          "page": 14,
          "quote": "D\n⟨∇u, ∇ϕ⟩dx + ε2\nZ"
        },
        {
          "blockId": "p14-1cf2fec128",
          "page": 14,
          "quote": "= −2ε\nZ"
        },
        {
          "blockId": "p14-9b869eba9c",
          "page": 14,
          "quote": "∂D\nϕ dσE + ε2\nZ"
        },
        {
          "blockId": "p14-85993fe6a5",
          "page": 14,
          "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
        },
        {
          "blockId": "p14-e7b31ba882",
          "page": 14,
          "quote": "Throughout the layer |ξ| ≤C|ε|, its perturbed energy density is"
        },
        {
          "blockId": "p14-0461573878",
          "page": 14,
          "quote": "(1 + |∇(F + εϕ)|2)j = 2 + 2εϕN + 2HEξ + O(ε2 + |εξ| + ξ2).\n(3.24)"
        },
        {
          "blockId": "p14-10f6d62795",
          "page": 14,
          "quote": "bε\n(1 + |∇(F + εϕ)|2)j dξ = −2bε −2εϕNbε −HEb2\nε + O(|ε|3)"
        },
        {
          "blockId": "p14-59fd4da02e",
          "page": 14,
          "quote": "= 2εϕ0 + ε2\n−2ϕ0ϕN + 2ϕ0ϕN −HEϕ2\n0\n\n+ O(|ε|3)"
        },
        {
          "blockId": "p14-b2fb2c5ad6",
          "page": 14,
          "quote": "= 2εϕ0 −ε2HEϕ2\n0 + O(|ε|3).\n(3.25)"
        },
        {
          "blockId": "p15-e8738a7c4a",
          "page": 15,
          "quote": "\n+ o(ε2)."
        },
        {
          "blockId": "p15-d3a05d2d6f",
          "page": 15,
          "quote": "J(uε) −J(u) = ε2\nZ"
        },
        {
          "blockId": "p15-d0552c7cd2",
          "page": 15,
          "quote": "The energies here are in a ball containing the modification. Minimality and division by ε2 prove\n(3.19)."
        },
        {
          "blockId": "p15-91cc1d2aa7",
          "page": 15,
          "quote": "R(−βψ + k−1ψ′)2 dτ, and\nR\nψψ′ = 0 by compact support.\nDivide the separated stability inequality by its positive radial factor and let k →∞. In tangent\ndimension m = 5, one has β2 = 4, proving (3.26). The same computation on an original link with\nmo = 4 would give 9/4; the working link always has m = 5.\nThe order of limits is fixed: for each angular test and each finite k, first take ε →0 in the second\nvariation, and then take k →∞. Angular cutoffs used later are removed only after (3.26) has been\nobtained for each fixed smooth test. The collar and the allowable size of ε may depend on these\nfixed supports."
        },
        {
          "blockId": "p33-b0e344b375",
          "page": 33,
          "quote": "Every cancellation above is made before removing the angular cutoff. For fixed j and radial\nindex k, the tests ζk(r)vj from Corollary 3.6 have compact support on the regular Euclidean cone.\nFirst take the variation parameter ε to zero on this fixed support; then let k →∞in Equation (3.27)\nto obtain the constant 4. The smooth extension collar and the allowed variation size may depend on\nj and k. We now take j →∞only in Equation (7.6), which contains the positive mass and three\ncutoff errors. The energy and boundary terms have already canceled for each fixed j."
        }
      ],
      "occurrenceCount": 17
    },
    {
      "symbol": "ζ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p15-aabfeea988",
        "page": 15,
        "quote": "Proof. More generally, consider a link component Ω⊂Sm with angular tests supported in its regular\nclosure, and write D = {ry : r > 0, y ∈Ω}. On its regular side the normal is tangent to Sm, and\n(3.2) gives HE(ry) = r−1H(y). Polar volume has factor rm dr and side area has factor rm−1 dr. For\nζ ∈C∞\nc ((0, ∞)) and an allowed angular test v, put ϕ(ry) = ζ(r)v(y) in Proposition 3.5. It gives\nZ ∞"
      },
      "occurrences": [
        {
          "blockId": "p15-aabfeea988",
          "page": 15,
          "quote": "Proof. More generally, consider a link component Ω⊂Sm with angular tests supported in its regular\nclosure, and write D = {ry : r > 0, y ∈Ω}. On its regular side the normal is tangent to Sm, and\n(3.2) gives HE(ry) = r−1H(y). Polar volume has factor rm dr and side area has factor rm−1 dr. For\nζ ∈C∞\nc ((0, ∞)) and an allowed angular test v, put ϕ(ry) = ζ(r)v(y) in Proposition 3.5. It gives\nZ ∞"
        },
        {
          "blockId": "p15-d356107f3e",
          "page": 15,
          "quote": "0\nrm−2ζ2 dr\n Z"
        },
        {
          "blockId": "p15-a298907820",
          "page": 15,
          "quote": "0\nrm−2ζ2 dr\n Z"
        },
        {
          "blockId": "p15-88096af21e",
          "page": 15,
          "quote": "0\nrm|ζ′|2 dr\n Z"
        },
        {
          "blockId": "p15-294693052d",
          "page": 15,
          "quote": "For each fixed ζ and v, the Euclidean support lies in a finite annulus and is separated from the\nsingular set and other components, so the preceding second variation applies.\nSet β = (m −1)/2. For fζ(r) = rβζ(r), integration of the cross term gives the radial Hardy\nidentity\nZ ∞"
        },
        {
          "blockId": "p15-22db447160",
          "page": 15,
          "quote": "0\nrm|ζ′|2 dr −β2\nZ ∞"
        },
        {
          "blockId": "p15-00dbd6f43a",
          "page": 15,
          "quote": "0\nrm−2ζ2 dr =\nZ ∞"
        },
        {
          "blockId": "p15-071055b17e",
          "page": 15,
          "quote": "0\nr|f′\nζ(r)|2 dr ≥0."
