{
  "schemaVersion": 1,
  "generator": "readable-paper 0.1.4",
  "paperId": "ccc4175b074f",
  "version": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
  "sha256": "46caabb46552913329e6a1a6226ad2dc196bd231d117562d0495f19a5832ae70",
  "title": "A counterexample to the Monge ansatz\nfor the three-marginal Coulomb cost",
  "status": "Unverified reading map",
  "nodes": [
    {
      "id": "theorem-1",
      "kind": "theorem",
      "label": "Theorem 1.1",
      "source": {
        "blockId": "p1-3eb947caad",
        "page": 1,
        "quote": "Theorem 1.1. There is a nonnegative density ρ on R3 such that\nZ"
      },
      "statement": [
        {
          "blockId": "p1-3eb947caad",
          "page": 1,
          "quote": "Theorem 1.1. There is a nonnegative density ρ on R3 such that\nZ"
        },
        {
          "blockId": "p1-b06be582b0",
          "page": 1,
          "quote": "R3 ρ(x) dx = 1,\nρ ∈C∞\nc (R3),\n√ρ ∈C∞\nc (R3),"
        },
        {
          "blockId": "p1-6b8d97f5fa",
          "page": 1,
          "quote": "with the following properties. For µ = ρ dx, the value E∗(µ) is finite and is attained by a coupling\nin Π(µ, µ, µ), but every pair of Borel maps T2, T3 : R3 →R3 satisfying (T2)#µ = (T3)#µ = µ obeys\nZ"
        },
        {
          "blockId": "p1-87b4f5f8c4",
          "page": 1,
          "quote": "R3 c\n\nx, T2(x), T3(x)\n\ndµ(x) > E∗(µ)."
        },
        {
          "blockId": "p1-509aaf5ec1",
          "page": 1,
          "quote": "The inequality permits the left-hand side to be infinite."
        }
      ],
      "proof": [
        {
          "blockId": "p7-4e494b71af",
          "page": 7,
          "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
        },
        {
          "blockId": "p7-5c7f30b349",
          "page": 7,
          "quote": "A smooth marginal.\nChoose a nonnegative function g ∈C∞\nc (B) with\nR"
        },
        {
          "blockId": "p7-ced4283f40",
          "page": 7,
          "quote": "B g2 dx = 1, and let\nν = g2 dx, regarded as a probability measure on R3. Define"
        },
        {
          "blockId": "p7-849909936f",
          "page": 7,
          "quote": "X"
        },
        {
          "blockId": "p7-fc3536fd08",
          "page": 7,
          "quote": "µ = 1"
        },
        {
          "blockId": "p7-cad5d5c99c",
          "page": 7,
          "quote": "3ν + 1"
        },
        {
          "blockId": "p7-288e67533e",
          "page": 7,
          "quote": "6"
        },
        {
          "blockId": "p7-2d7981f365",
          "page": 7,
          "quote": "H∈H\nH#ν.\n(9)"
        },
        {
          "blockId": "p7-dfbebdc7a0",
          "page": 7,
          "quote": "These coefficients sum to one. For H ∈H, define"
        },
        {
          "blockId": "p7-fb67544110",
          "page": 7,
          "quote": "gH(y) ="
        },
        {
          "blockId": "p7-892fb7f8e7",
          "page": 7,
          "quote": "(\ng(H−1(y)) | det DH−1(y)|1/2,\ny ∈H(B),\n0,\ny /∈H(B)."
        },
        {
          "blockId": "p7-927127455f",
          "page": 7,
          "quote": "The formula is smooth on H(B). Its support is contained in the compact set H(supp g) ⊂H(B), so\nextension by zero is smooth on R3. Thus gH ∈C∞\nc (R3), and change of variables gives H#ν = g2\nH dy.\nIt follows that µ = ρ dx, where"
        },
        {
          "blockId": "p7-1ea5a1b2fb",
          "page": 7,
          "quote": "X"
        },
        {
          "blockId": "p7-ada17ad119",
          "page": 7,
          "quote": "ρ = 1"
        },
        {
          "blockId": "p7-691e8d1283",
          "page": 7,
          "quote": "3g2 + 1"
        },
        {
          "blockId": "p7-237303e515",
          "page": 7,
          "quote": "6"
        },
        {
          "blockId": "p7-49c01783c1",
          "page": 7,
          "quote": "H∈H\ng2\nH."
        },
        {
          "blockId": "p8-5ae458df94",
          "page": 8,
          "quote": "The functions here are nonnegative and have pairwise disjoint supports, since their supports lie in\nthe five disjoint components. Consequently"
        },
        {
          "blockId": "p8-781a8924e4",
          "page": 8,
          "quote": "X"
        },
        {
          "blockId": "p8-31b96f11f6",
          "page": 8,
          "quote": "√ρ =\n1\n√"
        },
        {
          "blockId": "p8-bfa2aff894",
          "page": 8,
          "quote": "3g + 1\n√"
        },
        {
          "blockId": "p8-23b88e6388",
          "page": 8,
          "quote": "6"
        },
        {
          "blockId": "p8-c9906286cd",
          "page": 8,
          "quote": "H∈H\ngH."
        },
        {
          "blockId": "p8-88a56eff47",
          "page": 8,
          "quote": "This proves both asserted smoothness properties, compact support, and normalization.\nAlso\nµ(U) = 1 and µ|B = (1/3)ν."
        },
        {
          "blockId": "p8-3e0cd3485e",
          "page": 8,
          "quote": "A finite optimal coupling.\nSample x with law ν, choose k uniformly from {1, 2}, independently\nof x, and apply an independent uniform permutation to the three entries of"
        },
        {
          "blockId": "p8-65db13df65",
          "page": 8,
          "quote": "(x, Yk(x), Wk(x))."
        },
        {
          "blockId": "p8-73de921bb7",
          "page": 8,
          "quote": "Let π0 be the resulting law. In any fixed coordinate the central point has coefficient 1/3, and each\nspecified outer map has coefficient (1/2)(1/3) = 1/6. Hence every coordinate marginal is exactly µ.\nAll sampled triples satisfy (7) with equality. The boundedness of u therefore gives\nZ\nc dπ0 = 3\nZ"
        },
        {
          "blockId": "p8-2e7e6b930b",
          "page": 8,
          "quote": "U\nu dµ < ∞.\n(10)"
        },
        {
          "blockId": "p8-147638b826",
          "page": 8,
          "quote": "Conversely, every π ∈Π(µ, µ, µ) has π(U3) = 1: each of the three events xi /∈U has probability\nzero. Integrating (7) yields\nZ\nc dπ ≥3\nZ"
        },
        {
          "blockId": "p8-9f3c86b07c",
          "page": 8,
          "quote": "U\nu dµ."
        },
        {
          "blockId": "p8-7b7687a300",
          "page": 8,
          "quote": "Thus π0 attains E∗(µ), and the latter is finite. Only the inequality on U3 is used.\nFor any optimal π, the nonnegative slack c(x1, x2, x3)−P\ni u(xi) on U3 has integral zero. Indeed\nthe cost is finite and the potential sum is bounded, so subtraction and integration are legitimate. It\nfollows that π is concentrated on the equality triples in (8)."
        },
        {
          "blockId": "p8-9ce4cec472",
          "page": 8,
          "quote": "No optimal maps.\nSuppose that Borel maps T2, T3, each preserving µ, induce an optimal plan.\nThe preceding equality condition holds at (x, T2(x), T3(x)) for µ-almost every x, hence for ν-almost\nevery x ∈B.\nThe first coordinate x ∈B ⊂U0 is the unique central coordinate of an equality triple. The\nremaining coordinates lie one each in U+\nk and U−\nk for some k. If their values in those components\nare y, z, then x = Xk(y) and z = Zk(y). Injectivity of Xk forces"
        },
        {
          "blockId": "p8-ac03a08634",
          "page": 8,
          "quote": "y = Yk(x),\nz = Wk(x)."
        },
        {
          "blockId": "p8-8893c7b48e",
          "page": 8,
          "quote": "Allowing either order of these coordinates and either value of k, we obtain"
        },
        {
          "blockId": "p8-60ba255cef",
          "page": 8,
          "quote": "T2(x) ∈{Y1(x), W1(x), Y2(x), W2(x)}\nfor ν-almost every x ∈B."
        },
        {
          "blockId": "p8-76a40e8071",
          "page": 8,
          "quote": "This contradicts Lemma 2.1 applied to (9). Thus no Monge plan attains the minimum, completing\nthe proof of Theorem 1.1."
        }
      ],
      "guide": [
        {
          "text": "Read this as a nonattainment statement: an optimal coupling exists, but no pair of measure-preserving Borel maps attains that optimum. Equality of infima is a separate claim treated in Proposition 6.1.",
          "sources": [
            {
              "blockId": "p1-6b8d97f5fa",
              "page": 1,
              "quote": "with the following properties. For µ = ρ dx, the value E∗(µ) is finite and is attained by a coupling\nin Π(µ, µ, µ), but every pair of Borel maps T2, T3 : R3 →R3 satisfying (T2)#µ = (T3)#µ = µ obeys\nZ"
            },
            {
              "blockId": "p9-fd5691984e",
              "page": 9,
              "quote": "Proposition 6.1. For the probability measure µ of Theorem 1.1,"
            }
          ]
        },
        {
          "text": "The construction uses two families of triples sharing a central component. Proposition 4.1 provides a supporting potential, and Lemma 4.2 supplies the branch maps used to build the marginal.",
          "sources": [
            {
              "blockId": "p7-4e494b71af",
              "page": 7,
              "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
            },
            {
              "blockId": "p7-6fbfaf0c9c",
              "page": 7,
              "quote": "Proof. Both Xk(U+\nk ) are open neighborhoods of zero. Choose a positive-radius ball B about zero\ninside their intersection and U0. The inverse maps then take B into U+\nk , and Proposition 4.1 gives\nZk(Yk(B)) ⊂U−\nk . The inverse and composition of the relevant diffeomorphisms are diffeomorphisms\nonto open images. The identities Xk(Yk(x)) = x and Wk = Zk ◦Yk give equality by Proposition 4.1.\nDisjointness follows from the five component balls."
            }
          ]
        },
        {
          "text": "An explicit coupling attains the lower bound from the supporting inequality. Zero integrated slack then forces every optimal coupling onto the equality triples.",
          "sources": [
            {
              "blockId": "p8-7b7687a300",
              "page": 8,
              "quote": "Thus π0 attains E∗(µ), and the latter is finite. Only the inequality on U3 is used.\nFor any optimal π, the nonnegative slack c(x1, x2, x3)−P\ni u(xi) on U3 has integral zero. Indeed\nthe cost is finite and the potential sum is bounded, so subtraction and integration are legitimate. It\nfollows that π is concentrated on the equality triples in (8)."
            }
          ]
        },
        {
          "text": "A deterministic map would have to choose a branch. The image-measure mismatch in Lemma 2.1 rules out a preserving choice and completes the argument.",
          "sources": [
            {
              "blockId": "p8-76a40e8071",
              "page": 8,
              "quote": "This contradicts Lemma 2.1 applied to (9). Thus no Monge plan attains the minimum, completing\nthe proof of Theorem 1.1."
            },
            {
              "blockId": "p4-c5e9cbe917",
              "page": 4,
              "quote": "Thus ν(Sj) = 0 for all four indices, contradicting (3)."
            }
          ]
        }
      ],
      "guideStatus": "Agent-authored reading guide; not independently verified."
    },
    {
      "id": "lemma-2",
      "kind": "lemma",
      "label": "Lemma 2.1",
      "source": {
        "blockId": "p3-e8854be191",
        "page": 3,
        "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
      },
      "statement": [
        {
          "blockId": "p3-e8854be191",
          "page": 3,
          "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
        },
        {
          "blockId": "p3-3edd11fbf4",
          "page": 3,
          "quote": "4\nX"
        },
        {
          "blockId": "p3-83e41983b6",
          "page": 3,
          "quote": "µ = 1"
        },
        {
          "blockId": "p3-f71745260b",
          "page": 3,
          "quote": "3ν + 1"
        },
        {
          "blockId": "p3-3e787e84ff",
          "page": 3,
          "quote": "6"
        },
        {
          "blockId": "p3-3fbed76b34",
          "page": 3,
          "quote": "j=1\n(Hj)#ν.\n(2)"
        },
        {
          "blockId": "p4-f7601be3cf",
          "page": 4,
          "quote": "There is no Borel map T : Rd →Rd such that T#µ = µ and"
        },
        {
          "blockId": "p4-2492224c54",
          "page": 4,
          "quote": "T(x) ∈{H1(x), . . . , H4(x)}\nfor ν-almost every x ∈B.\n(3)"
        }
      ],
      "proof": [
        {
          "blockId": "p4-d4181fb059",
          "page": 4,
          "quote": "Proof. For each j, define the Borel set"
        },
        {
          "blockId": "p4-82db7befeb",
          "page": 4,
          "quote": "Sj = {x ∈B : T(x) = Hj(x)}."
        },
        {
          "blockId": "p4-4d920b5d57",
          "page": 4,
          "quote": "Its image Hj(Sj) is Borel. Pairwise disjointness of the component images and injectivity of Hj give"
        },
        {
          "blockId": "p4-9c8f651839",
          "page": 4,
          "quote": "µ(Sj) = 1"
        },
        {
          "blockId": "p4-0e53ba8b07",
          "page": 4,
          "quote": "3ν(Sj),\nµ(Hj(Sj)) = 1"
        },
        {
          "blockId": "p4-5b17f1aa6e",
          "page": 4,
          "quote": "6ν(Sj)."
        },
        {
          "blockId": "p4-21e05c47ed",
          "page": 4,
          "quote": "Since Sj ⊂T −1(Hj(Sj)), preservation of µ would imply"
        },
        {
          "blockId": "p4-27ba0f761e",
          "page": 4,
          "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
        },
        {
          "blockId": "p4-5236852130",
          "page": 4,
          "quote": "6ν(Sj)."
        },
        {
          "blockId": "p4-c5e9cbe917",
          "page": 4,
          "quote": "Thus ν(Sj) = 0 for all four indices, contradicting (3)."
        },
        {
          "blockId": "p4-44926de852",
          "page": 4,
          "quote": "The argument does not require ν to be uniform or even absolutely continuous. Additional\npoints mapped into Hj(Sj) can only increase its preimage mass. Matching the total mass of each\ncomponent by a partition therefore cannot remove the obstruction."
        }
      ],
      "guide": [
        {
          "text": "Partition the central points into sets Sj according to which branch Hj the proposed map chooses. Each Sj is Borel.",
          "sources": [
            {
              "blockId": "p4-d4181fb059",
              "page": 4,
              "quote": "Proof. For each j, define the Borel set"
            },
            {
              "blockId": "p4-82db7befeb",
              "page": 4,
              "quote": "Sj = {x ∈B : T(x) = Hj(x)}."
            }
          ]
        },
        {
          "text": "The source mass on Sj is ν(Sj)/3, while the target mass on Hj(Sj) is ν(Sj)/6. The PDF breaks these equations across several extracted blocks; inspect the rendered page.",
          "sources": [
            {
              "blockId": "p4-9c8f651839",
              "page": 4,
              "quote": "µ(Sj) = 1"
            },
            {
              "blockId": "p4-0e53ba8b07",
              "page": 4,
              "quote": "3ν(Sj),\nµ(Hj(Sj)) = 1"
            },
            {
              "blockId": "p4-5b17f1aa6e",
              "page": 4,
              "quote": "6ν(Sj)."
            }
          ]
        },
        {
          "text": "Because Sj lies inside the preimage of Hj(Sj), preservation would imply ν(Sj)/3 ≤ ν(Sj)/6. Nonnegativity forces ν(Sj)=0. All four sets having zero mass contradicts the almost-everywhere branch condition.",
          "sources": [
            {
              "blockId": "p4-21e05c47ed",
              "page": 4,
              "quote": "Since Sj ⊂T −1(Hj(Sj)), preservation of µ would imply"
            },
            {
              "blockId": "p4-27ba0f761e",
              "page": 4,
              "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
            },
            {
              "blockId": "p4-5236852130",
              "page": 4,
              "quote": "6ν(Sj)."
            },
            {
              "blockId": "p4-c5e9cbe917",
              "page": 4,
              "quote": "Thus ν(Sj) = 0 for all four indices, contradicting (3)."
            }
          ]
        }
      ],
      "guideStatus": "Agent-authored reading guide; not independently verified."
    },
    {
      "id": "lemma-3",
      "kind": "lemma",
      "label": "Lemma 3.1",
      "source": {
        "blockId": "p4-3754dd8622",
        "page": 4,
        "quote": "Lemma 3.1. There are a number K > 0, open neighborhoods P k\n0 , P +\nk , P −\nk of 0, ak, −ak, respectively,\nand smooth maps Xk, Zk : P +\nk →R3, for k = 1, 2, with the following properties:"
      },
      "statement": [
        {
          "blockId": "p4-3754dd8622",
          "page": 4,
          "quote": "Lemma 3.1. There are a number K > 0, open neighborhoods P k\n0 , P +\nk , P −\nk of 0, ak, −ak, respectively,\nand smooth maps Xk, Zk : P +\nk →R3, for k = 1, 2, with the following properties:"
        },
        {
          "blockId": "p4-ef4d01f9e4",
          "page": 4,
          "quote": "1. Each of Xk and Zk is a diffeomorphism onto its open image, and"
        },
        {
          "blockId": "p4-b3e0784aca",
          "page": 4,
          "quote": "Xk(ak) = 0,\nZk(ak) = −ak."
