{ "schemaVersion": 1, "generator": "readable-paper 0.1.4", "paperId": "9d567de1c376", "version": "adc7f1241b42e322a6451854ab7e4b4c146bf78a", "sha256": "4c8faec31a1af3d0c640485e464ecedd99fb984f4a2c3b457f032685013804ab", "title": "The Factor-Two Hardness Threshold for Vertex Cover", "status": "Unverified reading map", "nodes": [ { "id": "theorem-1", "kind": "theorem", "label": "Theorem 1.1", "source": { "blockId": "p1-9c682e0dc0", "page": 1, "quote": "Theorem 1.1 (Explicit cover gap). For every fixed integer m ≥4, there is a deterministic polynomial-\ntime reduction from 3SAT to finite simple undirected unweighted graphs with the following guarantees:" }, "statement": [ { "blockId": "p1-9c682e0dc0", "page": 1, "quote": "Theorem 1.1 (Explicit cover gap). For every fixed integer m ≥4, there is a deterministic polynomial-\ntime reduction from 3SAT to finite simple undirected unweighted graphs with the following guarantees:" }, { "blockId": "p1-0040e41d56", "page": 1, "quote": "the formula is satisfiable\n=⇒\nτ(G) <\n1" }, { "blockId": "p1-2332f4697a", "page": 1, "quote": "\n|V (G)|," }, { "blockId": "p1-ce989ee4c1", "page": 1, "quote": "2 + 1" }, { "blockId": "p1-ad2b7ed3cf", "page": 1, "quote": "m" }, { "blockId": "p1-b633dc36e4", "page": 1, "quote": "the formula is unsatisfiable\n=⇒\nτ(G) >\n\n1 −1" }, { "blockId": "p1-f68d367a10", "page": 1, "quote": "\n|V (G)|." }, { "blockId": "p1-298336383b", "page": 1, "quote": "m" }, { "blockId": "p1-42989c8e1e", "page": 1, "quote": "The output lists all vertices and edges explicitly." } ], "proof": [ { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" }, { "blockId": "p23-ed115d16da", "page": 23, "quote": "Suppose an algorithm always returns a vertex cover of size at most ατ(G). On a completeness\ninstance its output has size strictly less than" }, { "blockId": "p23-48253c9453", "page": 23, "quote": "α\n1" }, { "blockId": "p23-9426b4c8c9", "page": 23, "quote": "\nn <\n\n1 −1" }, { "blockId": "p23-18eb10d350", "page": 23, "quote": "\nn," }, { "blockId": "p23-8322ee646b", "page": 23, "quote": "2 + 1" }, { "blockId": "p23-684af55cb3", "page": 23, "quote": "m" }, { "blockId": "p23-bf4de5a0c5", "page": 23, "quote": "m" }, { "blockId": "p23-3d282cb669", "page": 23, "quote": "whereas on a soundness instance every cover has size strictly greater than (1−1/m)n. Comparing the\noutput cardinality with this rational threshold distinguishes the two source cases. The comparison" }, { "blockId": "p24-38dac4ff9d", "page": 24, "quote": "uses only integer arithmetic: multiply the cardinality by m and compare with (m −1)n. The real\nnumber α is used only to choose the fixed integer m; it is not an input to the reduction. This proves\nTheorem 1.2." }, { "blockId": "p24-8bda18762a", "page": 24, "quote": "For completeness, the matching upper bound on the approximation factor is elementary. Find a\nmaximal matching and return the set of all its endpoints. Maximality makes this set a vertex cover:\nan edge with neither endpoint selected could be added to the matching. Every vertex cover contains\nan endpoint of each matching edge, and these edges are disjoint. The returned cover therefore has\nsize at most twice optimum. Together with Theorem 1.2, this gives the constant approximation\nthreshold 2, unless P = NP." } ], "guide": [ { "text": "The result claims a gap between small vertex covers in satisfiable instances and large covers in unsatisfiable instances. Read both bounds and their quantifiers in the original statement.", "sources": [ { "blockId": "p1-9c682e0dc0", "page": 1, "quote": "Theorem 1.1 (Explicit cover gap). For every fixed integer m ≥4, there is a deterministic polynomial-\ntime reduction from 3SAT to finite simple undirected unweighted graphs with the following guarantees:" } ] }, { "text": "The final proof invokes Theorem 2.1 for the Label Cover hardness input. The nearby paragraph identifies the PCP and repetition references supporting that input. External statements have not been verified here.", "sources": [ { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." } ] }, { "text": "The graph must be constructible in polynomial time, then completeness and soundness provide the gap. The final proof calls the inputs “Theorems 3.3, 3.5 and 7.2”; their actual headings are Proposition 3.3, Proposition 3.5 and Corollary 7.2. Check these links rather than treating the label changes as new results.", "sources": [ { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, { "blockId": "p7-44df83b758", "page": 7, "quote": "Proposition 3.3 (Deterministic polynomial construction). For a fixed integer m ≥4, a fixed even\nsquare d satisfying Equation (2.1), and fixed source alphabets, the graph G can be output explicitly\nin deterministic time polynomial in the source bit length. Every edge test is exact." }, { "blockId": "p9-0650d48b6b", "page": 9, "quote": "Proposition 3.5 (Completeness). If val(Φ) = 1, then G has an independent set of size greater than\n(1/2 −1/m)n. Consequently," }, { "blockId": "p22-9e3b3dd195", "page": 22, "quote": "Corollary 7.2 (Soundness). For the graph constructed in Section 3, assume (6.2). If val(Φ) ≤σ,\nevery independent set has size strictly less than n/m. Consequently