        },
        {
          "blockId": "p15-12704680e3",
          "page": 15,
          "quote": "To approach its constant, choose nonzero ψ ∈C∞\nc (R) and set ζk(r) = r−βψ((log r)/k).\nThe\nsubstitution τ = (log r)/k yields\nR ∞\n0 rm|ζ′\nk|2 dr\nR ∞\n0 rm−2ζ2\nk dr = β2 + 1"
        },
        {
          "blockId": "p25-7670809c37",
          "page": 25,
          "quote": "5j(5 −j). Its square is 9/20 for j = 1, 4 and 1/30\nfor j = 2, 3, proving (6.10).\nTo prove (6.11), fix a vector ζ and symmetrize ζ ⊗A:"
        },
        {
          "blockId": "p26-8d1ed535fc",
          "page": 26,
          "quote": "Expanding the nine terms in |B|2 gives |B|2 = |ζ|2/3+2|Aζ|2/3. The symmetric trace-free projection\nis F = B −J v/7, in the notation of Lemma 6.1. Since |J v|2 = 21|v|2 and ⟨B, J v⟩= 3|v|2,"
        },
        {
          "blockId": "p26-bde1b7401b",
          "page": 26,
          "quote": "3|ζ|2 + 10"
        },
        {
          "blockId": "p26-d7432d582e",
          "page": 26,
          "quote": "7|ζ|2."
        },
        {
          "blockId": "p26-b8f9e5a948",
          "page": 26,
          "quote": "Symmetry and trace-freeness of X give ⟨X : A, ζ⟩= ⟨X, F⟩; Cauchy–Schwarz and duality prove the\nclaim.\nFinally put s = g2, t = |p|2. For h, b > 0, the maximum of shtb on s, t ≥0, s + t ≤1 lies on\ns + t = 1. Differentiating h log s + b log(1 −s) gives the unique interior maximum s = h/(h + b),\nproving (6.12). If either exponent is zero, the corresponding bound is simply 1."
        }
      ],
      "occurrenceCount": 14
    },
    {
      "symbol": "η",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p31-441dea3d25",
        "page": 31,
        "quote": "Proof. The boundary conclusion of Lemma 3.2 gives wo > 0 at every point of ∂Ωo. Compactness\nand continuity therefore give a boundary neighborhood on which wo has a positive lower bound.\nThe closed set Zo is thus compactly contained in the interior. If it is empty, wo has a positive\nminimum on the whole compact closure, and (7.1) follows immediately from the boundedness of Bo.\nOtherwise choose η ∈C∞\nc (Ωo) with 0 ≤η ≤1, equal to one on an open neighborhood of Zo.\nFor 0 < δ ≤1 put Wδ = w2\no + δ. Although wo need not be smooth at its zeros, Wδ is smooth and\nstrictly positive. The identities in Proposition 3.1 give"
      },
      "occurrences": [
        {
          "blockId": "p31-441dea3d25",
          "page": 31,
          "quote": "Proof. The boundary conclusion of Lemma 3.2 gives wo > 0 at every point of ∂Ωo. Compactness\nand continuity therefore give a boundary neighborhood on which wo has a positive lower bound.\nThe closed set Zo is thus compactly contained in the interior. If it is empty, wo has a positive\nminimum on the whole compact closure, and (7.1) follows immediately from the boundedness of Bo.\nOtherwise choose η ∈C∞\nc (Ωo) with 0 ≤η ≤1, equal to one on an open neighborhood of Zo.\nFor 0 < δ ≤1 put Wδ = w2\no + δ. Although wo need not be smooth at its zeros, Wδ is smooth and\nstrictly positive. The identities in Proposition 3.1 give"
        },
        {
          "blockId": "p31-a9e55efa94",
          "page": 31,
          "quote": "Ωo ηW a−1\nδ\n|Bo|2 dVo. Applying the chain rule to W a\nδ and integrating twice by parts against\nthe compactly supported η yields"
        },
        {
          "blockId": "p31-31447e09ef",
          "page": 31,
          "quote": "Ωo\n(∆η)W a\nδ dVo + (1 −a)\nZ"
        },
        {
          "blockId": "p31-8d32359266",
          "page": 31,
          "quote": "Ωo\n(∆η)W a\nδ dVo + 4(1 −a)Iδ."
        },
        {
          "blockId": "p31-a5342e9d9d",
          "page": 31,
          "quote": "a(M2\no + 1)a∥∆η∥L1(Ωo).\n(7.3)"
        },
        {
          "blockId": "p31-fd4079e419",
          "page": 31,
          "quote": "As δ ↓0, the nonnegative integrands restricted to Ωo,+ increase to ηw2a−2\no\n|Bo|2 because a < 1.\nMonotone convergence and (7.3) control that integral. The compact set supp(1 −η) ⊂Ωo is disjoint\nfrom Zo; writing bη > 0 for the minimum of wo there, the remaining integral is bounded by\nZ"
        },
        {
          "blockId": "p31-da05d9263f",
          "page": 31,
          "quote": "Ωo,+\n(1 −η)w2a−2\no\n|Bo|2 dVo ≤b2a−2\nη\n∥Bo∥2\nL2(Ωo)."
        },
        {
          "blockId": "p31-6e1b1f928c",
          "page": 31,
          "quote": "Thus all constants in this argument depend only on the fixed original component and its smooth\ndata, on a, and on the chosen interior cutoff η, and are independent of δ. Finally, on Ωo,+,"
        }
      ],
      "occurrenceCount": 8
    },
    {
      "symbol": "θ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p25-7204e1c5da",
        "page": 25,
        "quote": "Put θ = 4/5. We first establish the tensor estimates that will bound each scalar graph, retaining\nthe dependence between p and g."
      },
      "occurrences": [
        {
          "blockId": "p25-7204e1c5da",
          "page": 25,
          "quote": "Put θ = 4/5. We first establish the tensor estimates that will bound each scalar graph, retaining\nthe dependence between p and g."