        },
        {
          "blockId": "p4-9441961756",
          "page": 4,
          "quote": "2. Define"
        },
        {
          "blockId": "p4-0f278a759b",
          "page": 4,
          "quote": "u0(x) = 5"
        },
        {
          "blockId": "p4-d085ea5da9",
          "page": 4,
          "quote": "2 −K"
        },
        {
          "blockId": "p4-1aa298a712",
          "page": 4,
          "quote": "2 |x|2,\nvk(z) = 5"
        },
        {
          "blockId": "p4-0973954964",
          "page": 4,
          "quote": "4ak · (z + ak) −K"
        },
        {
          "blockId": "p4-fb0a7304a8",
          "page": 4,
          "quote": "2 |z + ak|2."
        },
        {
          "blockId": "p4-d468626056",
          "page": 4,
          "quote": "There is a smooth function wk : P +\nk →R, with wk(ak) = 0, such that on P k\n0 × P +\nk × P −\nk ,"
        },
        {
          "blockId": "p4-434aadaac6",
          "page": 4,
          "quote": "c(x, y, z) ≥u0(x) + wk(y) + vk(z).\n(5)"
        },
        {
          "blockId": "p4-eebdcc7708",
          "page": 4,
          "quote": "Equality holds if and only if (x, z) = (Xk(y), Zk(y))."
        }
      ],
      "proof": [
        {
          "blockId": "p5-66e92d6623",
          "page": 5,
          "quote": "Proof. For a parameter K > 0, use the potentials in the statement and set"
        },
        {
          "blockId": "p5-4f0708bcf7",
          "page": 5,
          "quote": "Fk(x, y, z) = c(x, y, z) −u0(x) −vk(z)."
        },
        {
          "blockId": "p5-3136c89e7b",
          "page": 5,
          "quote": "By (4), its (x, z)-gradient vanishes at the k-th center triple. We choose K to obtain a unique\nminimizing pair on a small product neighborhood for each fixed y, and to make each coordinate of\nthe resulting minimizing pair an invertible function of y.\nThe Coulomb cost is smooth near both center triples. At the k-th center, write"
        },
        {
          "blockId": "p5-aa841270ae",
          "page": 5,
          "quote": "Mk = D2\n(x,z)(x,z)c,\nNk = Dy∇(x,z)c."
        },
        {
          "blockId": "p5-c276c1f9c1",
          "page": 5,
          "quote": "The (x, z)-Hessian of Fk there is Mk + KI6, while its mixed derivative in y is Nk. For h(b) = |b|−1,\nb ̸= 0, differentiation gives"
        },
        {
          "blockId": "p5-91dd3c5657",
          "page": 5,
          "quote": "D2h(b) = |b|−3\n\n3bbT"
        },
        {
          "blockId": "p5-d57e57deae",
          "page": 5,
          "quote": "\n.\n(6)"
        },
        {
          "blockId": "p5-7f6afdbb72",
          "page": 5,
          "quote": "|b|2 −I3"
        },
        {
          "blockId": "p5-1f3374709b",
          "page": 5,
          "quote": "This matrix has eigenvalues 2|b|−3, −|b|−3, −|b|−3 and is therefore invertible. The two 3 × 3 blocks\nof Nk are\n−D2h(−ak),\n−D2h(−2ak),"
        },
        {
          "blockId": "p5-7409e62205",
          "page": 5,
          "quote": "and hence are both invertible. The matrices Mk are fixed symmetric 6 × 6 matrices. For all\nsufficiently large K, both Mk + KI6 are positive definite. Moreover,"
        },
        {
          "blockId": "p5-f71fff4cb8",
          "page": 5,
          "quote": "K(Mk + KI6)−1Nk = (I6 + K−1Mk)−1Nk −→Nk."
        },
        {
          "blockId": "p5-416df94a4d",
          "page": 5,
          "quote": "Invertibility is an open condition on 3 × 3 matrices, so we may choose a single K for which both\nblocks of (Mk + KI6)−1Nk are invertible for each k.\nFix such a K. The implicit function theorem applied to ∇(x,z)Fk = 0 gives a smooth stationary\nbranch\n∇(x,z)Fk(Xk(y), y, Zk(y)) = 0,"
        },
        {
          "blockId": "p5-2f8743dce6",
          "page": 5,
          "quote": "with the prescribed center values. Differentiating this equation at y = ak gives"
        },
        {
          "blockId": "p5-17960e6c6a",
          "page": 5,
          "quote": "D(Xk, Zk)(ak) = −(Mk + KI6)−1Nk."
        },
        {
          "blockId": "p5-1d44319a29",
          "page": 5,
          "quote": "The two block invertibilities allow the inverse function theorem to be applied separately to Xk and\nZk. After restricting to one sufficiently small neighborhood of ak, both maps are diffeomorphisms\nonto their open images.\nBy continuity, the (x, z)-Hessian of Fk remains positive definite on a product of sufficiently small\nopen balls around the three centers. Shrink the y-ball so that the stationary pair (Xk(y), Zk(y))\nbelongs to the product of the other two balls. For fixed y, this latter product is convex. The positive\ndefinite Hessian makes Fk(·, y, ·) strictly convex on it, so its stationary pair is its unique minimum\nthere.\nDefine its minimum value by"
        },
        {
          "blockId": "p5-997000a7e9",
          "page": 5,
          "quote": "wk(y) = Fk(Xk(y), y, Zk(y))."
        },
        {
          "blockId": "p5-f011e07e61",
          "page": 5,
          "quote": "The preceding minimum property gives (5) with exactly the stated equality set. The center values\nin (4) give wk(ak) = 0. All neighborhoods can be restricted further as needed to retain the separate\ndiffeomorphism properties."
        },
        {
          "blockId": "p6-627ccd39f8",
          "page": 6,
          "quote": "4\nA supporting inequality on five components"
        },
        {
          "blockId": "p6-3970a91003",
          "page": 6,
          "quote": "The two local inequalities use the same central potential. We can therefore combine their functions\ninto one potential on five disjoint components. To identify its entire equality set, we must also show\nstrict inequality for every component pattern other than permutations of the two opposite-pair\ntriples (0, ak, −ak)."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-4",
      "kind": "proposition",
      "label": "Proposition 4.1",
      "source": {
        "blockId": "p6-8eae37445a",
        "page": 6,
        "quote": "Proposition 4.1. There are pairwise disjoint open balls U0, U+\nk , U−\nk centered at 0, ak, −ak, respec-\ntively, and a bounded continuous function u : U →R, where"
      },
      "statement": [
        {
          "blockId": "p6-8eae37445a",
          "page": 6,
          "quote": "Proposition 4.1. There are pairwise disjoint open balls U0, U+\nk , U−\nk centered at 0, ak, −ak, respec-\ntively, and a bounded continuous function u : U →R, where"
        },
        {
          "blockId": "p6-077ea5e8d4",
          "page": 6,
          "quote": "U = U0 ∪U+\n1 ∪U−\n1 ∪U+\n2 ∪U−\n2 ,"
        },
        {
          "blockId": "p6-c280e38ab0",
          "page": 6,
          "quote": "such that"
        },
        {
          "blockId": "p6-0f636069cc",
          "page": 6,
          "quote": "3\nX"
        },
        {
          "blockId": "p6-b567e1219e",
          "page": 6,
          "quote": "c(x1, x2, x3) ≥"
        },
        {
          "blockId": "p6-bbd1947684",
          "page": 6,
          "quote": "i=1\nu(xi)\non U3.\n(7)"
        },
        {
          "blockId": "p6-66dd570b9e",
          "page": 6,
          "quote": "Equality holds precisely at the triples in U3 which, after a permutation, have the form"
        },
        {
          "blockId": "p6-92cf822209",
          "page": 6,
          "quote": "(Xk(y), y, Zk(y)) ∈U0 × U+\nk × U−\nk ,\ny ∈U+\nk ,\nk ∈{1, 2}.\n(8)"
        },
        {
          "blockId": "p6-eecde4ce7d",
          "page": 6,
          "quote": "The maps Xk, Zk are diffeomorphisms on U+\nk , with"
        },
        {
          "blockId": "p6-edc78b7a01",
          "page": 6,
          "quote": "Xk(U+\nk ) ⊂U0,\nZk(U+\nk ) ⊂U−\nk ."
        }
      ],
      "proof": [
        {
          "blockId": "p6-e46af3e914",
          "page": 6,
          "quote": "Proof. Use the functions and neighborhoods of Lemma 3.1. Choose small balls contained in these\nneighborhoods, taking U0 inside both central neighborhoods. On the respective components define"
        },
        {
          "blockId": "p6-9f2fa6fb18",
          "page": 6,
          "quote": "u = u0 on U0,\nu = wk on U+\nk ,\nu = vk on U−\nk ."
        },
        {
          "blockId": "p6-10b152e0ab",
          "page": 6,
          "quote": "The balls will be shrunk below. Since the functions are smooth near their centers, we can first\nrestrict to fixed preliminary balls on which their absolute values are bounded by one constant M > 0.\nTake the final balls sufficiently small that their closures are pairwise disjoint.\nAt the five centers the values of u are 5/2, 0, 0, 0, 0. For a triple of distinct centers other than a\npermutation of (0, ak, −ak), there are two possibilities. If zero is absent, its potential sum is zero\nand its cost is positive. If zero is present, the other two centers come from different opposite pairs.\nTheir cost is\n2 + 1\n√"
        },
        {
          "blockId": "p6-a49ff6ab70",
          "page": 6,
          "quote": "2,"
        },
        {
          "blockId": "p6-efab89c457",
          "page": 6,
          "quote": "2 > 5"
        },
        {
          "blockId": "p6-645759a579",
          "page": 6,
          "quote": "so again there is a strict gap. There are finitely many such triples, and the cost and local potentials\nare continuous near each one. All these strict inequalities therefore persist on the corresponding\nproducts after sufficiently small restrictions.\nFor a repeated-center type, two coordinates belong to the same ball. If all radii are at most r,\nthe distance between those two coordinates is at most 2r, and their pair cost is at least 1/(2r), with\nthe usual interpretation at coincidence. Taking r small enough that 1/(2r) > 3M makes the cost\nstrictly larger than the potential sum on every such product.\nThe inequalities of Lemma 3.1 remain valid when their domains are restricted, with the same\nequality characterization. Permuting a triple leaves both c and the sum of the three values of u\nunchanged. Thus the two prescribed component types and their permutations satisfy (7), with\nequality exactly as in (8); every other type has strict inequality. Finally, keep U0, U−\nk fixed and\nshrink each U+\nk once more. Since Xk(ak) = 0 and Zk(ak) = −ak, continuity ensures"
        },
        {
          "blockId": "p6-a40c491363",
          "page": 6,
          "quote": "Xk(U+\nk ) ⊂U0,\nZk(U+\nk ) ⊂U−\nk ."
        },
        {
          "blockId": "p7-e6d04fd3a5",
          "page": 7,
          "quote": "All inequalities and diffeomorphism properties survive this restriction. The function u is bounded\nand continuous on the final disjoint union U."
        },
        {
          "blockId": "p7-94baa1e105",
          "page": 7,
          "quote": "We now use the central point as parameter for both equality families. This gives four diffeomor-\nphisms on one common ball, as required by the selection obstruction and the construction of the\nsmooth marginal."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-5",
      "kind": "lemma",
      "label": "Lemma 4.2",
      "source": {
        "blockId": "p7-590711bf98",
        "page": 7,
        "quote": "Lemma 4.2. There is an open ball B of positive radius centered at zero, contained in U0, on which\nthe maps\nYk = (Xk|U+\nk )−1|B,\nWk = Zk ◦Yk\n(k = 1, 2)"
      },
      "statement": [
        {
          "blockId": "p7-590711bf98",
          "page": 7,
          "quote": "Lemma 4.2. There is an open ball B of positive radius centered at zero, contained in U0, on which\nthe maps\nYk = (Xk|U+\nk )−1|B,\nWk = Zk ◦Yk\n(k = 1, 2)"
        },
        {
          "blockId": "p7-266646cd67",
          "page": 7,
          "quote": "are well defined and are diffeomorphisms onto open images. They satisfy"
        },
        {
          "blockId": "p7-0b0266d320",
          "page": 7,
          "quote": "Yk(B) ⊂U+\nk ,\nWk(B) ⊂U−\nk ,"
        },
        {
          "blockId": "p7-af68789a88",
          "page": 7,
          "quote": "and every triple (x, Yk(x), Wk(x)), x ∈B, is an equality triple in (7). In particular, the five sets"
        },
        {
          "blockId": "p7-e935bf71a7",
          "page": 7,
          "quote": "B,\nY1(B),\nW1(B),\nY2(B),\nW2(B)"
        },
        {
          "blockId": "p7-47ea018b40",
          "page": 7,
          "quote": "are pairwise disjoint."
        }
      ],
      "proof": [
        {
          "blockId": "p7-6fbfaf0c9c",
          "page": 7,
          "quote": "Proof. Both Xk(U+\nk ) are open neighborhoods of zero. Choose a positive-radius ball B about zero\ninside their intersection and U0. The inverse maps then take B into U+\nk , and Proposition 4.1 gives\nZk(Yk(B)) ⊂U−\nk . The inverse and composition of the relevant diffeomorphisms are diffeomorphisms\nonto open images. The identities Xk(Yk(x)) = x and Wk = Zk ◦Yk give equality by Proposition 4.1.\nDisjointness follows from the five component balls."
        }
      ],
      "guide": []
    },
    {
      "id": "proposition-6",
      "kind": "proposition",
      "label": "Proposition 6.1",
      "source": {
        "blockId": "p9-fd5691984e",
        "page": 9,
        "quote": "Proposition 6.1. For the probability measure µ of Theorem 1.1,"
      },
      "statement": [
        {
          "blockId": "p9-fd5691984e",
          "page": 9,
          "quote": "Proposition 6.1. For the probability measure µ of Theorem 1.1,"
        },
        {
          "blockId": "p9-e5ffcd0470",
          "page": 9,
          "quote": "Z\nc(x, T2(x), T3(x)) dµ(x) = E∗(µ)."
        },
        {
          "blockId": "p9-b526ccd62f",
          "page": 9,
          "quote": "inf\nT2,T3:R3→R3 Borel\n(T2)#µ=(T3)#µ=µ"
        },
        {
          "blockId": "p9-860028ff5e",
          "page": 9,
          "quote": "Every pair in this infimum has cost strictly greater than the common infimum, but pairs with finite\ncost approach it."
        }
      ],
      "proof": [
        {
          "blockId": "p9-7bbd0ddf87",
          "page": 9,
          "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
        },
        {
          "blockId": "p9-b8dedfc947",
          "page": 9,
          "quote": "The optimal coupling π0 constructed in Section 5 is supported on products of three distinct\ncomponent sets.\nFix δ > 0. Partition each Ka into finitely many Borel cells of diameter at most δ, for example\nby a finite grid of half-open cubes. Discard the cells of zero µ-mass and denote the remaining cells\nby A1, . . . , AN. They partition a set of full µ-measure and each belongs to a single Ka. Put"
        },
        {
          "blockId": "p9-9e9d19809d",
          "page": 9,
          "quote": "qijk = π0(Ai × Aj × Ak)."
        },
        {
          "blockId": "p9-25f7e840f2",
          "page": 9,
          "quote": "The marginal constraints on π0 read\nX"
        },
        {
          "blockId": "p9-d493ef6808",
          "page": 9,
          "quote": "i,j\nqijk = µ(Ak)."
        },
        {
          "blockId": "p9-0e1bcd8931",
          "page": 9,
          "quote": "j,k\nqijk = µ(Ai),\nX"
        },
        {
          "blockId": "p9-590cb060b5",
          "page": 9,
          "quote": "i,k\nqijk = µ(Aj),\nX"
        },
        {
          "blockId": "p9-5997c90066",
          "page": 9,
          "quote": "For each fixed i, split Ai into Borel sets Bijk of masses qijk, ignoring zero-mass pieces. Such a split\ncan be made by successive first-coordinate slabs: the first-coordinate marginal of µ|Ai is absolutely\ncontinuous, so its cumulative distribution is continuous and takes every value between zero and\nµ(Ai). Endpoints of the slabs have zero mass; any unused null set is included in a fixed piece."
        },
        {
          "blockId": "p10-ffe9f03b89",
          "page": 10,
          "quote": "For each qijk > 0, Lemma 6.2 supplies maps on Bijk sending the probability q−1\nijkµ|Bijk, respec-\ntively, to\nµ|Aj\nµ(Aj)\nand\nµ|Ak\nµ(Ak)."
        },
        {
          "blockId": "p10-ff1f8e77c3",
          "page": 10,
          "quote": "Reset any exceptional values outside the indicated target cell to a fixed point in that cell. These\nchanges are Borel and occur on null sets. Glue the maps over the finitely many pieces to obtain\nT2, T3, and give them a fixed value on the remaining Borel null set in R3. Different source pieces\nmay use the same normalized target law. Their weights qijk add to the full mass of the target cell,\nso no target partition is needed. Indeed, for any Borel set D,"
        },
        {
          "blockId": "p10-71238b34a0",
          "page": 10,
          "quote": "µ(T −1\n2 (D)) =\nX"
        },
        {
          "blockId": "p10-fd5fd427a1",
          "page": 10,
          "quote": "µ(Aj)\n=\nX"
        },
        {
          "blockId": "p10-2b5b1cf1ec",
          "page": 10,
          "quote": "j\nµ(D ∩Aj) = µ(D),"
        },
        {
          "blockId": "p10-33c6b5f8a5",
          "page": 10,
          "quote": "i,j,k:qijk>0\nqijk\nµ(D ∩Aj)"
        },
        {
          "blockId": "p10-be5d523e56",
          "page": 10,
          "quote": "η2 ,\n|c(x) −c(y)| ≤6δ"
        },
        {
          "blockId": "p10-574128c370",
          "page": 10,
          "quote": "η2 ."