        },
        {
          "blockId": "p25-26685d701f",
          "page": 25,
          "quote": "∥A∥2\nop ≤θ,\n|Ak|2 ≤θk−1\n(k ≥1),\n(6.9)"
        },
        {
          "blockId": "p25-4e4810cb21",
          "page": 25,
          "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
        },
        {
          "blockId": "p26-93c0366bec",
          "page": 26,
          "quote": "Proof. First separate all scalar trace factors and use Lemma 6.3. A cubic tensor has at most one\nself-edge, since such an edge consumes two of its three incidences. An edge of power k ≥2 reduces\nthat tensor to the vector vi = Xijk(Ak)jk, for which |v| ≤|X| |Ak| ≤|X|θ(k−1)/2. For k = 1 use\n(6.11). These reductions leave at most two cubic vertices and no self-edge on either.\nWe list all possible connected components of the remaining graph. With zero cubic vertices, a\ncomponent is one edge joining two vectors; its contraction is bounded by the operator norm of its\nmatrix times the two vector norms. With one cubic vertex, all its three slots end at vector leaves.\nMove each edge matrix onto its vector, and use tensor Cauchy–Schwarz, obtaining the norm of X\ntimes the three vector norms and the product of the three operator norms.\nIf a component contains two cubic vertices, they are joined by r = 1, 2, or 3 edges; every other\nslot ends at a vector leaf. Contract these leaves first, obtaining rank-r tensors Y, Z with the same\npreceding norm bounds. Up to a permutation of their slots, the remaining contraction is\nD\nY, (Ak1 ⊗· · · ⊗Akr)Z\nE\n."
        }
      ],
      "occurrenceCount": 4
    },
    {
      "symbol": "κ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p31-3148e493f1",
        "page": 31,
        "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
      },
      "occurrences": [
        {
          "blockId": "p4-32abdf50fa",
          "page": 4,
          "quote": "Sources and applicability. We use the constant-coefficient theory with Λ = 1 in [17]. For each ball\nB ⋐D, the restriction of u to B minimizes against every H1\n0(B) competitor difference. Zero\nextension of differences therefore gives comparison on every O ⋐B, as required by Definitions 2.11–\n2.12 there. We apply the local results inside such balls. For global minimizers, the containing-ball\nargument above gives the comparison convention on all bounded open sets. Local Lipschitz continuity\nis Theorem 3.1. Harmonicity also follows directly: for a smooth test supported compactly in the\npositive phase, both signs of a sufficiently small additive variation preserve that phase, so the\nDirichlet first variation vanishes.\nThe Lipschitz bound and u(x0) = 0 give locally uniform compactness of the rescalings. Propo-\nsition 4.1 gives supBr(x0) u ≥κr for small r, with κ > 0, so a limit cannot vanish identically.\nProposition 6.2 gives strong Sobolev and positivity-indicator convergence and the full minimizing"
        },
        {
          "blockId": "p31-3148e493f1",
          "page": 31,
          "quote": "Let ϑ ∈C∞([0, ∞)) satisfy 0 ≤ϑ ≤1, ϑ = 0 on [0, 1], and ϑ = 1 on [2, ∞). Fix this profile once\nand for all, and set\nLϑ = ∥ϑ′∥∞,\nκ = 2Lϑ."
        },
        {
          "blockId": "p33-4ee4315d76",
          "page": 33,
          "quote": "|∇oαj| ≤κ|Bo|"
        },
        {
          "blockId": "p33-3cf50241b6",
          "page": 33,
          "quote": "|β′\nj(t)| ≤\nκ\nsin t.\n(7.8)"
        },
        {
          "blockId": "p33-5c3729d0d6",
          "page": 33,
          "quote": "1 + w + |X| ≤Kn(1 + bo),\n|∇ξj| ≤κ(1 + bo)."
        },
        {
          "blockId": "p33-4ebee114f5",
          "page": 33,
          "quote": "|E1,j| + |E2,j| + |E3,j| ≤C2\nF (κ2 + 4Knκ)Do.\n(7.9)"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-5876cae3e5",
          "page": 34,
          "quote": "|∇ξj| ≤κ"
        },
        {
          "blockId": "p34-12feee0731",
          "page": 34,
          "quote": "s4\nt\n\n|E1,j| + |E2,j| + |E3,j|\n\n≤C2\nF (κ2 + 4Ksκ)s2−2a\nt\nDo(y).\n(7.12)"
        }
      ],
      "occurrenceCount": 9
    },
    {
      "symbol": "λ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p36-ab20d66874",
        "page": 36,
        "quote": "In Table I the first field specifies the polynomial, the second its monomial, and the third its\ncoefficient numerator. The free-slot orders are Ujk, Klijk, Ml, and Pl, while f has rank zero. The\nnumbers of rows for (f, U, M, P, K) are (34, 57, 74, 32, 34), for a total of 231.\nEach line of Table II beginning with @ starts a separate block and gives its rank, projection\nlabel, and multiplier mode. For each coefficient column, sum the monomials with the coefficients\nin that column divided by Q to obtain a tensor Jc. For nonnegative rank the block contributes\nρ P\nc⟨Jc, ΠJc⟩to Γ, where the modes I and B mean ρ = 1 and ρ = χ = 1 −|p|2 −g2, respectively.\nThe labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not\ninstruct any further normalization of the coefficients. Identical headers in different positions still\ndefine distinct blocks.\nFor rank −1, the four code digits instead give the four individual exponents of x1, x2, x3, x4. The\nblock contributes to Λ the average over all 24 permutations of ρ P\nc J2\nc . Here I means ρ = 1, while H\nmeans ρ = P\ni xi = −λ. The averaging and boundary normalization R = P\ni x2\ni + (P\ni xi)2 = 1 are\nthose of Section 6.2. Table II contains 21 blocks: 19 interior blocks with 529 rows and two boundary\nblocks with 14 rows each, hence 557 coefficient rows in total. Each block retains its own number of\ncoefficient columns."
      },
      "occurrences": [
        {
          "blockId": "p11-a3924f6d0b",
          "page": 11,
          "quote": "A = T/w,\nλ = TNN/w = −H/w,\nS0 = λA −A2,"
        },
        {
          "blockId": "p11-7580df6f09",
          "page": 11,
          "quote": "Then |p| = 1, Ap = λp, λ < 0, S0p = 0, tr S0 = −1, and V is trace-free. The tensors B/w2 and V\nagree in every component with an odd number of normal slots. Moreover,"
        },
        {
          "blockId": "p11-f6ddc57c80",
          "page": 11,
          "quote": "VNjk = (S0 + p ⊗p)jk,\n∂Nw = w2\n2λ −tr A3\n.\n(3.13)"
        },
        {
          "blockId": "p12-a1dcbd3ddc",
          "page": 12,
          "quote": "∂Nw = w−1TjkBNjk = w2Ajk(S0 + p ⊗p)jk = w2(2λ −tr A3),"
        },
        {
          "blockId": "p16-57f3a15ba4",
          "page": 16,
          "quote": "g = 0,\n|p| = 1,\nAp = λp,\nλ < 0,"
        },
        {
          "blockId": "p19-dbc0d12dc6",
          "page": 19,
          "quote": "g = 0,\np = N,\n|p| = 1,\nAp = λp,\nλ = −H"
        },
        {
          "blockId": "p25-616b2bfb28",
          "page": 25,
          "quote": "and its boundary contribution is 2λ −tr A3. To expand a Gram block, sum over unordered pairs\nof rows s, t: the coefficient is P\nc cscctc/Q2, doubled when s ̸= t, and the graph polynomial is the\ncontraction of row t with Π applied to row s, multiplied by ρ. This uses self-adjointness of Π and\naccounts for every term of (6.2) exactly once."
        },
        {
          "blockId": "p28-5c345e0cc7",
          "page": 28,
          "quote": "A = diag(x1, x2, x3, x4, λ),\nλ = −"
        },
        {
          "blockId": "p30-b279fa0e2c",
          "page": 30,
          "quote": "Since P\ni xi = −λ > 0, (6.3) is nonnegative. It follows from (6.25) that"
        },
        {
          "blockId": "p30-31c4588c77",
          "page": 30,
          "quote": "This completes the proof of Proposition 4.1. The estimate also persists at λ = 0 by continuity;\nrepeated tangent eigenvalues cause no singularity in any of the formulas.\nThe pointwise comparisons and field bounds of Section 5 are therefore established. Section 7\nnow justifies their integration across Hessian zeros and near the suspension tips."
        },
        {
          "blockId": "p36-ab20d66874",
          "page": 36,
          "quote": "In Table I the first field specifies the polynomial, the second its monomial, and the third its\ncoefficient numerator. The free-slot orders are Ujk, Klijk, Ml, and Pl, while f has rank zero. The\nnumbers of rows for (f, U, M, P, K) are (34, 57, 74, 32, 34), for a total of 231.\nEach line of Table II beginning with @ starts a separate block and gives its rank, projection\nlabel, and multiplier mode. For each coefficient column, sum the monomials with the coefficients\nin that column divided by Q to obtain a tensor Jc. For nonnegative rank the block contributes\nρ P\nc⟨Jc, ΠJc⟩to Γ, where the modes I and B mean ρ = 1 and ρ = χ = 1 −|p|2 −g2, respectively.\nThe labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not\ninstruct any further normalization of the coefficients. Identical headers in different positions still\ndefine distinct blocks.\nFor rank −1, the four code digits instead give the four individual exponents of x1, x2, x3, x4. The\nblock contributes to Λ the average over all 24 permutations of ρ P\nc J2\nc . Here I means ρ = 1, while H\nmeans ρ = P\ni xi = −λ. The averaging and boundary normalization R = P\ni x2\ni + (P\ni xi)2 = 1 are\nthose of Section 6.2. Table II contains 21 blocks: 19 interior blocks with 529 rows and two boundary\nblocks with 14 rows each, hence 557 coefficient rows in total. Each block retains its own number of\ncoefficient columns."
        }
      ],
      "occurrenceCount": 11
    },
    {
      "symbol": "ν",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p25-4e4810cb21",
        "page": 25,
        "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
      },
      "occurrences": [
        {
          "blockId": "p25-4e4810cb21",
          "page": 25,
          "quote": "This proves the operator bound, and P\ni α2k\ni\n≤θk−1 P\ni α2\ni proves the Frobenius bound. Moreover\n| P\ni αk\ni | ≤P\ni |αi|k ≤θ(k−2)/2 P\ni α2\ni for k ≥2.\nFor the sharper cubic bound maximize and minimize P\ni α3\ni on the compact set P\ni αi = 0,\nP\ni α2\ni = 1. The gradients of the two constraints are independent there. Lagrange multipliers give\n3α2\ni = µ + 2ναi, so an extremum has two distinct eigenvalues (a single value would violate the\nconstraints). If their multiplicities are j, 5 −j, the values are, up to simultaneous sign,\ns"
        },
        {
          "blockId": "p28-f5c74d161c",
          "page": 28,
          "quote": "For example, this entrywise check consists of comparing each last-column integer in (6.18) with\nthe corresponding integer in (6.19) multiplied by 2Q2. Let ν = (2484, 1440, 996, 304)T . The exact\ndiagonal slacks are"
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "ξ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p14-85993fe6a5",
        "page": 14,
        "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
      },
      "occurrences": [
        {
          "blockId": "p14-193e7fe7bf",
          "page": 14,
          "quote": "Here harmonicity eliminates the interior term and the outward normal is −NE, so ∂−NEu = −1.\nAll integrations by parts have compact support on the regular portion.\nUse coordinates (y, ξ) obtained by flowing from the original boundary under"
        },
        {
          "blockId": "p14-5dd6d92075",
          "page": 14,
          "quote": "VF =\n∇F\n|∇F|2 ,\nξ = F."