        },
        {
          "blockId": "p10-257031fbc4",
          "page": 10,
          "quote": "and the same calculation proves (T3)#µ = µ. The graph coupling πδ = (Id, T2, T3)#µ gives each\nproduct Ai × Aj × Ak exactly the mass qijk.\nWhenever qijk > 0, the three cells lie in three distinct component sets. Thus all pairwise distances\nin their product are at least η, including after the redistribution within the cells. If x and y are two\ntriples in the same such product, then |xℓ−yℓ| ≤δ for each coordinate. Consequently\n\n1\n|xℓ−xr| −\n1\n|yℓ−yr|"
        },
        {
          "blockId": "p10-7f1304e36b",
          "page": 10,
          "quote": "≤2δ"
        },
        {
          "blockId": "p10-51b31c7d14",
          "page": 10,
          "quote": "Z\nc dπδ −\nZ\nc dπ0"
        },
        {
          "blockId": "p10-4705349f52",
          "page": 10,
          "quote": "η2 ."
        },
        {
          "blockId": "p10-5ba5f1ef95",
          "page": 10,
          "quote": "Both costs are finite, bounded by 3/η on all active products. Since πδ and π0 have identical masses\non these products, summing their oscillation bounds gives"
        },
        {
          "blockId": "p10-46a19c1389",
          "page": 10,
          "quote": "≤6δ"
        },
        {
          "blockId": "p10-068e0becae",
          "page": 10,
          "quote": "Letting δ ↓0 proves the required upper bound on the Monge infimum. The reverse bound follows\nbecause every admissible graph plan is a coupling. Strict nonattainment is Theorem 1.1."
        },
        {
          "blockId": "p10-cd46c4a83d",
          "page": 10,
          "quote": "The argument uses a uniform separation estimate on the approximating plans themselves; it\ndoes not pass the singular Coulomb cost through weak convergence. No classification or uniqueness\nof optimal couplings is needed."
        }
      ],
      "guide": []
    },
    {
      "id": "lemma-7",
      "kind": "lemma",
      "label": "Lemma 6.2",
      "source": {
        "blockId": "p9-01a654890a",
        "page": 9,
        "quote": "Lemma 6.2. Let d ≥2 be an integer. If α and β are absolutely continuous Borel probability\nmeasures on Rd, there is a Borel map Q : Rd →Rd such that Q#α = β."
      },
      "statement": [
        {
          "blockId": "p9-01a654890a",
          "page": 9,
          "quote": "Lemma 6.2. Let d ≥2 be an integer. If α and β are absolutely continuous Borel probability\nmeasures on Rd, there is a Borel map Q : Rd →Rd such that Q#α = β."
        }
      ],
      "proof": [
        {
          "blockId": "p10-ba787c9958",
          "page": 10,
          "quote": "Proof of Lemma 6.2. We construct a Borel map from each such probability to uniform measure on\n(0, 1)d, and a Borel map in the reverse direction. Let f be a nonnegative Borel probability density.\nThe forward map is Rosenblatt’s successive conditional-distribution transform [10]; we also give the\ninverse quantile construction and make the Borel choices explicit in both directions.\nThe forward map. We successively transform each coordinate by its conditional distribution\nfunction given the preceding coordinates. Write x1:j = (x1, . . . , xj), with x1:0 the empty prefix, and\nset\nf1:j(x1:j) =\nZ"
        },
        {
          "blockId": "p10-8dd4a0cd47",
          "page": 10,
          "quote": "Rd−j f(x1:j, z) dz\n(1 ≤j < d),\nf1:d = f,\nf1:0 = 1."
        },
        {
          "blockId": "p10-5351c6dae9",
          "page": 10,
          "quote": "The prefix densities are Borel, have integral one, and are finite almost everywhere. Tonelli’s theorem\ngives\nR"
        },
        {
          "blockId": "p10-a847e6fe15",
          "page": 10,
          "quote": "R f1:j(a, r) dr = f1:j−1(a) for every prefix a, including the empty prefix when j = 1. For\nj = 1, . . . , d, define"
        },
        {
          "blockId": "p10-775112f8b1",
          "page": 10,
          "quote": "R t\n−∞f1:j(a, r) dr"
        },
        {
          "blockId": "p10-a6ce5dfdc2",
          "page": 10,
          "quote": "Fj(a, t) ="
        },
        {
          "blockId": "p10-5d65cb1326",
          "page": 10,
          "quote": "f1:j−1(a)\n,\na ∈Rj−1."
        },
        {
          "blockId": "p11-aa87aa0373",
          "page": 11,
          "quote": "This formula is used where its denominator belongs to (0, ∞); on the remaining fibers use instead the\nfixed distribution function t 7→(1 + e−t)−1. Thus every fiber is a continuous distribution function\nwith limits zero and one, and all Fj are jointly Borel. For j ≥2, the exceptional prefix fibers have\nzero probability under f1:j−1(a) da; there are no exceptional fibers when j = 1.\nFor a random variable with a continuous distribution function F, the variable F(X) is uniform\non (0, 1). Indeed, continuity and the limits of F allow a cut at each probability level u ∈(0, 1); any\ninterval on which F = u has probability zero, and hence P(F(X) ≤u) = u. At stage j, this identity\nand Fubini show that, conditionally on the preceding original prefix, the new transformed coordinate\nis uniform. It is therefore independent of that prefix, and hence of all preceding transformed\ncoordinates. A finite induction over j = 1, . . . , d shows that the Borel map"
        },
        {
          "blockId": "p11-0fd591943c",
          "page": 11,
          "quote": "Rf(x) =\n\nFj(x1:j−1, xj)\nd\nj=1"
        },
        {
          "blockId": "p11-e6b921f6e6",
          "page": 11,
          "quote": "sends f dx to product uniform measure on (0, 1)d. Its values lie in [0, 1]d, including on exceptional\nfibers. This argument does not require a density to be positive everywhere or a distribution function\nto be strictly increasing.\nThe reverse map. Given independent uniform coordinates, we successively recover the original\nconditional laws by quantiles. For each j and 0 < u < 1, define"
        },
        {
          "blockId": "p11-e1902320ae",
          "page": 11,
          "quote": "qj(a, u) = inf{r ∈Q : Fj(a, r) ≥u}."
        },
        {
          "blockId": "p11-49f46b27ad",
          "page": 11,
          "quote": "This value is finite, and"
        },
        {
          "blockId": "p11-74a2b192fe",
          "page": 11,
          "quote": "{(a, u) : qj(a, u) < t} =\n["
        },
        {
          "blockId": "p11-5c99269ae4",
          "page": 11,
          "quote": "{(a, u) : Fj(a, r) ≥u}"
        },
        {
          "blockId": "p11-ed83e4f8c5",
          "page": 11,
          "quote": "r∈Q\nr<t"
        },
        {
          "blockId": "p11-1021423e11",
          "page": 11,
          "quote": "proves joint Borel measurability on 0 < u < 1. At u = 0, 1, define qj(a, u) = 0. For 0 < u < 1,\ncontinuity of Fj(a, ·) gives\nqj(a, u) ≤t\n⇐⇒\nu ≤Fj(a, t)."
        },
        {
          "blockId": "p11-65a64927e9",
          "page": 11,
          "quote": "Consequently qj(a, U) has distribution function Fj(a, ·) for uniform U. For u ∈[0, 1]d, recursively\nset\nzj = qj(z1:j−1, uj)\n(j = 1, . . . , d),\nSf(u) = (z1, . . . , zd),"
        },
        {
          "blockId": "p11-f9b5b634c4",
          "page": 11,
          "quote": "starting with the empty prefix. This finite recursion defines a Borel map. For independent uniform\ninputs, induction shows that after step j the output prefix has density f1:j: the preceding prefix has\ndensity f1:j−1, its exceptional fibers have probability zero, and on every other fiber the quantile has\nthe conditional distribution just defined. Thus Sf has output law f dx. If f and g are densities of α\nand β, respectively, then Sg ◦Rf has the required property."
        }
      ],
      "guide": []
    },
    {
      "id": "theorem-8",
      "kind": "theorem",
      "label": "Theorem 7.1",
      "source": {
        "blockId": "p12-7a431ba79d",
        "page": 12,
        "quote": "Theorem 7.1. For every integer d ≥2 and every real s > 0, there is a nonnegative density ρ on\nRd such that\nZ"
      },
      "statement": [
        {
          "blockId": "p12-7a431ba79d",
          "page": 12,
          "quote": "Theorem 7.1. For every integer d ≥2 and every real s > 0, there is a nonnegative density ρ on\nRd such that\nZ"
        },
        {
          "blockId": "p12-5f538e527b",
          "page": 12,
          "quote": "Rd ρ(x) dx = 1,\nρ ∈C∞\nc (Rd),\n√ρ ∈C∞\nc (Rd),"
        },
        {
          "blockId": "p12-a18b4af87d",
          "page": 12,
          "quote": "with the following properties. For µ = ρ dx, the value Es(µ) is finite and attained by a coupling, but\nevery pair of Borel maps T2, T3 : Rd →Rd with (T2)#µ = (T3)#µ = µ satisfies\nZ"
        },
        {
          "blockId": "p12-f7fe4fa7b7",
          "page": 12,
          "quote": "Rd cs\n\nx, T2(x), T3(x)\n\ndµ(x) > Es(µ)."
        },
        {
          "blockId": "p12-e2b2ffbe42",
          "page": 12,
          "quote": "The left-hand side may be infinite. Nevertheless,"
        },
        {
          "blockId": "p12-3ad9dff840",
          "page": 12,
          "quote": "Z\ncs(x, T2(x), T3(x)) dµ(x) = Es(µ),"
        },
        {
          "blockId": "p12-8a9ac73afd",
          "page": 12,
          "quote": "inf\nT2,T3:Rd→Rd Borel\n(T2)#µ=(T3)#µ=µ"
        },
        {
          "blockId": "p12-3542268785",
          "page": 12,
          "quote": "and pairs with finite cost approach this infimum."
        }
      ],
      "proof": [
        {
          "blockId": "p12-e8e995cecb",
          "page": 12,
          "quote": "Proof. Fix d ≥2 and s > 0, and put a1 = e1 and a2 = e2 in Rd. The measure argument already\napplies in this dimension. We verify the three properties of the cost that it needs: local equality\nfamilies given by diffeomorphisms, strict supporting inequalities on all other component products,\nand a uniform cost-oscillation bound on separated products."
        },
        {
          "blockId": "p12-7a79f58e63",
          "page": 12,
          "quote": "Local branches.\nFor hs(b) = |b|−s, b ̸= 0, the center jets are"
        },
        {
          "blockId": "p12-4bbd045149",
          "page": 12,
          "quote": "cs(0, ak, −ak) = 2 + 2−s,\n∇xcs(0, ak, −ak) = 0,\n∇zcs(0, ak, −ak) = s(1 + 2−s−1)ak."
        },
        {
          "blockId": "p12-edcc84f416",
          "page": 12,
          "quote": "As before, we keep the positive outer point y free and minimize in the other two coordinates. The\npotentials in Lemma 3.1 are replaced by"
        },
        {
          "blockId": "p12-ca7fb02386",
          "page": 12,
          "quote": "u0(x) = 2 + 2−s −K"
        },
        {
          "blockId": "p12-86e9a50c2b",
          "page": 12,
          "quote": "2 |x|2,\nvk(z) = s(1 + 2−s−1)ak · (z + ak) −K"
        },
        {
          "blockId": "p12-ef0374f13d",
          "page": 12,
          "quote": "2 |z + ak|2."
        },
        {
          "blockId": "p12-456db614e9",
          "page": 12,
          "quote": "The central potential is common to the two branches, and Fk = cs −u0 −vk has vanishing\n(x, z)-gradient at each center. Direct differentiation gives"
        },
        {
          "blockId": "p12-d761ee02cd",
          "page": 12,
          "quote": "\n."
        },
        {
          "blockId": "p12-d2ac4b25f6",
          "page": 12,
          "quote": "D2hs(b) = s|b|−s−2\n\n(s + 2)bbT"
        },
        {
          "blockId": "p12-83c74ac5bb",
          "page": 12,
          "quote": "|b|2 −Id"
        },
        {
          "blockId": "p12-6542b6bf81",
          "page": 12,
          "quote": "Its radial eigenvalue is s(s+1)|b|−s−2, and its tangential eigenvalues are −s|b|−s−2, so it is invertible.\nAt the k-th center, define Mk and Nk as in Section 3, with cs in place of c. The two d × d blocks\nof Nk are −D2hs(−ak) and −D2hs(−2ak), hence are invertible, while Mk is a symmetric 2d × 2d\nmatrix. For sufficiently large K, both Mk + KI2d are positive definite and both d × d blocks of\n(Mk + KI2d)−1Nk are invertible, because"
        },
        {
          "blockId": "p12-21aa52e93e",
          "page": 12,
          "quote": "K(Mk + KI2d)−1Nk = (I2d + K−1Mk)−1Nk −→Nk."
        },
        {
          "blockId": "p12-23d3ec1e7b",
          "page": 12,
          "quote": "One K works for the two branches. The implicit and inverse function arguments of Lemma 3.1 now\ngive separate local diffeomorphisms Xk, Zk on d-dimensional neighborhoods. The cost is smooth\nnear the collision-free center triples; after shrinking the product balls, the (x, z)-Hessian of Fk\nremains positive definite. Strict convexity on the product of the x- and z-balls therefore gives the\nsame local supporting inequality and exact equality set, with wk(ak) = 0."
        },
        {
          "blockId": "p13-e4f359f719",
          "page": 13,
          "quote": "The supporting inequality and the marginal.\nThe five center values of the potential are now\n2 + 2−s, 0, 0, 0, 0. A distinct-center triple containing zero but not an opposite pair has cost"
        },
        {
          "blockId": "p13-6bec47e3e1",
          "page": 13,
          "quote": "2 + 2−s/2 > 2 + 2−s."
        },
        {
          "blockId": "p13-edc3e66ccb",
          "page": 13,
          "quote": "A distinct-center triple without zero has positive cost and potential sum zero. These finitely many\nstrict gaps persist on small products by continuity. If two points lie in one ball of radius at most r,\ntheir pair cost is at least (2r)−s. This exceeds 3M for sufficiently small r, where M bounds the\nabsolute values of the preliminary potentials. Positivity of the other pair terms and permutation\nsymmetry of cs therefore give the full supporting inequality and exact contact set described in\nProposition 4.1, now in Rd. The proof of Lemma 4.2 then also applies in Rd.\nOn that common ball B, choose a nonnegative g ∈C∞\nc (B) with\nR"
        },
        {
          "blockId": "p13-3e919ebd7e",
          "page": 13,
          "quote": "B g2 dx = 1, and form ν and µ\nby (9) using the four resulting diffeomorphisms. The change-of-variables construction in Section 5,\nnow with d-dimensional Jacobians, gives the same smooth compactly supported density and square\nroot: each image function is supported compactly inside its open component, and the five supports\nare disjoint. The same random choice of a branch and permutation gives an optimal coupling π0\nwith the three marginals equal to µ. The bounded supporting potential proves its cost is finite and\nforces every optimal coupling onto the contact set. Applying Lemma 2.1 to the four branch maps\nthen excludes every marginal-preserving Borel pair, exactly as in Section 5."
        },
        {
          "blockId": "p13-56dc16656f",
          "page": 13,
          "quote": "Approximation by preserving maps.\nThe five compact component sets have a positive mutual\nseparation η, and π0 uses only products of three distinct components. Repeat the cell partition\nand redistribution from the proof of Proposition 6.1 in Rd, using Lemma 6.2. The first-coordinate\nmarginals used to split the cells are absolutely continuous by Fubini in every such dimension. The\nredistribution gives a graph coupling πδ of a preserving Borel pair with the same mass as π0 on\neach product of cells of diameter at most δ. Every active product still has all pair distances at least\nη. For two triples x, y in one such product, the scalar mean value theorem gives\n|xℓ−xr|−s −|yℓ−yr|−s ≤2sδ η−s−1,\n|cs(x) −cs(y)| ≤6sδ η−s−1."
        },
        {
          "blockId": "p13-85c2b519ef",
          "page": 13,
          "quote": "Z\ncs dπδ −\nZ\ncs dπ0"
        },
        {
          "blockId": "p13-7ff671788f",
          "page": 13,
          "quote": "Both costs are bounded by 3η−s on the active products. Summing their oscillations against the\ncommon product masses yields"
        },
        {
          "blockId": "p13-9d0ee63a5c",
          "page": 13,
          "quote": "≤6sδ η−s−1 −→0."
        },
        {
          "blockId": "p13-9928d49f9c",
          "page": 13,
          "quote": "The reverse inequality for the infima holds because every graph plan is a coupling. This proves\nthe final assertion. In particular, no restriction relating s to d is needed: the optimizer and these\napproximating plans stay away from all collisions."