        },
        {
          "blockId": "p14-534134ce25",
          "page": 14,
          "quote": "The denominator is nonzero on a sufficiently small compact collar. At ξ = 0, VF = NE. If j(y, ξ) is\nthe volume Jacobian relative to dσE(y) dξ, then"
        },
        {
          "blockId": "p14-ee2a00e317",
          "page": 14,
          "quote": "∂ξ|∇F|2\nξ=0 = 2FNENE = −2HE,"
        },
        {
          "blockId": "p14-7772931587",
          "page": 14,
          "quote": "and hence\n∂ξ\n\n(1 + |∇F|2)j\n\nξ=0 = 2HE.\n(3.22)"
        },
        {
          "blockId": "p14-85993fe6a5",
          "page": 14,
          "quote": "Write ϕ0 = ϕ(y, 0) and ϕN = ∂NEϕ(y, 0). The new zero surface has coordinate bε(y) determined\nby bε + εϕ(y, bε) = 0. The implicit function theorem and ∂ξϕ(y, 0) = ϕN give, uniformly on the\ncompact patch,\nbε = −εϕ0 + ε2ϕ0ϕN + O(|ε|3).\n(3.23)"
        },
        {
          "blockId": "p14-e7b31ba882",
          "page": 14,
          "quote": "Throughout the layer |ξ| ≤C|ε|, its perturbed energy density is"
        },
        {
          "blockId": "p14-0461573878",
          "page": 14,
          "quote": "(1 + |∇(F + εϕ)|2)j = 2 + 2εϕN + 2HEξ + O(ε2 + |εξ| + ξ2).\n(3.24)"
        },
        {
          "blockId": "p32-0405f3618c",
          "page": 32,
          "quote": "The constant depends only on a, the dimension 5, the finitely many coefficients defining f, G, M, P,\nand the positive lower bound f ≥ϵ∗. In particular CF is independent of the cone, w, X, and j.\nFor each fixed j, define vj = ξjh and Fj = ξ2\nj Z on Ω+, and extend them by zero where w = 0.\nThese are smooth on the regular closure of Ωand are supported away from the tips when present.\nThe same neighborhood argument as for αj proves smoothness of these extensions; it does not\nrequire h or Z themselves to extend smoothly across w = 0. All products below containing one of\nthese cutoff factors or its derivative are likewise understood to be zero on w = 0. The mass w2a,\nwhose exponent is positive, has its ordinary value zero there and involves no normalized tensor."
        },
        {
          "blockId": "p32-e3897bb493",
          "page": 32,
          "quote": "Ω\nξ2\nj w2a dV ≤\nZ"
        },
        {
          "blockId": "p33-27f1dface5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV."
        },
        {
          "blockId": "p33-86d871abd5",
          "page": 33,
          "quote": "Ω\nξ2\nj (4h2 + |∇h|2 + ϵ∗w2a) dV ≤\nZ"
        },
        {
          "blockId": "p33-cc573589a3",
          "page": 33,
          "quote": "∂Ω\nξ2\nj ⟨N, Z⟩dσ −2\nZ"
        },
        {
          "blockId": "p33-a9615cafa5",
          "page": 33,
          "quote": "Ω\nξ2\nj div Z dV = −\nZ"
        },
        {
          "blockId": "p33-9f2f452bde",
          "page": 33,
          "quote": "\n|∇(ξjh)|2 + 4ξ2\nj h2\ndV."
        },
        {
          "blockId": "p34-10a8b72510",
          "page": 34,
          "quote": "Ω\nξ2\nj w2a dV = Io."
        },
        {
          "blockId": "p35-1f9109f274",
          "page": 35,
          "quote": "Ω\nξ2\nj w2a dV = Io"
        }
      ],
      "occurrenceCount": 17
    },
    {
      "symbol": "π",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p22-070d569830",
        "page": 22,
        "quote": "For an arbitrary rank-r tensor F, let F π denote slot permutation by π. We use the averaged\nsymmetrization Sym F = (r!)−1 P\nπ∈Sr F π. The operator Trab contracts slots a, b with the Euclidean\nmetric, leaving the remaining slots in their original order. The label none means Π = Id. The label\nanti means\nΠF =\nX"
      },
      "occurrences": [
        {
          "blockId": "p8-7722e556c0",
          "page": 8,
          "quote": "eΩ=\n\nΦ(t, y) := (sin t y, cos t) : 0 < t < π, y ∈Ωo\n\t\n,\neg(Φ(t, y)) = sin t go(y).\n(2.9)"
        },
        {
          "blockId": "p8-9d96cf4b31",
          "page": 8,
          "quote": "Its boundary consists of the smooth side Φ((0, π) × ∂Ωo) and the two singular tips p+ = (0, 1) and\np−= (0, −1)."
        },
        {
          "blockId": "p9-6632f13d2e",
          "page": 9,
          "quote": "This is nonflat, so no axis point is regular. The axis meets S5 only in p+ and p−.\nAway from these two points, Φ is a smooth product coordinate map from (0, π) × S4 to\nS5 \\ {p+, p−}. Positivity in these coordinates is precisely go(y) > 0, so connected components are\nas in (2.9). Since Ωo is compact, its suspended closure is obtained by adding the two endpoints of\nthe interval, with each endpoint collapsed to its tip. Its remaining boundary is exactly the stated\nsmooth side."
        },
        {
          "blockId": "p9-6c6e1e3c68",
          "page": 9,
          "quote": "All spherical boundary values below are taken from the positive side. Tests on Ωwill be smooth and\nsupported away from Σ. Every compact portion of the selected Euclidean cone’s closure that avoids\nthe origin and, in the suspended case, the axis is separated from all other positive components. This\nfollows from the disjoint compact original closures, and in the suspension from the same fact on any\nclosed subinterval of (0, π). Consequently the local variations on this component can be made as\nvariations of the global minimizer. No minimizing property of the selected component extended by\nzero is needed."
        },
        {
          "blockId": "p12-9d3f73e267",
          "page": 12,
          "quote": "(t, y) 7−→(ρy, c),\n0 < t < π,\ny ∈Ωo."
        },
        {
          "blockId": "p22-62ce869654",
          "page": 22,
          "quote": "π∈S4\nF(xπ(1), . . . , xπ(4))."
        },
        {
          "blockId": "p22-070d569830",
          "page": 22,
          "quote": "For an arbitrary rank-r tensor F, let F π denote slot permutation by π. We use the averaged\nsymmetrization Sym F = (r!)−1 P\nπ∈Sr F π. The operator Trab contracts slots a, b with the Euclidean\nmetric, leaving the remaining slots in their original order. The label none means Π = Id. The label\nanti means\nΠF =\nX"
        },
        {
          "blockId": "p22-0f32a430b8",
          "page": 22,
          "quote": "π∈Sr\nsgn(π)F π."