        }
      ],
      "guide": []
    },
    {
      "id": "external-9",
      "kind": "external",
      "label": "Reference [1]",
      "source": {
        "blockId": "p13-2fb9257484",
        "page": 13,
        "quote": "[1] Ugo Bindini, Luigi De Pascale, and Anna Kausamo. On deterministic solutions for multi-\nmarginal optimal transport with Coulomb cost. Communications on Pure and Applied Analysis,\n21(4):1189–1208, 2022. doi:10.3934/cpaa.2022015. Preprint: arXiv:2011.05063v2."
      },
      "statement": [
        {
          "blockId": "p13-2fb9257484",
          "page": 13,
          "quote": "[1] Ugo Bindini, Luigi De Pascale, and Anna Kausamo. On deterministic solutions for multi-\nmarginal optimal transport with Coulomb cost. Communications on Pure and Applied Analysis,\n21(4):1189–1208, 2022. doi:10.3934/cpaa.2022015. Preprint: arXiv:2011.05063v2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-10",
      "kind": "external",
      "label": "Reference [2]",
      "source": {
        "blockId": "p13-bbcc1b1308",
        "page": 13,
        "quote": "[2] Maria Colombo, Luigi De Pascale, and Simone Di Marino. Multimarginal optimal transport\nmaps for one-dimensional repulsive costs. Canadian Journal of Mathematics, 67(2):350–368,\n2015. doi:10.4153/CJM-2014-011-x."
      },
      "statement": [
        {
          "blockId": "p13-bbcc1b1308",
          "page": 13,
          "quote": "[2] Maria Colombo, Luigi De Pascale, and Simone Di Marino. Multimarginal optimal transport\nmaps for one-dimensional repulsive costs. Canadian Journal of Mathematics, 67(2):350–368,\n2015. doi:10.4153/CJM-2014-011-x."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-11",
      "kind": "external",
      "label": "Reference [3]",
      "source": {
        "blockId": "p14-d94543915b",
        "page": 14,
        "quote": "[3] Maria Colombo and Simone Di Marino. Equality between Monge and Kantorovich multi-\nmarginal problems with Coulomb cost. Annali di Matematica Pura ed Applicata, 194:307–320,\n2015. doi:10.1007/s10231-013-0376-0. Author version, April 11, 2013: https://cvgmt.sns.it/\nmedia/doc/paper/2127/MultiColDim.pdf."
      },
      "statement": [
        {
          "blockId": "p14-d94543915b",
          "page": 14,
          "quote": "[3] Maria Colombo and Simone Di Marino. Equality between Monge and Kantorovich multi-\nmarginal problems with Coulomb cost. Annali di Matematica Pura ed Applicata, 194:307–320,\n2015. doi:10.1007/s10231-013-0376-0. Author version, April 11, 2013: https://cvgmt.sns.it/\nmedia/doc/paper/2127/MultiColDim.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-12",
      "kind": "external",
      "label": "Reference [4]",
      "source": {
        "blockId": "p14-c88646507c",
        "page": 14,
        "quote": "[4] Maria Colombo and Federico Stra. Counterexamples in multimarginal optimal transport with\nCoulomb cost and spherically symmetric data. Mathematical Models and Methods in Applied\nSciences, 26(6):1025–1049, 2016. doi:10.1142/S021820251650024X."
      },
      "statement": [
        {
          "blockId": "p14-c88646507c",
          "page": 14,
          "quote": "[4] Maria Colombo and Federico Stra. Counterexamples in multimarginal optimal transport with\nCoulomb cost and spherically symmetric data. Mathematical Models and Methods in Applied\nSciences, 26(6):1025–1049, 2016. doi:10.1142/S021820251650024X."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-13",
      "kind": "external",
      "label": "Reference [5]",
      "source": {
        "blockId": "p14-d1f9236ad4",
        "page": 14,
        "quote": "[5] Codina Cotar, Gero Friesecke, and Claudia Kl¨uppelberg.\nDensity functional theory and\noptimal transportation with Coulomb cost. Communications on Pure and Applied Mathematics,\n66(4):548–599, 2013. doi:10.1002/cpa.21437. Preprint: arXiv:1104.0603v1."
      },
      "statement": [
        {
          "blockId": "p14-d1f9236ad4",
          "page": 14,
          "quote": "[5] Codina Cotar, Gero Friesecke, and Claudia Kl¨uppelberg.\nDensity functional theory and\noptimal transportation with Coulomb cost. Communications on Pure and Applied Mathematics,\n66(4):548–599, 2013. doi:10.1002/cpa.21437. Preprint: arXiv:1104.0603v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-14",
      "kind": "external",
      "label": "Reference [6]",
      "source": {
        "blockId": "p14-95561ab43e",
        "page": 14,
        "quote": "[6] Gero Friesecke. Mass splitting in the time-discrete generalized Euler equations and non-Monge\nsolutions in multi-marginal optimal transport, 2026. arXiv:2601.02616v1."
      },
      "statement": [
        {
          "blockId": "p14-95561ab43e",
          "page": 14,
          "quote": "[6] Gero Friesecke. Mass splitting in the time-discrete generalized Euler equations and non-Monge\nsolutions in multi-marginal optimal transport, 2026. arXiv:2601.02616v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-15",
      "kind": "external",
      "label": "Reference [7]",
      "source": {
        "blockId": "p14-a172d288f0",
        "page": 14,
        "quote": "[7] Gero Friesecke, Augusto Gerolin, and Paola Gori-Giorgi. The strong-interaction limit of density\nfunctional theory. In Eric Canc`es and Gero Friesecke, editors, Density Functional Theory:\nModeling, Mathematical Analysis, Computational Methods, and Applications, Mathematics and\nMolecular Modeling, pages 183–266. Springer, Cham, 2023. doi:10.1007/978-3-031-22340-2 4.\nPreprint version (2022): arXiv:2202.09760v1."
      },
      "statement": [
        {
          "blockId": "p14-a172d288f0",
          "page": 14,
          "quote": "[7] Gero Friesecke, Augusto Gerolin, and Paola Gori-Giorgi. The strong-interaction limit of density\nfunctional theory. In Eric Canc`es and Gero Friesecke, editors, Density Functional Theory:\nModeling, Mathematical Analysis, Computational Methods, and Applications, Mathematics and\nMolecular Modeling, pages 183–266. Springer, Cham, 2023. doi:10.1007/978-3-031-22340-2 4.\nPreprint version (2022): arXiv:2202.09760v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-16",
      "kind": "external",
      "label": "Reference [8]",
      "source": {
        "blockId": "p14-bbf2edb314",
        "page": 14,
        "quote": "[8] Augusto Gerolin, Anna Kausamo, and Tapio Rajala. Nonexistence of optimal transport maps\nfor the multimarginal repulsive harmonic cost. SIAM Journal on Mathematical Analysis,\n51(3):2359–2371, 2019. doi:10.1137/18M1186514."
      },
      "statement": [
        {
          "blockId": "p14-bbf2edb314",
          "page": 14,
          "quote": "[8] Augusto Gerolin, Anna Kausamo, and Tapio Rajala. Nonexistence of optimal transport maps\nfor the multimarginal repulsive harmonic cost. SIAM Journal on Mathematical Analysis,\n51(3):2359–2371, 2019. doi:10.1137/18M1186514."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-17",
      "kind": "external",
      "label": "Reference [9]",
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        "quote": "[9] Brendan Pass.\nRemarks on the semi-classical Hohenberg–Kohn functional.\nNonlinearity,\n26(9):2731–2744, 2013. doi:10.1088/0951-7715/26/9/2731. Preprint: arXiv:1211.2766."
      },
      "statement": [
        {
          "blockId": "p14-678209f460",
          "page": 14,
          "quote": "[9] Brendan Pass.\nRemarks on the semi-classical Hohenberg–Kohn functional.\nNonlinearity,\n26(9):2731–2744, 2013. doi:10.1088/0951-7715/26/9/2731. Preprint: arXiv:1211.2766."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-18",
      "kind": "external",
      "label": "Reference [10]",
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        "page": 14,
        "quote": "[10] Murray Rosenblatt. Remarks on a multivariate transformation. The Annals of Mathematical\nStatistics, 23(3):470–472, 1952. doi:10.1214/aoms/1177729394."
      },
      "statement": [
        {
          "blockId": "p14-88cbfeb8ac",
          "page": 14,
          "quote": "[10] Murray Rosenblatt. Remarks on a multivariate transformation. The Annals of Mathematical\nStatistics, 23(3):470–472, 1952. doi:10.1214/aoms/1177729394."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "external-19",
      "kind": "external",
      "label": "Reference [11]",
      "source": {
        "blockId": "p14-254bfe3db3",
        "page": 14,
        "quote": "[11] Michael Seidl, Paola Gori-Giorgi, and Andreas Savin. Strictly correlated electrons in density-\nfunctional theory: A general formulation with applications to spherical densities. Physical\nReview A, 75(4):042511, 2007. arXiv:cond-mat/0701025v2."
      },
      "statement": [
        {
          "blockId": "p14-254bfe3db3",
          "page": 14,
          "quote": "[11] Michael Seidl, Paola Gori-Giorgi, and Andreas Savin. Strictly correlated electrons in density-\nfunctional theory: A general formulation with applications to spherical densities. Physical\nReview A, 75(4):042511, 2007. arXiv:cond-mat/0701025v2."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "Bibliographic entry only; external source not retrieved."
    },
    {
      "id": "equation-20",
      "kind": "equation",
      "label": "Equation (1)",
      "source": {
        "blockId": "p1-8f9be33b4d",
        "page": 1,
        "quote": "c(x1, x2, x3) =\n1\n|x1 −x2| +\n1\n|x1 −x3| +\n1\n|x2 −x3|,\n(1)"
      },
      "statement": [
        {
          "blockId": "p1-8f9be33b4d",
          "page": 1,
          "quote": "c(x1, x2, x3) =\n1\n|x1 −x2| +\n1\n|x1 −x3| +\n1\n|x2 −x3|,\n(1)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-21",
      "kind": "equation",
      "label": "Equation (2)",
      "source": {
        "blockId": "p3-3fbed76b34",
        "page": 3,
        "quote": "j=1\n(Hj)#ν.\n(2)"
      },
      "statement": [
        {
          "blockId": "p3-3fbed76b34",
          "page": 3,
          "quote": "j=1\n(Hj)#ν.\n(2)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-22",
      "kind": "equation",
      "label": "Equation (3)",
      "source": {
        "blockId": "p4-2492224c54",
        "page": 4,
        "quote": "T(x) ∈{H1(x), . . . , H4(x)}\nfor ν-almost every x ∈B.\n(3)"
      },
      "statement": [
        {
          "blockId": "p4-2492224c54",
          "page": 4,
          "quote": "T(x) ∈{H1(x), . . . , H4(x)}\nfor ν-almost every x ∈B.\n(3)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-23",
      "kind": "equation",
      "label": "Equation (4)",
      "source": {
        "blockId": "p4-b1127417ba",
        "page": 4,
        "quote": "4ak.\n(4)"
      },
      "statement": [
        {
          "blockId": "p4-b1127417ba",
          "page": 4,
          "quote": "4ak.\n(4)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-24",
      "kind": "equation",
      "label": "Equation (5)",
      "source": {
        "blockId": "p4-434aadaac6",
        "page": 4,
        "quote": "c(x, y, z) ≥u0(x) + wk(y) + vk(z).\n(5)"
      },
      "statement": [
        {
          "blockId": "p4-434aadaac6",
          "page": 4,
          "quote": "c(x, y, z) ≥u0(x) + wk(y) + vk(z).\n(5)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-25",
      "kind": "equation",
      "label": "Equation (6)",
      "source": {
        "blockId": "p5-d57e57deae",
        "page": 5,
        "quote": "\n.\n(6)"
      },
      "statement": [
        {
          "blockId": "p5-d57e57deae",
          "page": 5,
          "quote": "\n.\n(6)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-26",
      "kind": "equation",
      "label": "Equation (7)",
      "source": {
        "blockId": "p6-bbd1947684",
        "page": 6,
        "quote": "i=1\nu(xi)\non U3.\n(7)"
      },
      "statement": [
        {
          "blockId": "p6-bbd1947684",
          "page": 6,
          "quote": "i=1\nu(xi)\non U3.\n(7)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-27",
      "kind": "equation",
      "label": "Equation (8)",
      "source": {
        "blockId": "p6-92cf822209",
        "page": 6,
        "quote": "(Xk(y), y, Zk(y)) ∈U0 × U+\nk × U−\nk ,\ny ∈U+\nk ,\nk ∈{1, 2}.\n(8)"
      },
      "statement": [
        {
          "blockId": "p6-92cf822209",
          "page": 6,
          "quote": "(Xk(y), y, Zk(y)) ∈U0 × U+\nk × U−\nk ,\ny ∈U+\nk ,\nk ∈{1, 2}.\n(8)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-28",
      "kind": "equation",
      "label": "Equation (9)",
      "source": {
        "blockId": "p7-2d7981f365",
        "page": 7,
        "quote": "H∈H\nH#ν.\n(9)"
      },
      "statement": [
        {
          "blockId": "p7-2d7981f365",
          "page": 7,
          "quote": "H∈H\nH#ν.\n(9)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-29",
      "kind": "equation",
      "label": "Equation (10)",
      "source": {
        "blockId": "p8-2e7e6b930b",
        "page": 8,
        "quote": "U\nu dµ < ∞.\n(10)"
      },
      "statement": [
        {
          "blockId": "p8-2e7e6b930b",
          "page": 8,
          "quote": "U\nu dµ < ∞.\n(10)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "equation-30",
      "kind": "equation",
      "label": "Equation (11)",
      "source": {
        "blockId": "p11-71dd884c97",
        "page": 11,
        "quote": "1≤i<j≤3\n|xi −xj|−s,\n(11)"
      },
      "statement": [
        {
          "blockId": "p11-71dd884c97",
          "page": 11,
          "quote": "1≤i<j≤3\n|xi −xj|−s,\n(11)"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "external-31",
      "kind": "external",
      "label": "[3] Theorem 1.1",
      "source": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "statement": [
        {
          "blockId": "p2-c37de2ec8b",
          "page": 2,
          "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
        },
        {
          "blockId": "p14-d94543915b",
          "page": 14,
          "quote": "[3] Maria Colombo and Simone Di Marino. Equality between Monge and Kantorovich multi-\nmarginal problems with Coulomb cost. Annali di Matematica Pura ed Applicata, 194:307–320,\n2015. doi:10.1007/s10231-013-0376-0. Author version, April 11, 2013: https://cvgmt.sns.it/\nmedia/doc/paper/2127/MultiColDim.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-32",
      "kind": "external",
      "label": "[5] Theorem 3.6",
      "source": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "statement": [
        {
          "blockId": "p2-c37de2ec8b",
          "page": 2,
          "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
        },
        {
          "blockId": "p14-d1f9236ad4",
          "page": 14,
          "quote": "[5] Codina Cotar, Gero Friesecke, and Claudia Kl¨uppelberg.\nDensity functional theory and\noptimal transportation with Coulomb cost. Communications on Pure and Applied Mathematics,\n66(4):548–599, 2013. doi:10.1002/cpa.21437. Preprint: arXiv:1104.0603v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-33",
      "kind": "external",
      "label": "[2] Theorem 1.1",
      "source": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "statement": [
        {
          "blockId": "p2-c37de2ec8b",
          "page": 2,
          "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
        },
        {
          "blockId": "p13-bbcc1b1308",
          "page": 13,
          "quote": "[2] Maria Colombo, Luigi De Pascale, and Simone Di Marino. Multimarginal optimal transport\nmaps for one-dimensional repulsive costs. Canadian Journal of Mathematics, 67(2):350–368,\n2015. doi:10.4153/CJM-2014-011-x."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-34",
      "kind": "external",
      "label": "[7] Theorem 2.4",
      "source": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "statement": [
        {
          "blockId": "p2-c37de2ec8b",
          "page": 2,
          "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
        },
        {
          "blockId": "p14-a172d288f0",
          "page": 14,
          "quote": "[7] Gero Friesecke, Augusto Gerolin, and Paola Gori-Giorgi. The strong-interaction limit of density\nfunctional theory. In Eric Canc`es and Gero Friesecke, editors, Density Functional Theory:\nModeling, Mathematical Analysis, Computational Methods, and Applications, Mathematics and\nMolecular Modeling, pages 183–266. Springer, Cham, 2023. doi:10.1007/978-3-031-22340-2 4.\nPreprint version (2022): arXiv:2202.09760v1."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-35",
      "kind": "external",
      "label": "[3] Lemma 2.3",
      "source": {
        "blockId": "p9-b57243d837",
        "page": 9,
        "quote": "The obstruction above concerns attainment, not the value of the Monge infimum. We prove this\ndirectly for the constructed marginal. The partition and cost-oscillation argument adapts the\napproximation method of Colombo and Di Marino [3, author version, Lemma 2.3 and proof of\nTheorem 2.1]. Their general theorem treats the Coulomb singularity and cyclic maps for identical\nnonatomic marginals. Here the separated components allow an elementary proof, including the\nrequired Borel maps."
      },
      "statement": [
        {
          "blockId": "p9-b57243d837",
          "page": 9,
          "quote": "The obstruction above concerns attainment, not the value of the Monge infimum. We prove this\ndirectly for the constructed marginal. The partition and cost-oscillation argument adapts the\napproximation method of Colombo and Di Marino [3, author version, Lemma 2.3 and proof of\nTheorem 2.1]. Their general theorem treats the Coulomb singularity and cyclic maps for identical\nnonatomic marginals. Here the separated components allow an elementary proof, including the\nrequired Borel maps."
        },
        {
          "blockId": "p14-d94543915b",
          "page": 14,
          "quote": "[3] Maria Colombo and Simone Di Marino. Equality between Monge and Kantorovich multi-\nmarginal problems with Coulomb cost. Annali di Matematica Pura ed Applicata, 194:307–320,\n2015. doi:10.1007/s10231-013-0376-0. Author version, April 11, 2013: https://cvgmt.sns.it/\nmedia/doc/paper/2127/MultiColDim.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "external-36",
      "kind": "external",
      "label": "[3] Theorem 2.1",
      "source": {
        "blockId": "p9-b57243d837",
        "page": 9,
        "quote": "The obstruction above concerns attainment, not the value of the Monge infimum. We prove this\ndirectly for the constructed marginal. The partition and cost-oscillation argument adapts the\napproximation method of Colombo and Di Marino [3, author version, Lemma 2.3 and proof of\nTheorem 2.1]. Their general theorem treats the Coulomb singularity and cyclic maps for identical\nnonatomic marginals. Here the separated components allow an elementary proof, including the\nrequired Borel maps."
      },
      "statement": [
        {
          "blockId": "p9-b57243d837",
          "page": 9,
          "quote": "The obstruction above concerns attainment, not the value of the Monge infimum. We prove this\ndirectly for the constructed marginal. The partition and cost-oscillation argument adapts the\napproximation method of Colombo and Di Marino [3, author version, Lemma 2.3 and proof of\nTheorem 2.1]. Their general theorem treats the Coulomb singularity and cyclic maps for identical\nnonatomic marginals. Here the separated components allow an elementary proof, including the\nrequired Borel maps."
        },
        {
          "blockId": "p14-d94543915b",
          "page": 14,
          "quote": "[3] Maria Colombo and Simone Di Marino. Equality between Monge and Kantorovich multi-\nmarginal problems with Coulomb cost. Annali di Matematica Pura ed Applicata, 194:307–320,\n2015. doi:10.1007/s10231-013-0376-0. Author version, April 11, 2013: https://cvgmt.sns.it/\nmedia/doc/paper/2127/MultiColDim.pdf."
        }
      ],
      "proof": [],
      "guide": [],
      "externalStatus": "External result cited by this paper; statement and proof not retrieved."
    },
    {
      "id": "curated-37",
      "kind": "definition",
      "label": "Couplings Π(µ, µ, µ)",
      "source": {
        "blockId": "p1-3adbcf9fd6",
        "page": 1,
        "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
      },
      "statement": [
        {
          "blockId": "p1-3adbcf9fd6",
          "page": 1,
          "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
        },
        {
          "blockId": "p1-9ee395c707",
          "page": 1,
          "quote": "with all three coordinate marginals equal to µ. We use the Coulomb cost"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "curated-38",
      "kind": "definition",
      "label": "Monge plan and pushforward",
      "source": {
        "blockId": "p1-93cb8bc6c3",
        "page": 1,
        "quote": "Here and below | · | is the Euclidean norm. A Monge plan is a coupling of the form (Id, T2, T3)#µ,\nwhere T2, T3 : R3 →R3 are Borel maps and (T2)#µ = (T3)#µ = µ. For a Borel map T, the\npushforward is defined by T#µ(A) = µ(T −1(A)). For a map defined on a set of full measure, we use\nthe same notation for the pushforward of the measure restricted to that set.\nThe question is whether absolute continuity of µ, together with E∗(µ) < ∞, guarantees an\noptimal Monge plan. We give a counterexample with additional regularity."
      },
      "statement": [
        {
          "blockId": "p1-93cb8bc6c3",
          "page": 1,
          "quote": "Here and below | · | is the Euclidean norm. A Monge plan is a coupling of the form (Id, T2, T3)#µ,\nwhere T2, T3 : R3 →R3 are Borel maps and (T2)#µ = (T3)#µ = µ. For a Borel map T, the\npushforward is defined by T#µ(A) = µ(T −1(A)). For a map defined on a set of full measure, we use\nthe same notation for the pushforward of the measure restricted to that set.\nThe question is whether absolute continuity of µ, together with E∗(µ) < ∞, guarantees an\noptimal Monge plan. We give a counterexample with additional regularity."
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "curated-39",
      "kind": "definition",
      "label": "Coulomb cost",
      "source": {
        "blockId": "p1-8f9be33b4d",
        "page": 1,
        "quote": "c(x1, x2, x3) =\n1\n|x1 −x2| +\n1\n|x1 −x3| +\n1\n|x2 −x3|,\n(1)"
      },
      "statement": [
        {
          "blockId": "p1-8f9be33b4d",
          "page": 1,
          "quote": "c(x1, x2, x3) =\n1\n|x1 −x2| +\n1\n|x1 −x3| +\n1\n|x2 −x3|,\n(1)"
        },
        {
          "blockId": "p1-54bddfaa9c",
          "page": 1,
          "quote": "where c = +∞if two coordinates coincide, and write"
        }
      ],
      "proof": [],
      "guide": []
    },
    {
      "id": "curated-40",
      "kind": "example",
      "label": "Illustration: mass imbalance",
      "source": {
        "blockId": "p4-27ba0f761e",
        "page": 4,
        "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
      },
      "statement": [
        {
          "blockId": "p4-27ba0f761e",
          "page": 4,
          "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
        }
      ],
      "proof": [],
      "guide": [
        {
          "text": "Agent-created numerical illustration, not an example printed in the paper: if ν(Sj)=0.6, the selected source set has mass 0.2 while its branch image has mass 0.1. A measure-preserving map cannot send all of that source set into that image.",
          "sources": [
            {
              "blockId": "p4-27ba0f761e",
              "page": 4,
              "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
            },
            {
              "blockId": "p4-5236852130",
              "page": 4,
              "quote": "6ν(Sj)."
            }
          ]
        },
        {
          "text": "This arithmetic illustrates the contradiction inside the proof. It does not construct a separate example satisfying every topological hypothesis of the lemma.",
          "sources": [
            {
              "blockId": "p3-e8854be191",
              "page": 3,
              "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
            }
          ]
        }
      ],
      "guideStatus": "Agent-authored reading guide; not independently verified."
    }
  ],
  "edges": [
    {
      "id": "edge-1",
      "from": "equation-21",
      "to": "lemma-2",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p3-3fbed76b34",
        "page": 3,
        "quote": "j=1\n(Hj)#ν.\n(2)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-2",
      "from": "equation-22",
      "to": "lemma-2",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p4-2492224c54",
        "page": 4,
        "quote": "T(x) ∈{H1(x), . . . , H4(x)}\nfor ν-almost every x ∈B.\n(3)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-3",
      "from": "equation-24",
      "to": "lemma-3",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p4-434aadaac6",
        "page": 4,
        "quote": "c(x, y, z) ≥u0(x) + wk(y) + vk(z).\n(5)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-4",
      "from": "equation-25",
      "to": "lemma-3",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p5-d57e57deae",
        "page": 5,
        "quote": "\n.\n(6)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-5",
      "from": "equation-26",
      "to": "proposition-4",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p6-bbd1947684",
        "page": 6,
        "quote": "i=1\nu(xi)\non U3.\n(7)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-6",
      "from": "equation-27",
      "to": "proposition-4",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p6-92cf822209",
        "page": 6,
        "quote": "(Xk(y), y, Zk(y)) ∈U0 × U+\nk × U−\nk ,\ny ∈U+\nk ,\nk ∈{1, 2}.\n(8)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-7",
      "from": "equation-28",
      "to": "theorem-1",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p7-2d7981f365",
        "page": 7,
        "quote": "H∈H\nH#ν.\n(9)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-8",
      "from": "equation-29",
      "to": "theorem-1",
      "relation": "occurs-in",
      "basis": "inferred",
      "evidence": {
        "blockId": "p8-2e7e6b930b",
        "page": 8,
        "quote": "U\nu dµ < ∞.\n(10)"
      },
      "note": "Placement in a statement or nearby proof; not a verified logical dependency."
    },
    {
      "id": "edge-9",
      "from": "external-19",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-10",
      "from": "external-15",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-11",
      "from": "external-31",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-12",
      "from": "external-32",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-13",
      "from": "external-33",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-14",
      "from": "external-17",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-15",
      "from": "external-12",
      "to": "theorem-1",
      "relation": "mentioned-in",
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      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-16",
      "from": "external-9",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-17",
      "from": "external-34",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "External result named in a citation. Its source has not been checked."
    },
    {
      "id": "edge-18",
      "from": "external-14",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-19",
      "from": "external-16",
      "to": "theorem-1",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p2-c37de2ec8b",
        "page": 2,
        "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-20",
      "from": "lemma-3",
      "to": "proposition-4",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p6-e46af3e914",
        "page": 6,
        "quote": "Proof. Use the functions and neighborhoods of Lemma 3.1. Choose small balls contained in these\nneighborhoods, taking U0 inside both central neighborhoods. On the respective components define"
      },
      "note": ""
    },
    {
      "id": "edge-21",
      "from": "lemma-3",
      "to": "proposition-4",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p6-645759a579",
        "page": 6,
        "quote": "so again there is a strict gap. There are finitely many such triples, and the cost and local potentials\nare continuous near each one. All these strict inequalities therefore persist on the corresponding\nproducts after sufficiently small restrictions.\nFor a repeated-center type, two coordinates belong to the same ball. If all radii are at most r,\nthe distance between those two coordinates is at most 2r, and their pair cost is at least 1/(2r), with\nthe usual interpretation at coincidence. Taking r small enough that 1/(2r) > 3M makes the cost\nstrictly larger than the potential sum on every such product.\nThe inequalities of Lemma 3.1 remain valid when their domains are restricted, with the same\nequality characterization. Permuting a triple leaves both c and the sum of the three values of u\nunchanged. Thus the two prescribed component types and their permutations satisfy (7), with\nequality exactly as in (8); every other type has strict inequality. Finally, keep U0, U−\nk fixed and\nshrink each U+\nk once more. Since Xk(ak) = 0 and Zk(ak) = −ak, continuity ensures"
      },
      "note": ""
    },
    {
      "id": "edge-22",
      "from": "proposition-4",
      "to": "lemma-5",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-6fbfaf0c9c",
        "page": 7,
        "quote": "Proof. Both Xk(U+\nk ) are open neighborhoods of zero. Choose a positive-radius ball B about zero\ninside their intersection and U0. The inverse maps then take B into U+\nk , and Proposition 4.1 gives\nZk(Yk(B)) ⊂U−\nk . The inverse and composition of the relevant diffeomorphisms are diffeomorphisms\nonto open images. The identities Xk(Yk(x)) = x and Wk = Zk ◦Yk give equality by Proposition 4.1.\nDisjointness follows from the five component balls."
      },
      "note": ""
    },
    {
      "id": "edge-23",
      "from": "proposition-4",
      "to": "theorem-1",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-4e494b71af",
        "page": 7,
        "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
      },
      "note": ""
    },
    {
      "id": "edge-24",
      "from": "lemma-5",
      "to": "theorem-1",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p7-4e494b71af",
        "page": 7,
        "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
      },
      "note": ""
    },
    {
      "id": "edge-25",
      "from": "lemma-2",
      "to": "theorem-1",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p8-76a40e8071",
        "page": 8,
        "quote": "This contradicts Lemma 2.1 applied to (9). Thus no Monge plan attains the minimum, completing\nthe proof of Theorem 1.1."
      },
      "note": ""
    },
    {
      "id": "edge-26",
      "from": "theorem-1",
      "to": "proposition-6",
      "relation": "mentioned-in",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p9-fd5691984e",
        "page": 9,
        "quote": "Proposition 6.1. For the probability measure µ of Theorem 1.1,"
      },
      "note": ""
    },
    {
      "id": "edge-27",
      "from": "lemma-7",
      "to": "proposition-6",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p10-ffe9f03b89",
        "page": 10,
        "quote": "For each qijk > 0, Lemma 6.2 supplies maps on Bijk sending the probability q−1\nijkµ|Bijk, respec-\ntively, to\nµ|Aj\nµ(Aj)\nand\nµ|Ak\nµ(Ak)."
      },
      "note": ""
    },
    {
      "id": "edge-28",
      "from": "theorem-1",
      "to": "proposition-6",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p10-068e0becae",
        "page": 10,
        "quote": "Letting δ ↓0 proves the required upper bound on the Monge infimum. The reverse bound follows\nbecause every admissible graph plan is a coupling. Strict nonattainment is Theorem 1.1."
      },
      "note": ""
    },
    {
      "id": "edge-29",
      "from": "external-18",
      "to": "lemma-7",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p10-ba787c9958",
        "page": 10,
        "quote": "Proof of Lemma 6.2. We construct a Borel map from each such probability to uniform measure on\n(0, 1)d, and a Borel map in the reverse direction. Let f be a nonnegative Borel probability density.\nThe forward map is Rosenblatt’s successive conditional-distribution transform [10]; we also give the\ninverse quantile construction and make the Borel choices explicit in both directions.\nThe forward map. We successively transform each coordinate by its conditional distribution\nfunction given the preceding coordinates. Write x1:j = (x1, . . . , xj), with x1:0 the empty prefix, and\nset\nf1:j(x1:j) =\nZ"
      },
      "note": "Citation establishes a textual reference, not a verified dependency on the external theorem."
    },
    {
      "id": "edge-30",
      "from": "lemma-3",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-edcc84f416",
        "page": 12,
        "quote": "As before, we keep the positive outer point y free and minimize in the other two coordinates. The\npotentials in Lemma 3.1 are replaced by"
      },
      "note": ""
    },
    {
      "id": "edge-31",
      "from": "lemma-3",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p12-23d3ec1e7b",
        "page": 12,
        "quote": "One K works for the two branches. The implicit and inverse function arguments of Lemma 3.1 now\ngive separate local diffeomorphisms Xk, Zk on d-dimensional neighborhoods. The cost is smooth\nnear the collision-free center triples; after shrinking the product balls, the (x, z)-Hessian of Fk\nremains positive definite. Strict convexity on the product of the x- and z-balls therefore gives the\nsame local supporting inequality and exact equality set, with wk(ak) = 0."
      },
      "note": ""
    },
    {
      "id": "edge-32",
      "from": "proposition-4",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-edc3e66ccb",
        "page": 13,
        "quote": "A distinct-center triple without zero has positive cost and potential sum zero. These finitely many\nstrict gaps persist on small products by continuity. If two points lie in one ball of radius at most r,\ntheir pair cost is at least (2r)−s. This exceeds 3M for sufficiently small r, where M bounds the\nabsolute values of the preliminary potentials. Positivity of the other pair terms and permutation\nsymmetry of cs therefore give the full supporting inequality and exact contact set described in\nProposition 4.1, now in Rd. The proof of Lemma 4.2 then also applies in Rd.\nOn that common ball B, choose a nonnegative g ∈C∞\nc (B) with\nR"
      },
      "note": ""
    },
    {
      "id": "edge-33",
      "from": "lemma-5",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-edc3e66ccb",
        "page": 13,
        "quote": "A distinct-center triple without zero has positive cost and potential sum zero. These finitely many\nstrict gaps persist on small products by continuity. If two points lie in one ball of radius at most r,\ntheir pair cost is at least (2r)−s. This exceeds 3M for sufficiently small r, where M bounds the\nabsolute values of the preliminary potentials. Positivity of the other pair terms and permutation\nsymmetry of cs therefore give the full supporting inequality and exact contact set described in\nProposition 4.1, now in Rd. The proof of Lemma 4.2 then also applies in Rd.\nOn that common ball B, choose a nonnegative g ∈C∞\nc (B) with\nR"
      },
      "note": ""
    },
    {
      "id": "edge-34",
      "from": "lemma-2",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-3e919ebd7e",
        "page": 13,
        "quote": "B g2 dx = 1, and form ν and µ\nby (9) using the four resulting diffeomorphisms. The change-of-variables construction in Section 5,\nnow with d-dimensional Jacobians, gives the same smooth compactly supported density and square\nroot: each image function is supported compactly inside its open component, and the five supports\nare disjoint. The same random choice of a branch and permutation gives an optimal coupling π0\nwith the three marginals equal to µ. The bounded supporting potential proves its cost is finite and\nforces every optimal coupling onto the contact set. Applying Lemma 2.1 to the four branch maps\nthen excludes every marginal-preserving Borel pair, exactly as in Section 5."
      },
      "note": ""
    },
    {
      "id": "edge-35",
      "from": "proposition-6",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-56dc16656f",
        "page": 13,
        "quote": "Approximation by preserving maps.\nThe five compact component sets have a positive mutual\nseparation η, and π0 uses only products of three distinct components. Repeat the cell partition\nand redistribution from the proof of Proposition 6.1 in Rd, using Lemma 6.2. The first-coordinate\nmarginals used to split the cells are absolutely continuous by Fubini in every such dimension. The\nredistribution gives a graph coupling πδ of a preserving Borel pair with the same mass as π0 on\neach product of cells of diameter at most δ. Every active product still has all pair distances at least\nη. For two triples x, y in one such product, the scalar mean value theorem gives\n|xℓ−xr|−s −|yℓ−yr|−s ≤2sδ η−s−1,\n|cs(x) −cs(y)| ≤6sδ η−s−1."