        },
        {
          "blockId": "p22-1092f766ba",
          "page": 22,
          "quote": "For sym, put F ′ = P\nπ∈Sr F π and v = Tr12 F ′. The operators used in Table II are"
        },
        {
          "blockId": "p22-8a978235ac",
          "page": 22,
          "quote": "π∈S3\nF π,\nv = Tr12 F ′,"
        },
        {
          "blockId": "p23-e7f9509e86",
          "page": 23,
          "quote": "π∈S3\nπ,\nP0 = S12 −S."
        },
        {
          "blockId": "p32-873251d495",
          "page": 32,
          "quote": "0\nπ"
        },
        {
          "blockId": "p32-024527fd90",
          "page": 32,
          "quote": "t = π −arcsin(2/j)"
        },
        {
          "blockId": "p33-d88ba6541b",
          "page": 33,
          "quote": "At every fixed point of Ωo,+, αj = 1 and ∇oαj = 0 for all sufficiently large j. At every fixed t ∈(0, π)\nthe corresponding statements hold for βj.\nWrite bo = |Bo|/wo on Ωo,+. Let E1,j, E2,j, and E3,j denote, in order, the three signed integrands\non the right of (7.6).\nFirst suppose mo = 5, so Ω= Ωo, and set Kn = 1 + Mo. Since X = Bo/wo,"
        },
        {
          "blockId": "p34-a6c4f669ee",
          "page": 34,
          "quote": "Indeed the three coefficient bounds are respectively C2\nF κ2, 2C2\nF Knκ, and 2CF Knκ, and CF ≥1.\nThe dominator is integrable by Lemma 7.1. Each error is eventually zero at every point of Ωo,+,\nand is defined to be zero on Zo. Dominated convergence therefore makes the right side of (7.6) tend\nto zero.\nNow suppose mo = 4. Write st = sin t and use the suspension coordinates (t, y) ∈(0, π) × Ωo.\nThe exact formulas in Proposition 3.4 give"
        },
        {
          "blockId": "p34-2041107dfc",
          "page": 34,
          "quote": "In particular Ω+ = (0, π) × Ωo,+, and there"
        },
        {
          "blockId": "p34-b878115b78",
          "page": 34,
          "quote": "The right side is integrable with respect to dt dVo, because 21/25 > −1 and (7.1) holds. Every\nerror is eventually zero at each fixed (t, y) ∈(0, π) × Ωo,+. Dominated convergence applied to (7.12)\ntherefore makes the right side of (7.6) tend to zero also in the suspension."
        },
        {
          "blockId": "p35-9d857d7a46",
          "page": 35,
          "quote": "Z π"
        },
        {
          "blockId": "p35-2be4139c3b",
          "page": 35,
          "quote": "Z π"
        }
      ],
      "occurrenceCount": 19
    },
    {
      "symbol": "ρ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p12-6e66e62020",
        "page": 12,
        "quote": "Proposition 3.4 (Suspended data). If mo = 4, write ρ = sin t, c = cos t and parameterize the\nregular part of the suspended component by"
      },
      "occurrences": [
        {
          "blockId": "p12-6e66e62020",
          "page": 12,
          "quote": "Proposition 3.4 (Suspended data). If mo = 4, write ρ = sin t, c = cos t and parameterize the\nregular part of the suspended component by"
        },
        {
          "blockId": "p12-31d748db9a",
          "page": 12,
          "quote": "Its metric is dt2 + ρ2ds2\nS4, and eg = ρgo. With E0 = ∂t and Eα = ρ−1eα for a lifted orthonormal\nframe on S4, the data satisfy"
        },
        {
          "blockId": "p12-b9af30c473",
          "page": 12,
          "quote": "ep0 = cgo,\nepα = (po)α,\nes2 = s2\no,\n(3.14)\neT00 = 0,\neT0α = 0,\neTαβ = ρ−1(To)αβ.\n(3.15)"
        },
        {
          "blockId": "p12-fe39066800",
          "page": 12,
          "quote": "eBαβγ = ρ−2(Bo)αβγ,\neB0αβ = eBα0β = eBαβ0 = −cρ−2(To)αβ.\n(3.16)"
        },
        {
          "blockId": "p12-8b2672b64e",
          "page": 12,
          "quote": "ew = ρ−1wo,\n| eB|2 = ρ−4\n|Bo|2 + 3c2w2\no\n\n,\neH = ρ−1Ho\non the regular side.\n(3.17)"
        },
        {
          "blockId": "p12-8a8302c5ce",
          "page": 12,
          "quote": "dV = ρ4 dt dVo,\ndσ = ρ3 dt dσo.\n(3.18)"
        },
        {
          "blockId": "p13-1921e32005",
          "page": 13,
          "quote": "∇E0E0 = ∇E0Eα = 0,\n∇EαE0 = (c/ρ)Eα,\n∇EαEβ = −(c/ρ)δαβE0."
        },
        {
          "blockId": "p13-61831a7400",
          "page": 13,
          "quote": "The horizontal part of the last expression at other points is the base connection with factor ρ−1.\nDifferentiating eg = ρgo gives (3.14). Its Hessian has entries"
        },
        {
          "blockId": "p13-13d5fe94da",
          "page": 13,
          "quote": "(∇2eg)00 = −ρgo,\n(∇2eg)0α = 0,\n(∇2eg)αβ = ρ−1(∇2\nogo)αβ + (c2/ρ)goδαβ."
        },
        {
          "blockId": "p13-aca19449d5",
          "page": 13,
          "quote": "Adding eg Id proves (3.15). Differentiating that tensor with the displayed connection gives (3.16);\nentries with two or three zero slots vanish. Each term with exactly one zero slot occurs in three\npositions, which proves the factor 3 in (3.17).\nThe inward normal to the side is the horizontal lift of No. Thus the normal-normal entry of\n(3.15) gives eH = ρ−1Ho. The metric on the side is dt2 + ρ2ds2\n∂Ωo, proving (3.18). The trace of (3.15)\nis zero, so ∆eg = −5eg. The interior identities follow locally from Proposition 3.1. The bounds for s2,\nH and w follow from the original compact component and (3.14)–(3.17); no maximum principle at a\ntip is used. The local boundary hypotheses and hence Proposition 3.3 hold on the regular side."