      },
      "note": ""
    },
    {
      "id": "edge-36",
      "from": "lemma-7",
      "to": "theorem-8",
      "relation": "cited-in-proof",
      "basis": "explicit-reference",
      "evidence": {
        "blockId": "p13-56dc16656f",
        "page": 13,
        "quote": "Approximation by preserving maps.\nThe five compact component sets have a positive mutual\nseparation η, and π0 uses only products of three distinct components. Repeat the cell partition\nand redistribution from the proof of Proposition 6.1 in Rd, using Lemma 6.2. The first-coordinate\nmarginals used to split the cells are absolutely continuous by Fubini in every such dimension. The\nredistribution gives a graph coupling πδ of a preserving Borel pair with the same mass as π0 on\neach product of cells of diameter at most δ. Every active product still has all pair distances at least\nη. For two triples x, y in one such product, the scalar mean value theorem gives\n|xℓ−xr|−s −|yℓ−yr|−s ≤2sδ η−s−1,\n|cs(x) −cs(y)| ≤6sδ η−s−1."
      },
      "note": ""
    },
    {
      "id": "edge-37",
      "from": "proposition-4",
      "to": "theorem-1",
      "relation": "supports",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p7-4e494b71af",
        "page": 7,
        "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
      },
      "note": "Agent-proposed dependency from the cited passage; not proof verification."
    },
    {
      "id": "edge-38",
      "from": "lemma-5",
      "to": "theorem-1",
      "relation": "supports",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p7-4e494b71af",
        "page": 7,
        "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
      },
      "note": "Agent-proposed dependency from the cited passage; not proof verification."
    },
    {
      "id": "edge-39",
      "from": "lemma-2",
      "to": "theorem-1",
      "relation": "supports",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p8-76a40e8071",
        "page": 8,
        "quote": "This contradicts Lemma 2.1 applied to (9). Thus no Monge plan attains the minimum, completing\nthe proof of Theorem 1.1."
      },
      "note": "Agent-proposed dependency from the cited passage; not proof verification."
    },
    {
      "id": "edge-40",
      "from": "curated-37",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-6b8d97f5fa",
        "page": 1,
        "quote": "with the following properties. For µ = ρ dx, the value E∗(µ) is finite and is attained by a coupling\nin Π(µ, µ, µ), but every pair of Borel maps T2, T3 : R3 →R3 satisfying (T2)#µ = (T3)#µ = µ obeys\nZ"
      },
      "note": "Agent-proposed dependency from the cited passage; not proof verification."
    },
    {
      "id": "edge-41",
      "from": "curated-38",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-93cb8bc6c3",
        "page": 1,
        "quote": "Here and below | · | is the Euclidean norm. A Monge plan is a coupling of the form (Id, T2, T3)#µ,\nwhere T2, T3 : R3 →R3 are Borel maps and (T2)#µ = (T3)#µ = µ. For a Borel map T, the\npushforward is defined by T#µ(A) = µ(T −1(A)). For a map defined on a set of full measure, we use\nthe same notation for the pushforward of the measure restricted to that set.\nThe question is whether absolute continuity of µ, together with E∗(µ) < ∞, guarantees an\noptimal Monge plan. We give a counterexample with additional regularity."
      },
      "note": "Agent-proposed dependency from the cited passage; not proof verification."
    },
    {
      "id": "edge-42",
      "from": "curated-39",
      "to": "theorem-1",
      "relation": "defines",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p1-8f9be33b4d",
        "page": 1,
        "quote": "c(x1, x2, x3) =\n1\n|x1 −x2| +\n1\n|x1 −x3| +\n1\n|x2 −x3|,\n(1)"
      },
      "note": "Agent-proposed dependency from the cited passage; not proof verification."
    },
    {
      "id": "edge-43",
      "from": "curated-40",
      "to": "lemma-2",
      "relation": "illustrates",
      "basis": "agent-inferred",
      "evidence": {
        "blockId": "p4-27ba0f761e",
        "page": 4,
        "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
      },
      "note": "Agent-created illustration of the displayed inequality; not an example in the source."
    }
  ],
  "notation": [
    {
      "symbol": "Yk, Wk",
      "meaning": "Local branch diffeomorphisms from the common central ball.",
      "status": "source-located",
      "scope": "Lemma 4.2, k = 1, 2.",
      "source": {
        "blockId": "p7-590711bf98",
        "page": 7,
        "quote": "Lemma 4.2. There is an open ball B of positive radius centered at zero, contained in U0, on which\nthe maps\nYk = (Xk|U+\nk )−1|B,\nWk = Zk ◦Yk\n(k = 1, 2)"
      },
      "occurrences": [
        {
          "blockId": "p7-590711bf98",
          "page": 7,
          "quote": "Lemma 4.2. There is an open ball B of positive radius centered at zero, contained in U0, on which\nthe maps\nYk = (Xk|U+\nk )−1|B,\nWk = Zk ◦Yk\n(k = 1, 2)"
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "B",
      "meaning": "Open central set in the abstract obstruction.",
      "status": "source-located",
      "scope": "Lemma 2.1; later chosen as a small ball for the construction.",
      "source": {
        "blockId": "p3-e8854be191",
        "page": 3,
        "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
      },
      "occurrences": [
        {
          "blockId": "p3-e8854be191",
          "page": 3,
          "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "µ",
      "meaning": "Common probability marginal; later constructed using central and branch measures.",
      "status": "source-located",
      "scope": "Generic in Section 1; constructed in Lemma 2.1 and Section 5.",
      "source": {
        "blockId": "p1-3adbcf9fd6",
        "page": 1,
        "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
      },
      "occurrences": [
        {
          "blockId": "p1-3adbcf9fd6",
          "page": 1,
          "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "T#µ",
      "meaning": "Pushforward of µ under the Borel map T.",
      "status": "source-located",
      "scope": "Defined in Section 1, including the full-measure-domain convention.",
      "source": {
        "blockId": "p1-93cb8bc6c3",
        "page": 1,
        "quote": "Here and below | · | is the Euclidean norm. A Monge plan is a coupling of the form (Id, T2, T3)#µ,\nwhere T2, T3 : R3 →R3 are Borel maps and (T2)#µ = (T3)#µ = µ. For a Borel map T, the\npushforward is defined by T#µ(A) = µ(T −1(A)). For a map defined on a set of full measure, we use\nthe same notation for the pushforward of the measure restricted to that set.\nThe question is whether absolute continuity of µ, together with E∗(µ) < ∞, guarantees an\noptimal Monge plan. We give a counterexample with additional regularity."
      },
      "occurrences": [
        {
          "blockId": "p1-93cb8bc6c3",
          "page": 1,
          "quote": "Here and below | · | is the Euclidean norm. A Monge plan is a coupling of the form (Id, T2, T3)#µ,\nwhere T2, T3 : R3 →R3 are Borel maps and (T2)#µ = (T3)#µ = µ. For a Borel map T, the\npushforward is defined by T#µ(A) = µ(T −1(A)). For a map defined on a set of full measure, we use\nthe same notation for the pushforward of the measure restricted to that set.\nThe question is whether absolute continuity of µ, together with E∗(µ) < ∞, guarantees an\noptimal Monge plan. We give a counterexample with additional regularity."
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "E∗(µ)",
      "meaning": "Infimum of the Coulomb transport cost over couplings.",
      "status": "source-located",
      "scope": "Sections 1–6. Formula spans extracted blocks; consult the PDF.",
      "source": {
        "blockId": "p1-835858d883",
        "page": 1,
        "quote": "E∗(µ) =\ninf\nπ∈Π(µ,µ,µ)"
      },
      "occurrences": [
        {
          "blockId": "p1-835858d883",
          "page": 1,
          "quote": "E∗(µ) =\ninf\nπ∈Π(µ,µ,µ)"
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "Π(µ, µ, µ)",
      "meaning": "Couplings with all three coordinate marginals equal to µ.",
      "status": "source-located",
      "scope": "R³ in Section 1; reintroduced on Rᵈ in Section 7.",
      "source": {
        "blockId": "p1-3adbcf9fd6",
        "page": 1,
        "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
      },
      "occurrences": [
        {
          "blockId": "p1-3adbcf9fd6",
          "page": 1,
          "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
        }
      ],
      "occurrenceCount": 1
    },
    {
      "symbol": "H",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-3d48e476fb",
        "page": 3,
        "quote": "The triples (0, ek, −ek) have the same Coulomb cost and the same central gradient. This allows\nthe two local families to share a supporting potential on B. Completing this potential on the four\nouter components gives an inequality c(x1, x2, x3) ≥u(x1) + u(x2) + u(x3), with equality precisely\non the two families. Strict cost gaps exclude the other triples of distinct centers; the Coulomb\nsingularity excludes repeated components after the neighborhoods have been made small enough.\nFor a smooth probability measure ν on B, we form the marginal µ by giving the central\ncomponent measure ν/3 and each outer component measure (Yk)#ν/6 or (Wk)#ν/6. Randomly\nchoosing a branch and permuting its three coordinates then gives an optimal coupling with this\ncommon marginal. Every optimizer must saturate the supporting inequality. In particular, on\nthe central component, a deterministic map must choose one of the four branch images. If it\nchooses a branch H on a Borel set S ⊂B, it sends mass ν(S)/3 into H(S), which has marginal\nmass only ν(S)/6. This contradicts preservation on every selection set of positive mass. The\nexact image-measure relation, rather than the component totals alone, excludes arbitrarily irregular\nchoices of branch.\nSections 2–5 establish the mass obstruction, construct and certify the two local families, and\nform the smooth marginal and an explicit optimizer. Section 6 constructs preserving maps whose\ncosts approach that optimum. Section 7 extends the construction to every inverse-power Riesz cost\nin every Euclidean dimension at least two."
      },
      "occurrences": [
        {
          "blockId": "p2-c37de2ec8b",
          "page": 2,
          "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
        },
        {
          "blockId": "p3-3d48e476fb",
          "page": 3,
          "quote": "The triples (0, ek, −ek) have the same Coulomb cost and the same central gradient. This allows\nthe two local families to share a supporting potential on B. Completing this potential on the four\nouter components gives an inequality c(x1, x2, x3) ≥u(x1) + u(x2) + u(x3), with equality precisely\non the two families. Strict cost gaps exclude the other triples of distinct centers; the Coulomb\nsingularity excludes repeated components after the neighborhoods have been made small enough.\nFor a smooth probability measure ν on B, we form the marginal µ by giving the central\ncomponent measure ν/3 and each outer component measure (Yk)#ν/6 or (Wk)#ν/6. Randomly\nchoosing a branch and permuting its three coordinates then gives an optimal coupling with this\ncommon marginal. Every optimizer must saturate the supporting inequality. In particular, on\nthe central component, a deterministic map must choose one of the four branch images. If it\nchooses a branch H on a Borel set S ⊂B, it sends mass ν(S)/3 into H(S), which has marginal\nmass only ν(S)/6. This contradicts preservation on every selection set of positive mass. The\nexact image-measure relation, rather than the component totals alone, excludes arbitrarily irregular\nchoices of branch.\nSections 2–5 establish the mass obstruction, construct and certify the two local families, and\nform the smooth marginal and an explicit optimizer. Section 6 constructs preserving maps whose\ncosts approach that optimum. Section 7 extends the construction to every inverse-power Riesz cost\nin every Euclidean dimension at least two."
        },
        {
          "blockId": "p3-e8854be191",
          "page": 3,
          "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
        },
        {
          "blockId": "p3-44e8fa9e28",
          "page": 3,
          "quote": "Figure 1: Schematic positions of the five disjoint components, with their marginal masses. Every\noptimal plan is concentrated on triples joining a central point to its two images with the same\nsubscript. The arrows denote local diffeomorphisms, not straight transport trajectories. For each\nindicated map H and Borel set S ⊂B, µ(H(S)) = µ(S)/2."
        },
        {
          "blockId": "p4-2492224c54",
          "page": 4,
          "quote": "T(x) ∈{H1(x), . . . , H4(x)}\nfor ν-almost every x ∈B.\n(3)"
        },
        {
          "blockId": "p7-4e494b71af",
          "page": 7,
          "quote": "We now prove Theorem 1.1. Let the balls, maps, and potential be as in Proposition 4.1 and\nLemma 4.2. Set\nH = {Y1, W1, Y2, W2}."
        },
        {
          "blockId": "p7-2d7981f365",
          "page": 7,
          "quote": "H∈H\nH#ν.\n(9)"
        },
        {
          "blockId": "p7-dfbebdc7a0",
          "page": 7,
          "quote": "These coefficients sum to one. For H ∈H, define"
        },
        {
          "blockId": "p7-892fb7f8e7",
          "page": 7,
          "quote": "(\ng(H−1(y)) | det DH−1(y)|1/2,\ny ∈H(B),\n0,\ny /∈H(B)."
        },
        {
          "blockId": "p7-927127455f",
          "page": 7,
          "quote": "The formula is smooth on H(B). Its support is contained in the compact set H(supp g) ⊂H(B), so\nextension by zero is smooth on R3. Thus gH ∈C∞\nc (R3), and change of variables gives H#ν = g2\nH dy.\nIt follows that µ = ρ dx, where"
        },
        {
          "blockId": "p7-49c01783c1",
          "page": 7,
          "quote": "H∈H\ng2\nH."
        },
        {
          "blockId": "p8-c9906286cd",
          "page": 8,
          "quote": "H∈H\ngH."
        },
        {
          "blockId": "p9-7bbd0ddf87",
          "page": 9,
          "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
        }
      ],
      "occurrenceCount": 13
    },
    {
      "symbol": "K0",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p9-7bbd0ddf87",
        "page": 9,
        "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
      },
      "occurrences": [
        {
          "blockId": "p9-7bbd0ddf87",
          "page": 9,
          "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
        }
      ],
      "occurrenceCount": 1,
      "latex": "K_{0}",
      "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": "p3-3d48e476fb",
        "page": 3,
        "quote": "The triples (0, ek, −ek) have the same Coulomb cost and the same central gradient. This allows\nthe two local families to share a supporting potential on B. Completing this potential on the four\nouter components gives an inequality c(x1, x2, x3) ≥u(x1) + u(x2) + u(x3), with equality precisely\non the two families. Strict cost gaps exclude the other triples of distinct centers; the Coulomb\nsingularity excludes repeated components after the neighborhoods have been made small enough.\nFor a smooth probability measure ν on B, we form the marginal µ by giving the central\ncomponent measure ν/3 and each outer component measure (Yk)#ν/6 or (Wk)#ν/6. Randomly\nchoosing a branch and permuting its three coordinates then gives an optimal coupling with this\ncommon marginal. Every optimizer must saturate the supporting inequality. In particular, on\nthe central component, a deterministic map must choose one of the four branch images. If it\nchooses a branch H on a Borel set S ⊂B, it sends mass ν(S)/3 into H(S), which has marginal\nmass only ν(S)/6. This contradicts preservation on every selection set of positive mass. The\nexact image-measure relation, rather than the component totals alone, excludes arbitrarily irregular\nchoices of branch.\nSections 2–5 establish the mass obstruction, construct and certify the two local families, and\nform the smooth marginal and an explicit optimizer. Section 6 constructs preserving maps whose\ncosts approach that optimum. Section 7 extends the construction to every inverse-power Riesz cost\nin every Euclidean dimension at least two."
      },
      "occurrences": [
        {
          "blockId": "p3-3d48e476fb",
          "page": 3,
          "quote": "The triples (0, ek, −ek) have the same Coulomb cost and the same central gradient. This allows\nthe two local families to share a supporting potential on B. Completing this potential on the four\nouter components gives an inequality c(x1, x2, x3) ≥u(x1) + u(x2) + u(x3), with equality precisely\non the two families. Strict cost gaps exclude the other triples of distinct centers; the Coulomb\nsingularity excludes repeated components after the neighborhoods have been made small enough.\nFor a smooth probability measure ν on B, we form the marginal µ by giving the central\ncomponent measure ν/3 and each outer component measure (Yk)#ν/6 or (Wk)#ν/6. Randomly\nchoosing a branch and permuting its three coordinates then gives an optimal coupling with this\ncommon marginal. Every optimizer must saturate the supporting inequality. In particular, on\nthe central component, a deterministic map must choose one of the four branch images. If it\nchooses a branch H on a Borel set S ⊂B, it sends mass ν(S)/3 into H(S), which has marginal\nmass only ν(S)/6. This contradicts preservation on every selection set of positive mass. The\nexact image-measure relation, rather than the component totals alone, excludes arbitrarily irregular\nchoices of branch.\nSections 2–5 establish the mass obstruction, construct and certify the two local families, and\nform the smooth marginal and an explicit optimizer. Section 6 constructs preserving maps whose\ncosts approach that optimum. Section 7 extends the construction to every inverse-power Riesz cost\nin every Euclidean dimension at least two."
        },
        {
          "blockId": "p3-44e8fa9e28",
          "page": 3,
          "quote": "Figure 1: Schematic positions of the five disjoint components, with their marginal masses. Every\noptimal plan is concentrated on triples joining a central point to its two images with the same\nsubscript. The arrows denote local diffeomorphisms, not straight transport trajectories. For each\nindicated map H and Borel set S ⊂B, µ(H(S)) = µ(S)/2."
        },
        {
          "blockId": "p9-7bbd0ddf87",
          "page": 9,
          "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
        }
      ],
      "occurrenceCount": 3
    },
    {
      "symbol": "a1",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p4-b26085603f",
        "page": 4,
        "quote": "Put a1 = e1 = (1, 0, 0) and a2 = e2 = (0, 1, 0). At each of the two triples (0, ak, −ak),"
      },
      "occurrences": [
        {
          "blockId": "p4-b26085603f",
          "page": 4,
          "quote": "Put a1 = e1 = (1, 0, 0) and a2 = e2 = (0, 1, 0). At each of the two triples (0, ak, −ak),"
        },
        {
          "blockId": "p12-e8e995cecb",
          "page": 12,
          "quote": "Proof. Fix d ≥2 and s > 0, and put a1 = e1 and a2 = e2 in Rd. The measure argument already\napplies in this dimension. We verify the three properties of the cost that it needs: local equality\nfamilies given by diffeomorphisms, strict supporting inequalities on all other component products,\nand a uniform cost-oscillation bound on separated products."