        },
        {
          "blockId": "p22-3af64900bc",
          "page": 22,
          "quote": "ρ\nX"
        },
        {
          "blockId": "p22-9fbca11b8f",
          "page": 22,
          "quote": "c\n⟨Jc, ΠJc⟩,\nρ ="
        },
        {
          "blockId": "p22-1fc3863172",
          "page": 22,
          "quote": "ρ(x)\nX"
        },
        {
          "blockId": "p22-bf52c99b3d",
          "page": 22,
          "quote": ",\nρ(x) ="
        },
        {
          "blockId": "p25-616b2bfb28",
          "page": 25,
          "quote": "and its boundary contribution is 2λ −tr A3. To expand a Gram block, sum over unordered pairs\nof rows s, t: the coefficient is P\nc cscctc/Q2, doubled when s ̸= t, and the graph polynomial is the\ncontraction of row t with Π applied to row s, multiplied by ρ. This uses self-adjointness of Π and\naccounts for every term of (6.2) exactly once."
        },
        {
          "blockId": "p36-ab20d66874",
          "page": 36,
          "quote": "In Table I the first field specifies the polynomial, the second its monomial, and the third its\ncoefficient numerator. The free-slot orders are Ujk, Klijk, Ml, and Pl, while f has rank zero. The\nnumbers of rows for (f, U, M, P, K) are (34, 57, 74, 32, 34), for a total of 231.\nEach line of Table II beginning with @ starts a separate block and gives its rank, projection\nlabel, and multiplier mode. For each coefficient column, sum the monomials with the coefficients\nin that column divided by Q to obtain a tensor Jc. For nonnegative rank the block contributes\nρ P\nc⟨Jc, ΠJc⟩to Γ, where the modes I and B mean ρ = 1 and ρ = χ = 1 −|p|2 −g2, respectively.\nThe labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not\ninstruct any further normalization of the coefficients. Identical headers in different positions still\ndefine distinct blocks.\nFor rank −1, the four code digits instead give the four individual exponents of x1, x2, x3, x4. The\nblock contributes to Λ the average over all 24 permutations of ρ P\nc J2\nc . Here I means ρ = 1, while H\nmeans ρ = P\ni xi = −λ. The averaging and boundary normalization R = P\ni x2\ni + (P\ni xi)2 = 1 are\nthose of Section 6.2. Table II contains 21 blocks: 19 interior blocks with 529 rows and two boundary\nblocks with 14 rows each, hence 557 coefficient rows in total. Each block retains its own number of\ncoefficient columns."
        }
      ],
      "occurrenceCount": 16
    },
    {
      "symbol": "τ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-05b288455c",
        "page": 6,
        "quote": "for small τ > 0. Thus a nonzero minimizing cone has just one positive slope. After reflection write\nu(x) = bx+, b > 0. The competitors vt(x) = bR(x −t)+/(R −t) on (−R, R), |t| < R, have the"
      },
      "occurrences": [
        {
          "blockId": "p6-7aa4a96b63",
          "page": 6,
          "quote": "J(u + τϕ; (−R, R)) −J(u; (−R, R)) = −2τ(b+ + b−)ϕ(0) + τ 2\nZ R"
        },
        {
          "blockId": "p6-05b288455c",
          "page": 6,
          "quote": "for small τ > 0. Thus a nonzero minimizing cone has just one positive slope. After reflection write\nu(x) = bx+, b > 0. The competitors vt(x) = bR(x −t)+/(R −t) on (−R, R), |t| < R, have the"
        },
        {
          "blockId": "p11-48cf81081f",
          "page": 11,
          "quote": "Proof. The boundary conditions give s2\no = 1 on ∂Ωo. By (3.6), s2\no is subharmonic. The maximum\nprinciple on the compact closure gives s2\no ≤1. If equality held at an interior point, the strong\nmaximum principle would make s2\no constant on the connected component, and then ∆s2\no = 2w2\no\nwould imply To ≡0. Thus s2\no < 1 inside.\nThe smooth boundary has the interior sphere property at each point. The Hopf boundary\nlemma therefore gives ∂Nos2\no < 0. Since po = No there, (3.6) identifies this derivative as 2To(No, No),\nproving Ho > 0. For a tangent vector τ to ∂Ωo, tangential differentiation of s2\no = 1 similarly gives\nTo(τ, No) = 0. This proves the eigenvector assertion and wo ≥Ho > 0. Continuity and compactness\nof the boundary give a positive minimum of wo and a neighborhood of the boundary disjoint from\nits zero set."
        },
        {
          "blockId": "p15-12704680e3",
          "page": 15,
          "quote": "To approach its constant, choose nonzero ψ ∈C∞\nc (R) and set ζk(r) = r−βψ((log r)/k).\nThe\nsubstitution τ = (log r)/k yields\nR ∞\n0 rm|ζ′\nk|2 dr\nR ∞\n0 rm−2ζ2\nk dr = β2 + 1"
        }
      ],
      "occurrenceCount": 4
    },
    {
      "symbol": "χ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p16-6e69ee52c4",
        "page": 16,
        "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
      },
      "occurrences": [
        {
          "blockId": "p16-6e69ee52c4",
          "page": 16,
          "quote": "Let A be a symmetric trace-free matrix with |A|2 = tr A2 = 1, and let X be a symmetric\ntrace-free three-tensor. Set\nql = XlijAij,\nχ = 1 −|p|2 −g2,"
        },
        {
          "blockId": "p16-5a9c71c879",
          "page": 16,
          "quote": "For every choice of the data above with χ ≥0,"
        },
        {
          "blockId": "p22-1fe5a2118c",
          "page": 22,
          "quote": "(\n1,\nmode = I,\nχ,\nmode = B.\n(6.2)"
        },
        {
          "blockId": "p23-0a22271950",
          "page": 23,
          "quote": "Lemma 6.1. All these operators are self-adjoint and positive semidefinite on their entire tensor\nspaces. In particular, Γ ≥0 whenever χ ≥0."