        }
      ],
      "occurrenceCount": 2,
      "latex": "a_{1}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "f",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p10-ba787c9958",
        "page": 10,
        "quote": "Proof of Lemma 6.2. We construct a Borel map from each such probability to uniform measure on\n(0, 1)d, and a Borel map in the reverse direction. Let f be a nonnegative Borel probability density.\nThe forward map is Rosenblatt’s successive conditional-distribution transform [10]; we also give the\ninverse quantile construction and make the Borel choices explicit in both directions.\nThe forward map. We successively transform each coordinate by its conditional distribution\nfunction given the preceding coordinates. Write x1:j = (x1, . . . , xj), with x1:0 the empty prefix, and\nset\nf1:j(x1:j) =\nZ"
      },
      "occurrences": [
        {
          "blockId": "p10-ba787c9958",
          "page": 10,
          "quote": "Proof of Lemma 6.2. We construct a Borel map from each such probability to uniform measure on\n(0, 1)d, and a Borel map in the reverse direction. Let f be a nonnegative Borel probability density.\nThe forward map is Rosenblatt’s successive conditional-distribution transform [10]; we also give the\ninverse quantile construction and make the Borel choices explicit in both directions.\nThe forward map. We successively transform each coordinate by its conditional distribution\nfunction given the preceding coordinates. Write x1:j = (x1, . . . , xj), with x1:0 the empty prefix, and\nset\nf1:j(x1:j) =\nZ"
        },
        {
          "blockId": "p10-8dd4a0cd47",
          "page": 10,
          "quote": "Rd−j f(x1:j, z) dz\n(1 ≤j < d),\nf1:d = f,\nf1:0 = 1."
        },
        {
          "blockId": "p10-a847e6fe15",
          "page": 10,
          "quote": "R f1:j(a, r) dr = f1:j−1(a) for every prefix a, including the empty prefix when j = 1. For\nj = 1, . . . , d, define"
        },
        {
          "blockId": "p10-775112f8b1",
          "page": 10,
          "quote": "R t\n−∞f1:j(a, r) dr"
        },
        {
          "blockId": "p10-5d65cb1326",
          "page": 10,
          "quote": "f1:j−1(a)\n,\na ∈Rj−1."
        },
        {
          "blockId": "p11-aa87aa0373",
          "page": 11,
          "quote": "This formula is used where its denominator belongs to (0, ∞); on the remaining fibers use instead the\nfixed distribution function t 7→(1 + e−t)−1. Thus every fiber is a continuous distribution function\nwith limits zero and one, and all Fj are jointly Borel. For j ≥2, the exceptional prefix fibers have\nzero probability under f1:j−1(a) da; there are no exceptional fibers when j = 1.\nFor a random variable with a continuous distribution function F, the variable F(X) is uniform\non (0, 1). Indeed, continuity and the limits of F allow a cut at each probability level u ∈(0, 1); any\ninterval on which F = u has probability zero, and hence P(F(X) ≤u) = u. At stage j, this identity\nand Fubini show that, conditionally on the preceding original prefix, the new transformed coordinate\nis uniform. It is therefore independent of that prefix, and hence of all preceding transformed\ncoordinates. A finite induction over j = 1, . . . , d shows that the Borel map"
        },
        {
          "blockId": "p11-e6b921f6e6",
          "page": 11,
          "quote": "sends f dx to product uniform measure on (0, 1)d. Its values lie in [0, 1]d, including on exceptional\nfibers. This argument does not require a density to be positive everywhere or a distribution function\nto be strictly increasing.\nThe reverse map. Given independent uniform coordinates, we successively recover the original\nconditional laws by quantiles. For each j and 0 < u < 1, define"
        },
        {
          "blockId": "p11-f9b5b634c4",
          "page": 11,
          "quote": "starting with the empty prefix. This finite recursion defines a Borel map. For independent uniform\ninputs, induction shows that after step j the output prefix has density f1:j: the preceding prefix has\ndensity f1:j−1, its exceptional fibers have probability zero, and on every other fiber the quantile has\nthe conditional distribution just defined. Thus Sf has output law f dx. If f and g are densities of α\nand β, respectively, then Sg ◦Rf has the required property."
        }
      ],
      "occurrenceCount": 8
    },
    {
      "symbol": "zj",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p11-65a64927e9",
        "page": 11,
        "quote": "Consequently qj(a, U) has distribution function Fj(a, ·) for uniform U. For u ∈[0, 1]d, recursively\nset\nzj = qj(z1:j−1, uj)\n(j = 1, . . . , d),\nSf(u) = (z1, . . . , zd),"
      },
      "occurrences": [
        {
          "blockId": "p11-65a64927e9",
          "page": 11,
          "quote": "Consequently qj(a, U) has distribution function Fj(a, ·) for uniform U. For u ∈[0, 1]d, recursively\nset\nzj = qj(z1:j−1, uj)\n(j = 1, . . . , d),\nSf(u) = (z1, . . . , zd),"
        }
      ],
      "occurrenceCount": 1,
      "latex": "z_{j}",
      "latexBasis": "pdf-script-position"
    },
    {
      "symbol": "Π",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p1-3adbcf9fd6",
        "page": 1,
        "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
      },
      "occurrences": [
        {
          "blockId": "p1-3adbcf9fd6",
          "page": 1,
          "quote": "For a Borel probability measure µ on R3, let Π(µ, µ, µ) denote the probability measures on (R3)3"
        },
        {
          "blockId": "p1-835858d883",
          "page": 1,
          "quote": "E∗(µ) =\ninf\nπ∈Π(µ,µ,µ)"
        },
        {
          "blockId": "p1-6b8d97f5fa",
          "page": 1,
          "quote": "with the following properties. For µ = ρ dx, the value E∗(µ) is finite and is attained by a coupling\nin Π(µ, µ, µ), but every pair of Borel maps T2, T3 : R3 →R3 satisfying (T2)#µ = (T3)#µ = µ obeys\nZ"
        },
        {
          "blockId": "p8-147638b826",
          "page": 8,
          "quote": "Conversely, every π ∈Π(µ, µ, µ) has π(U3) = 1: each of the three events xi /∈U has probability\nzero. Integrating (7) yields\nZ\nc dπ ≥3\nZ"
        },
        {
          "blockId": "p11-d998b56636",
          "page": 11,
          "quote": "with cs = +∞if two coordinates coincide. For a Borel probability measure µ on Rd, let Π(µ, µ, µ)\nnow denote the couplings on (Rd)3, and write"
        },
        {
          "blockId": "p11-c88396987b",
          "page": 11,
          "quote": "Es(µ) =\ninf\nπ∈Π(µ,µ,µ)"
        }
      ],
      "occurrenceCount": 6
    },
    {
      "symbol": "α",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p9-01a654890a",
        "page": 9,
        "quote": "Lemma 6.2. Let d ≥2 be an integer. If α and β are absolutely continuous Borel probability\nmeasures on Rd, there is a Borel map Q : Rd →Rd such that Q#α = β."
      },
      "occurrences": [
        {
          "blockId": "p9-01a654890a",
          "page": 9,
          "quote": "Lemma 6.2. Let d ≥2 be an integer. If α and β are absolutely continuous Borel probability\nmeasures on Rd, there is a Borel map Q : Rd →Rd such that Q#α = β."
        },
        {
          "blockId": "p11-f9b5b634c4",
          "page": 11,
          "quote": "starting with the empty prefix. This finite recursion defines a Borel map. For independent uniform\ninputs, induction shows that after step j the output prefix has density f1:j: the preceding prefix has\ndensity f1:j−1, its exceptional fibers have probability zero, and on every other fiber the quantile has\nthe conditional distribution just defined. Thus Sf has output law f dx. If f and g are densities of α\nand β, respectively, then Sg ◦Rf has the required property."
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "β",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p9-01a654890a",
        "page": 9,
        "quote": "Lemma 6.2. Let d ≥2 be an integer. If α and β are absolutely continuous Borel probability\nmeasures on Rd, there is a Borel map Q : Rd →Rd such that Q#α = β."
      },
      "occurrences": [
        {
          "blockId": "p9-01a654890a",
          "page": 9,
          "quote": "Lemma 6.2. Let d ≥2 be an integer. If α and β are absolutely continuous Borel probability\nmeasures on Rd, there is a Borel map Q : Rd →Rd such that Q#α = β."
        },
        {
          "blockId": "p11-f9b5b634c4",
          "page": 11,
          "quote": "starting with the empty prefix. This finite recursion defines a Borel map. For independent uniform\ninputs, induction shows that after step j the output prefix has density f1:j: the preceding prefix has\ndensity f1:j−1, its exceptional fibers have probability zero, and on every other fiber the quantile has\nthe conditional distribution just defined. Thus Sf has output law f dx. If f and g are densities of α\nand β, respectively, then Sg ◦Rf has the required property."
        }
      ],
      "occurrenceCount": 2
    },
    {
      "symbol": "δ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p9-b8dedfc947",
        "page": 9,
        "quote": "The optimal coupling π0 constructed in Section 5 is supported on products of three distinct\ncomponent sets.\nFix δ > 0. Partition each Ka into finitely many Borel cells of diameter at most δ, for example\nby a finite grid of half-open cubes. Discard the cells of zero µ-mass and denote the remaining cells\nby A1, . . . , AN. They partition a set of full µ-measure and each belongs to a single Ka. Put"
      },
      "occurrences": [
        {
          "blockId": "p9-b8dedfc947",
          "page": 9,
          "quote": "The optimal coupling π0 constructed in Section 5 is supported on products of three distinct\ncomponent sets.\nFix δ > 0. Partition each Ka into finitely many Borel cells of diameter at most δ, for example\nby a finite grid of half-open cubes. Discard the cells of zero µ-mass and denote the remaining cells\nby A1, . . . , AN. They partition a set of full µ-measure and each belongs to a single Ka. Put"
        },
        {
          "blockId": "p10-be5d523e56",
          "page": 10,
          "quote": "η2 ,\n|c(x) −c(y)| ≤6δ"
        },
        {
          "blockId": "p10-257031fbc4",
          "page": 10,
          "quote": "and the same calculation proves (T3)#µ = µ. The graph coupling πδ = (Id, T2, T3)#µ gives each\nproduct Ai × Aj × Ak exactly the mass qijk.\nWhenever qijk > 0, the three cells lie in three distinct component sets. Thus all pairwise distances\nin their product are at least η, including after the redistribution within the cells. If x and y are two\ntriples in the same such product, then |xℓ−yℓ| ≤δ for each coordinate. Consequently\n\n1\n|xℓ−xr| −\n1\n|yℓ−yr|"
        },
        {
          "blockId": "p10-7f1304e36b",
          "page": 10,
          "quote": "≤2δ"
        },
        {
          "blockId": "p10-51b31c7d14",
          "page": 10,
          "quote": "Z\nc dπδ −\nZ\nc dπ0"
        },
        {
          "blockId": "p10-5ba5f1ef95",
          "page": 10,
          "quote": "Both costs are finite, bounded by 3/η on all active products. Since πδ and π0 have identical masses\non these products, summing their oscillation bounds gives"
        },
        {
          "blockId": "p10-46a19c1389",
          "page": 10,
          "quote": "≤6δ"
        },
        {
          "blockId": "p10-068e0becae",
          "page": 10,
          "quote": "Letting δ ↓0 proves the required upper bound on the Monge infimum. The reverse bound follows\nbecause every admissible graph plan is a coupling. Strict nonattainment is Theorem 1.1."
        },
        {
          "blockId": "p13-56dc16656f",
          "page": 13,
          "quote": "Approximation by preserving maps.\nThe five compact component sets have a positive mutual\nseparation η, and π0 uses only products of three distinct components. Repeat the cell partition\nand redistribution from the proof of Proposition 6.1 in Rd, using Lemma 6.2. The first-coordinate\nmarginals used to split the cells are absolutely continuous by Fubini in every such dimension. The\nredistribution gives a graph coupling πδ of a preserving Borel pair with the same mass as π0 on\neach product of cells of diameter at most δ. Every active product still has all pair distances at least\nη. For two triples x, y in one such product, the scalar mean value theorem gives\n|xℓ−xr|−s −|yℓ−yr|−s ≤2sδ η−s−1,\n|cs(x) −cs(y)| ≤6sδ η−s−1."
        },
        {
          "blockId": "p13-85c2b519ef",
          "page": 13,
          "quote": "Z\ncs dπδ −\nZ\ncs dπ0"
        }
      ],
      "occurrenceCount": 10
    },
    {
      "symbol": "η",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p9-7bbd0ddf87",
        "page": 9,
        "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
      },
      "occurrences": [
        {
          "blockId": "p9-7bbd0ddf87",
          "page": 9,
          "quote": "Proof of Proposition 6.1. Write K0 = supp g, and let K1, . . . , K4 be its images under the four\nmaps in H. These are nonempty compact sets in the five disjoint components, and µ(K) = 1 for\nK = S4\na=0 Ka. In particular,\nη = min\na̸=b dist(Ka, Kb) > 0."
        },
        {
          "blockId": "p10-be5d523e56",
          "page": 10,
          "quote": "η2 ,\n|c(x) −c(y)| ≤6δ"
        },
        {
          "blockId": "p10-574128c370",
          "page": 10,
          "quote": "η2 ."
        },
        {
          "blockId": "p10-257031fbc4",
          "page": 10,
          "quote": "and the same calculation proves (T3)#µ = µ. The graph coupling πδ = (Id, T2, T3)#µ gives each\nproduct Ai × Aj × Ak exactly the mass qijk.\nWhenever qijk > 0, the three cells lie in three distinct component sets. Thus all pairwise distances\nin their product are at least η, including after the redistribution within the cells. If x and y are two\ntriples in the same such product, then |xℓ−yℓ| ≤δ for each coordinate. Consequently\n\n1\n|xℓ−xr| −\n1\n|yℓ−yr|"
        },
        {
          "blockId": "p10-4705349f52",
          "page": 10,
          "quote": "η2 ."
        },
        {
          "blockId": "p10-5ba5f1ef95",
          "page": 10,
          "quote": "Both costs are finite, bounded by 3/η on all active products. Since πδ and π0 have identical masses\non these products, summing their oscillation bounds gives"
        },
        {
          "blockId": "p13-56dc16656f",
          "page": 13,
          "quote": "Approximation by preserving maps.\nThe five compact component sets have a positive mutual\nseparation η, and π0 uses only products of three distinct components. Repeat the cell partition\nand redistribution from the proof of Proposition 6.1 in Rd, using Lemma 6.2. The first-coordinate\nmarginals used to split the cells are absolutely continuous by Fubini in every such dimension. The\nredistribution gives a graph coupling πδ of a preserving Borel pair with the same mass as π0 on\neach product of cells of diameter at most δ. Every active product still has all pair distances at least\nη. For two triples x, y in one such product, the scalar mean value theorem gives\n|xℓ−xr|−s −|yℓ−yr|−s ≤2sδ η−s−1,\n|cs(x) −cs(y)| ≤6sδ η−s−1."
        },
        {
          "blockId": "p13-7ff671788f",
          "page": 13,
          "quote": "Both costs are bounded by 3η−s on the active products. Summing their oscillations against the\ncommon product masses yields"
        },
        {
          "blockId": "p13-9d0ee63a5c",
          "page": 13,
          "quote": "≤6sδ η−s−1 −→0."
        }
      ],
      "occurrenceCount": 9
    },
    {
      "symbol": "ν",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p3-3d48e476fb",
        "page": 3,
        "quote": "The triples (0, ek, −ek) have the same Coulomb cost and the same central gradient. This allows\nthe two local families to share a supporting potential on B. Completing this potential on the four\nouter components gives an inequality c(x1, x2, x3) ≥u(x1) + u(x2) + u(x3), with equality precisely\non the two families. Strict cost gaps exclude the other triples of distinct centers; the Coulomb\nsingularity excludes repeated components after the neighborhoods have been made small enough.\nFor a smooth probability measure ν on B, we form the marginal µ by giving the central\ncomponent measure ν/3 and each outer component measure (Yk)#ν/6 or (Wk)#ν/6. Randomly\nchoosing a branch and permuting its three coordinates then gives an optimal coupling with this\ncommon marginal. Every optimizer must saturate the supporting inequality. In particular, on\nthe central component, a deterministic map must choose one of the four branch images. If it\nchooses a branch H on a Borel set S ⊂B, it sends mass ν(S)/3 into H(S), which has marginal\nmass only ν(S)/6. This contradicts preservation on every selection set of positive mass. The\nexact image-measure relation, rather than the component totals alone, excludes arbitrarily irregular\nchoices of branch.\nSections 2–5 establish the mass obstruction, construct and certify the two local families, and\nform the smooth marginal and an explicit optimizer. Section 6 constructs preserving maps whose\ncosts approach that optimum. Section 7 extends the construction to every inverse-power Riesz cost\nin every Euclidean dimension at least two."