        },
        {
          "blockId": "p24-508d7d7d45",
          "page": 24,
          "quote": "Proof. The incidence assertions hold for each table row by its decoding. A disjoint union preserves\nthem. A permutation only renames free slots. Contracting two free slots removes their unique\nincidences and joins the remaining ends, or closes a trace; it does not change the number of slots of\nany cubic tensor. The derivative insertion described above replaces one edge by two incidences of a\nnew X and supplies its third incidence as a new free slot. Trace differentiation does the same with\na self-edge. The ql term supplies two incidences on its self-edge and the third at l. Derivatives of p\nand g supply a free derivative slot without introducing a cubic tensor. These statements prove the\nincidence assertions by induction over the operations.\nTable I has degree zero and its derivative has degree one in the indicated variables. The formula\nfor I multiplies such derivatives by one further factor, or a degree-zero coefficient by two factors.\nEach Table II row has degree one; projection, contraction, and multiplication by χ preserve that\ndegree. Squaring produces degree two. In particular no operation used here needs three copies of X.\nTermination is immediate from the finite tables, finite permutation sums, and the finite product\nrule; every contraction reduces the number of free slots by two."
        },
        {
          "blockId": "p36-ab20d66874",
          "page": 36,
          "quote": "In Table I the first field specifies the polynomial, the second its monomial, and the third its\ncoefficient numerator. The free-slot orders are Ujk, Klijk, Ml, and Pl, while f has rank zero. The\nnumbers of rows for (f, U, M, P, K) are (34, 57, 74, 32, 34), for a total of 231.\nEach line of Table II beginning with @ starts a separate block and gives its rank, projection\nlabel, and multiplier mode. For each coefficient column, sum the monomials with the coefficients\nin that column divided by Q to obtain a tensor Jc. For nonnegative rank the block contributes\nρ P\nc⟨Jc, ΠJc⟩to Γ, where the modes I and B mean ρ = 1 and ρ = χ = 1 −|p|2 −g2, respectively.\nThe labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not\ninstruct any further normalization of the coefficients. Identical headers in different positions still\ndefine distinct blocks.\nFor rank −1, the four code digits instead give the four individual exponents of x1, x2, x3, x4. The\nblock contributes to Λ the average over all 24 permutations of ρ P\nc J2\nc . Here I means ρ = 1, while H\nmeans ρ = P\ni xi = −λ. The averaging and boundary normalization R = P\ni x2\ni + (P\ni xi)2 = 1 are\nthose of Section 6.2. Table II contains 21 blocks: 19 interior blocks with 529 rows and two boundary\nblocks with 14 rows each, hence 557 coefficient rows in total. Each block retains its own number of\ncoefficient columns."
        }
      ],
      "occurrenceCount": 6
    },
    {
      "symbol": "ψ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p6-9d760e4b18",
        "page": 6,
        "quote": "Let ψ be the zero extension of q −v. For L > 1, choose ηL : R →[0, 1] equal to 1 on [−L, L], equal\nto 0 outside [−L −1, L + 1], and linear on each intervening interval. Define"
      },
      "occurrences": [
        {
          "blockId": "p6-9d760e4b18",
          "page": 6,
          "quote": "Let ψ be the zero extension of q −v. For L > 1, choose ηL : R →[0, 1] equal to 1 on [−L, L], equal\nto 0 outside [−L −1, L + 1], and linear on each intervening interval. Define"
        },
        {
          "blockId": "p6-2f894589b6",
          "page": 6,
          "quote": "VL(y, s) = U(y, s) + ηL(s)ψ(y)."
        },
        {
          "blockId": "p15-888fe1eb00",
          "page": 15,
          "quote": "R |ψ′|2 dτ\nR"
        },
        {
          "blockId": "p15-51cc79bd7d",
          "page": 15,
          "quote": "R ψ2 dτ .\n(3.27)"
        },
        {
          "blockId": "p15-12704680e3",
          "page": 15,
          "quote": "To approach its constant, choose nonzero ψ ∈C∞\nc (R) and set ζk(r) = r−βψ((log r)/k).\nThe\nsubstitution τ = (log r)/k yields\nR ∞\n0 rm|ζ′\nk|2 dr\nR ∞\n0 rm−2ζ2\nk dr = β2 + 1"
        },
        {
          "blockId": "p15-91cc1d2aa7",
          "page": 15,
          "quote": "R(−βψ + k−1ψ′)2 dτ, and\nR\nψψ′ = 0 by compact support.\nDivide the separated stability inequality by its positive radial factor and let k →∞. In tangent\ndimension m = 5, one has β2 = 4, proving (3.26). The same computation on an original link with\nmo = 4 would give 9/4; the working link always has m = 5.\nThe order of limits is fixed: for each angular test and each finite k, first take ε →0 in the second\nvariation, and then take k →∞. Angular cutoffs used later are removed only after (3.26) has been\nobtained for each fixed smooth test. The collar and the allowable size of ε may depend on these\nfixed supports."
        }
      ],
      "occurrenceCount": 6
    }
  ],
  "issues": [
    {
      "id": "issue-1",
      "kind": "extraction-review",
      "message": "Fractions, integrals, exponents and cases can appear in a different order in extracted text. Statement boundaries were checked against rendered pages, including Lemma 7.1 across pages 30–31; mathematical typography must be read from the original PDF.",
      "source": {
        "blockId": "p25-66881bf618",
        "page": 25,
        "quote": "Lemma 6.3. For the constrained matrix A and symmetric trace-free tensor X,"
      },
      "related": []
    }
  ],
  "coverage": {
    "pages": 50,
    "blocks": 794,
    "method": "Numbered headings, explicit cross-references, nearby proof boundaries, and candidate notation contexts. PDF text can be incomplete.",
    "externalSourcesRetrieved": false,
    "enrichment": "Agent-authored source-linked notes on selected results. Remaining extraction is automatic.",
    "reviewedReferenceCorrections": 11
  }
}