      },
      "occurrences": [
        {
          "blockId": "p3-3d48e476fb",
          "page": 3,
          "quote": "The triples (0, ek, −ek) have the same Coulomb cost and the same central gradient. This allows\nthe two local families to share a supporting potential on B. Completing this potential on the four\nouter components gives an inequality c(x1, x2, x3) ≥u(x1) + u(x2) + u(x3), with equality precisely\non the two families. Strict cost gaps exclude the other triples of distinct centers; the Coulomb\nsingularity excludes repeated components after the neighborhoods have been made small enough.\nFor a smooth probability measure ν on B, we form the marginal µ by giving the central\ncomponent measure ν/3 and each outer component measure (Yk)#ν/6 or (Wk)#ν/6. Randomly\nchoosing a branch and permuting its three coordinates then gives an optimal coupling with this\ncommon marginal. Every optimizer must saturate the supporting inequality. In particular, on\nthe central component, a deterministic map must choose one of the four branch images. If it\nchooses a branch H on a Borel set S ⊂B, it sends mass ν(S)/3 into H(S), which has marginal\nmass only ν(S)/6. This contradicts preservation on every selection set of positive mass. The\nexact image-measure relation, rather than the component totals alone, excludes arbitrarily irregular\nchoices of branch.\nSections 2–5 establish the mass obstruction, construct and certify the two local families, and\nform the smooth marginal and an explicit optimizer. Section 6 constructs preserving maps whose\ncosts approach that optimum. Section 7 extends the construction to every inverse-power Riesz cost\nin every Euclidean dimension at least two."
        },
        {
          "blockId": "p3-e8854be191",
          "page": 3,
          "quote": "Lemma 2.1. Let d ≥2 be an integer, let B ⊂Rd be open, let ν be a Borel probability measure\nwith ν(B) = 1, and let H1, . . . , H4 : B →Rd be homeomorphisms onto open images. Suppose that\nB, H1(B), . . . , H4(B) are pairwise disjoint. Define"
        },
        {
          "blockId": "p3-f71745260b",
          "page": 3,
          "quote": "3ν + 1"
        },
        {
          "blockId": "p3-3fbed76b34",
          "page": 3,
          "quote": "j=1\n(Hj)#ν.\n(2)"
        },
        {
          "blockId": "p4-2492224c54",
          "page": 4,
          "quote": "T(x) ∈{H1(x), . . . , H4(x)}\nfor ν-almost every x ∈B.\n(3)"
        },
        {
          "blockId": "p4-0e53ba8b07",
          "page": 4,
          "quote": "3ν(Sj),\nµ(Hj(Sj)) = 1"
        },
        {
          "blockId": "p4-5b17f1aa6e",
          "page": 4,
          "quote": "6ν(Sj)."
        },
        {
          "blockId": "p4-27ba0f761e",
          "page": 4,
          "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
        },
        {
          "blockId": "p4-5236852130",
          "page": 4,
          "quote": "6ν(Sj)."
        },
        {
          "blockId": "p4-c5e9cbe917",
          "page": 4,
          "quote": "Thus ν(Sj) = 0 for all four indices, contradicting (3)."
        },
        {
          "blockId": "p4-44926de852",
          "page": 4,
          "quote": "The argument does not require ν to be uniform or even absolutely continuous. Additional\npoints mapped into Hj(Sj) can only increase its preimage mass. Matching the total mass of each\ncomponent by a partition therefore cannot remove the obstruction."
        },
        {
          "blockId": "p7-ced4283f40",
          "page": 7,
          "quote": "B g2 dx = 1, and let\nν = g2 dx, regarded as a probability measure on R3. Define"
        },
        {
          "blockId": "p7-cad5d5c99c",
          "page": 7,
          "quote": "3ν + 1"
        },
        {
          "blockId": "p7-2d7981f365",
          "page": 7,
          "quote": "H∈H\nH#ν.\n(9)"
        },
        {
          "blockId": "p7-927127455f",
          "page": 7,
          "quote": "The formula is smooth on H(B). Its support is contained in the compact set H(supp g) ⊂H(B), so\nextension by zero is smooth on R3. Thus gH ∈C∞\nc (R3), and change of variables gives H#ν = g2\nH dy.\nIt follows that µ = ρ dx, where"
        },
        {
          "blockId": "p8-88a56eff47",
          "page": 8,
          "quote": "This proves both asserted smoothness properties, compact support, and normalization.\nAlso\nµ(U) = 1 and µ|B = (1/3)ν."
        },
        {
          "blockId": "p8-3e0cd3485e",
          "page": 8,
          "quote": "A finite optimal coupling.\nSample x with law ν, choose k uniformly from {1, 2}, independently\nof x, and apply an independent uniform permutation to the three entries of"
        },
        {
          "blockId": "p8-9ce4cec472",
          "page": 8,
          "quote": "No optimal maps.\nSuppose that Borel maps T2, T3, each preserving µ, induce an optimal plan.\nThe preceding equality condition holds at (x, T2(x), T3(x)) for µ-almost every x, hence for ν-almost\nevery x ∈B.\nThe first coordinate x ∈B ⊂U0 is the unique central coordinate of an equality triple. The\nremaining coordinates lie one each in U+\nk and U−\nk for some k. If their values in those components\nare y, z, then x = Xk(y) and z = Zk(y). Injectivity of Xk forces"
        },
        {
          "blockId": "p8-60ba255cef",
          "page": 8,
          "quote": "T2(x) ∈{Y1(x), W1(x), Y2(x), W2(x)}\nfor ν-almost every x ∈B."
        },
        {
          "blockId": "p13-3e919ebd7e",
          "page": 13,
          "quote": "B g2 dx = 1, and form ν and µ\nby (9) using the four resulting diffeomorphisms. The change-of-variables construction in Section 5,\nnow with d-dimensional Jacobians, gives the same smooth compactly supported density and square\nroot: each image function is supported compactly inside its open component, and the five supports\nare disjoint. The same random choice of a branch and permutation gives an optimal coupling π0\nwith the three marginals equal to µ. The bounded supporting potential proves its cost is finite and\nforces every optimal coupling onto the contact set. Applying Lemma 2.1 to the four branch maps\nthen excludes every marginal-preserving Borel pair, exactly as in Section 5."
        }
      ],
      "occurrenceCount": 20
    },
    {
      "symbol": "π",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p8-73de921bb7",
        "page": 8,
        "quote": "Let π0 be the resulting law. In any fixed coordinate the central point has coefficient 1/3, and each\nspecified outer map has coefficient (1/2)(1/3) = 1/6. Hence every coordinate marginal is exactly µ.\nAll sampled triples satisfy (7) with equality. The boundedness of u therefore gives\nZ\nc dπ0 = 3\nZ"
      },
      "occurrences": [
        {
          "blockId": "p1-835858d883",
          "page": 1,
          "quote": "E∗(µ) =\ninf\nπ∈Π(µ,µ,µ)"
        },
        {
          "blockId": "p8-73de921bb7",
          "page": 8,
          "quote": "Let π0 be the resulting law. In any fixed coordinate the central point has coefficient 1/3, and each\nspecified outer map has coefficient (1/2)(1/3) = 1/6. Hence every coordinate marginal is exactly µ.\nAll sampled triples satisfy (7) with equality. The boundedness of u therefore gives\nZ\nc dπ0 = 3\nZ"
        },
        {
          "blockId": "p8-147638b826",
          "page": 8,
          "quote": "Conversely, every π ∈Π(µ, µ, µ) has π(U3) = 1: each of the three events xi /∈U has probability\nzero. Integrating (7) yields\nZ\nc dπ ≥3\nZ"
        },
        {
          "blockId": "p8-7b7687a300",
          "page": 8,
          "quote": "Thus π0 attains E∗(µ), and the latter is finite. Only the inequality on U3 is used.\nFor any optimal π, the nonnegative slack c(x1, x2, x3)−P\ni u(xi) on U3 has integral zero. Indeed\nthe cost is finite and the potential sum is bounded, so subtraction and integration are legitimate. It\nfollows that π is concentrated on the equality triples in (8)."
        },
        {
          "blockId": "p9-ded9131c04",
          "page": 9,
          "quote": "The approximation keeps the finite component structure of the optimizer π0. We partition the\ncomponents into small cells and prescribe the mass of a graph plan in each product of three cells\nto equal that of π0. Within each product the two plans may differ, but the components remain\nseparated and the cost oscillation tends to zero with the cell diameter. To realize the prescribed\nmasses, we split each source cell into pieces and send each piece into two specified target cells, with\nthe normalized restriction of µ as each target law. The following elementary transport fact supplies\nthose maps; its proof is given at the end of the section."
        },
        {
          "blockId": "p9-b8dedfc947",
          "page": 9,
          "quote": "The optimal coupling π0 constructed in Section 5 is supported on products of three distinct\ncomponent sets.\nFix δ > 0. Partition each Ka into finitely many Borel cells of diameter at most δ, for example\nby a finite grid of half-open cubes. Discard the cells of zero µ-mass and denote the remaining cells\nby A1, . . . , AN. They partition a set of full µ-measure and each belongs to a single Ka. Put"
        },
        {
          "blockId": "p9-9e9d19809d",
          "page": 9,
          "quote": "qijk = π0(Ai × Aj × Ak)."
        },
        {
          "blockId": "p9-25f7e840f2",
          "page": 9,
          "quote": "The marginal constraints on π0 read\nX"
        },
        {
          "blockId": "p10-257031fbc4",
          "page": 10,
          "quote": "and the same calculation proves (T3)#µ = µ. The graph coupling πδ = (Id, T2, T3)#µ gives each\nproduct Ai × Aj × Ak exactly the mass qijk.\nWhenever qijk > 0, the three cells lie in three distinct component sets. Thus all pairwise distances\nin their product are at least η, including after the redistribution within the cells. If x and y are two\ntriples in the same such product, then |xℓ−yℓ| ≤δ for each coordinate. Consequently\n\n1\n|xℓ−xr| −\n1\n|yℓ−yr|"
        },
        {
          "blockId": "p10-5ba5f1ef95",
          "page": 10,
          "quote": "Both costs are finite, bounded by 3/η on all active products. Since πδ and π0 have identical masses\non these products, summing their oscillation bounds gives"
        },
        {
          "blockId": "p11-c88396987b",
          "page": 11,
          "quote": "Es(µ) =\ninf\nπ∈Π(µ,µ,µ)"
        },
        {
          "blockId": "p13-3e919ebd7e",
          "page": 13,
          "quote": "B g2 dx = 1, and form ν and µ\nby (9) using the four resulting diffeomorphisms. The change-of-variables construction in Section 5,\nnow with d-dimensional Jacobians, gives the same smooth compactly supported density and square\nroot: each image function is supported compactly inside its open component, and the five supports\nare disjoint. The same random choice of a branch and permutation gives an optimal coupling π0\nwith the three marginals equal to µ. The bounded supporting potential proves its cost is finite and\nforces every optimal coupling onto the contact set. Applying Lemma 2.1 to the four branch maps\nthen excludes every marginal-preserving Borel pair, exactly as in Section 5."
        },
        {
          "blockId": "p13-56dc16656f",
          "page": 13,
          "quote": "Approximation by preserving maps.\nThe five compact component sets have a positive mutual\nseparation η, and π0 uses only products of three distinct components. Repeat the cell partition\nand redistribution from the proof of Proposition 6.1 in Rd, using Lemma 6.2. The first-coordinate\nmarginals used to split the cells are absolutely continuous by Fubini in every such dimension. The\nredistribution gives a graph coupling πδ of a preserving Borel pair with the same mass as π0 on\neach product of cells of diameter at most δ. Every active product still has all pair distances at least\nη. For two triples x, y in one such product, the scalar mean value theorem gives\n|xℓ−xr|−s −|yℓ−yr|−s ≤2sδ η−s−1,\n|cs(x) −cs(y)| ≤6sδ η−s−1."
        }
      ],
      "occurrenceCount": 13
    },
    {
      "symbol": "ρ",
      "meaning": "Candidate definition context; inspect the source.",
      "status": "candidate",
      "scope": "Unresolved; symbols can be rebound in different sections.",
      "source": {
        "blockId": "p7-927127455f",
        "page": 7,
        "quote": "The formula is smooth on H(B). Its support is contained in the compact set H(supp g) ⊂H(B), so\nextension by zero is smooth on R3. Thus gH ∈C∞\nc (R3), and change of variables gives H#ν = g2\nH dy.\nIt follows that µ = ρ dx, where"
      },
      "occurrences": [
        {
          "blockId": "p1-3eb947caad",
          "page": 1,
          "quote": "Theorem 1.1. There is a nonnegative density ρ on R3 such that\nZ"
        },
        {
          "blockId": "p1-b06be582b0",
          "page": 1,
          "quote": "R3 ρ(x) dx = 1,\nρ ∈C∞\nc (R3),\n√ρ ∈C∞\nc (R3),"
        },
        {
          "blockId": "p1-6b8d97f5fa",
          "page": 1,
          "quote": "with the following properties. For µ = ρ dx, the value E∗(µ) is finite and is attained by a coupling\nin Π(µ, µ, µ), but every pair of Borel maps T2, T3 : R3 →R3 satisfying (T2)#µ = (T3)#µ = µ obeys\nZ"
        },
        {
          "blockId": "p2-c37de2ec8b",
          "page": 2,
          "quote": "History and significance.\nCoulomb multi-marginal transport arises in the strong-interaction\nlimit of density functional theory. In this setting the description of optimal configurations by measure-\npreserving maps is usually called the Monge or co-motion ansatz. The co-motion formulation was\ndeveloped in the strictly correlated electron literature; see Seidl, Gori-Giorgi and Savin [11] and the\naccount by Friesecke, Gerolin and Gori-Giorgi [7]. The distinction between equality of optimal values\nand attainment by maps is essential. Colombo and Di Marino [3, author version, Theorem 1.1]\nproved that the Coulomb Kantorovich minimum equals the infimum over measure-preserving cyclic\nmaps for identical non-atomic marginals. This provides approximating maps without asserting\nthat a minimizing map exists. Section 6 gives a direct proof of equality of the two infima for the\nparticular marginal constructed here.\nFor two particles, Cotar, Friesecke and Kl¨uppelberg [5, arXiv version, Theorem 3.6] proved\nexistence and uniqueness of an optimal map for absolutely continuous marginals in every dimension.\nFor arbitrary particle number on the line, Colombo, De Pascale and Di Marino [2, Theorem 1.1]\nconstructed optimal cyclic maps for identical nonatomic marginals with finite transport value.\nPass [9] constructed Coulomb optimizers that are not induced by maps for radial data; the existence\nof such an optimizer does not exclude a different optimizer of Monge form. Colombo and Stra [4]\ngave counterexamples to the prescribed radial quantile-cyclic constructions. Bindini, De Pascale and\nKausamo [1] subsequently constructed smooth densities for the reduced radial cost of the planar\nthree-particle problem for which none of the four prescribed maps III, DDI, DID and IDD is optimal.\nThis radial cost is obtained by minimizing the Coulomb interaction over angles; it is different from\nthe Coulomb cost on the line.\nThe general higher-dimensional map-attainment question with at least three particles is recorded\nin the 2022 preprint of [7, arXiv version, Theorem 2.4 and the following discussion]. The three-\nparticle Coulomb question in R3 was still stated as open in January 2026 by Friesecke [6, Section 3.2].\nTheorem 1.1 excludes every preserving Borel pair for a smooth identical marginal. With the\nparticle-density normalization n = 3ρ, one has\nR\nn = 3 and √n ∈H1(R3). Thus the example\nbelongs to the physical density class D3 of [7, arXiv version, Eq. (12)]. This is a density-regularity\nstatement; no external potential or ground-state realization is asserted.\nOur construction adapts the shared-component splitting mechanism of Gerolin, Kausamo and\nRajala [8] for the repulsive harmonic cost −P\ni<j |xi −xj|2. For the Coulomb interaction, the\nadditional task is to build two families of triples with a common supporting potential. We obtain\nthese families and their supporting inequalities by local calculus."
        },
        {
          "blockId": "p7-927127455f",
          "page": 7,
          "quote": "The formula is smooth on H(B). Its support is contained in the compact set H(supp g) ⊂H(B), so\nextension by zero is smooth on R3. Thus gH ∈C∞\nc (R3), and change of variables gives H#ν = g2\nH dy.\nIt follows that µ = ρ dx, where"
        },
        {
          "blockId": "p7-ada17ad119",
          "page": 7,
          "quote": "ρ = 1"
        },
        {
          "blockId": "p8-31b96f11f6",
          "page": 8,
          "quote": "√ρ =\n1\n√"
        },
        {
          "blockId": "p12-7a431ba79d",
          "page": 12,
          "quote": "Theorem 7.1. For every integer d ≥2 and every real s > 0, there is a nonnegative density ρ on\nRd such that\nZ"
        },
        {
          "blockId": "p12-5f538e527b",
          "page": 12,
          "quote": "Rd ρ(x) dx = 1,\nρ ∈C∞\nc (Rd),\n√ρ ∈C∞\nc (Rd),"
        },
        {
          "blockId": "p12-a18b4af87d",
          "page": 12,
          "quote": "with the following properties. For µ = ρ dx, the value Es(µ) is finite and attained by a coupling, but\nevery pair of Borel maps T2, T3 : Rd →Rd with (T2)#µ = (T3)#µ = µ satisfies\nZ"
        }
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      "occurrenceCount": 10
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      "source": {
        "blockId": "p4-27ba0f761e",
        "page": 4,
        "quote": "1\n3ν(Sj) ≤µ(T −1(Hj(Sj))) = µ(Hj(Sj)) = 1"
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          "quote": "6ν(Sj)."
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