τ(G) > (1 −1/m)n." } ] } ], "guideStatus": "Agent-authored reading guide; not independently verified." }, { "id": "corollary-2", "kind": "corollary", "label": "Corollary 1.2", "source": { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." }, "statement": [ { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." } ], "proof": [ { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" }, { "blockId": "p23-ed115d16da", "page": 23, "quote": "Suppose an algorithm always returns a vertex cover of size at most ατ(G). On a completeness\ninstance its output has size strictly less than" }, { "blockId": "p23-48253c9453", "page": 23, "quote": "α\n1" }, { "blockId": "p23-9426b4c8c9", "page": 23, "quote": "\nn <\n\n1 −1" }, { "blockId": "p23-18eb10d350", "page": 23, "quote": "\nn," }, { "blockId": "p23-8322ee646b", "page": 23, "quote": "2 + 1" }, { "blockId": "p23-684af55cb3", "page": 23, "quote": "m" }, { "blockId": "p23-bf4de5a0c5", "page": 23, "quote": "m" }, { "blockId": "p23-3d282cb669", "page": 23, "quote": "whereas on a soundness instance every cover has size strictly greater than (1−1/m)n. Comparing the\noutput cardinality with this rational threshold distinguishes the two source cases. The comparison" }, { "blockId": "p24-38dac4ff9d", "page": 24, "quote": "uses only integer arithmetic: multiply the cardinality by m and compare with (m −1)n. The real\nnumber α is used only to choose the fixed integer m; it is not an input to the reduction. This proves\nTheorem 1.2." }, { "blockId": "p24-8bda18762a", "page": 24, "quote": "For completeness, the matching upper bound on the approximation factor is elementary. Find a\nmaximal matching and return the set of all its endpoints. Maximality makes this set a vertex cover:\nan edge with neither endpoint selected could be added to the matching. Every vertex cover contains\nan endpoint of each matching edge, and these edges are disjoint. The returned cover therefore has\nsize at most twice optimum. Together with Theorem 1.2, this gives the constant approximation\nthreshold 2, unless P = NP." } ], "guide": [] }, { "id": "theorem-3", "kind": "theorem", "label": "Theorem 2.1", "source": { "blockId": "p4-15866c4f93", "page": 4, "quote": "Theorem 2.1 (Perfect-completeness Label Cover hardness). For every fixed rational σ ∈(0, 1),\nthere are nonempty constant alphabets and a deterministic polynomial-time reduction from 3SAT to\ninstances of the form above such that satisfiable formulas give val(Φ) = 1, and unsatisfiable formulas\ngive val(Φ) ≤σ." }, "statement": [ { "blockId": "p4-15866c4f93", "page": 4, "quote": "Theorem 2.1 (Perfect-completeness Label Cover hardness). For every fixed rational σ ∈(0, 1),\nthere are nonempty constant alphabets and a deterministic polynomial-time reduction from 3SAT to\ninstances of the form above such that satisfiable formulas give val(Φ) = 1, and unsatisfiable formulas\ngive val(Φ) ≤σ." } ], "proof": [], "guide": [] }, { "id": "lemma-4", "kind": "lemma", "label": "Lemma 3.1", "source": { "blockId": "p6-5b94c9fe95", "page": 6, "quote": "Lemma 3.1 (Compatibility norm). The family I is hereditary and contains every coordinate\nsingleton. Formula (3.1) defines a norm. For each fixed query i, the increment is linear in s and\nobeys\n∥z(i, s)∥∗≤∥s∥∞.\n(3.2)" }, "statement": [ { "blockId": "p6-5b94c9fe95", "page": 6, "quote": "Lemma 3.1 (Compatibility norm). The family I is hereditary and contains every coordinate\nsingleton. Formula (3.1) defines a norm. For each fixed query i, the increment is linear in s and\nobeys\n∥z(i, s)∥∗≤∥s∥∞.\n(3.2)" } ], "proof": [ { "blockId": "p6-4581aaf54a", "page": 6, "quote": "Proof. Removing labels preserves agreement and can only shrink the scope union. Every constraint\ninternal to the smaller union was therefore satisfied before removal. This proves heredity. Every\nsingleton belongs to I by the definition of a valid query label. The maximum of absolute linear forms\nis absolutely homogeneous and satisfies the triangle inequality; the singleton forms ensure that it\nvanishes only at zero. If the coordinate set is empty, this is the unique norm on the zero-dimensional\nspace. Finally, a compatible selection contains at most one coordinate of the i-block, proving (3.2).\nLinearity follows from the coordinate formula." } ], "guide": [] }, { "id": "lemma-5", "kind": "lemma", "label": "Lemma 3.2", "source": { "blockId": "p7-bd0f6459db", "page": 7, "quote": "Lemma 3.2 (Local evaluation of the norm). Let B ⊆Q be a set of query blocks supporting z ∈RP.\nThen" }, "statement": [ { "blockId": "p7-bd0f6459db", "page": 7, "quote": "Lemma 3.2 (Local evaluation of the norm). Let B ⊆Q be a set of query blocks supporting z ∈RP.\nThen" }, { "blockId": "p7-cca964d42f", "page": 7, "quote": "X" }, { "blockId": "p7-65cbc81ba7", "page": 7, "quote": "(i,λ)∈I\nzi,λ" }, { "blockId": "p7-182f70c256", "page": 7, "quote": "∥z∥∗=\nmax\nI∈I:\n(i,λ)∈I =⇒i∈B" }, { "blockId": "p7-e0da83d6df", "page": 7, "quote": ".\n(3.7)" }, { "blockId": "p7-fddd405aad", "page": 7, "quote": "For z = Sv + Sw, this maximum can be evaluated by at most (1 + r)2d choices, each checked by a\nscan of the source constraints." } ], "proof": [ { "blockId": "p7-2caab3f05c", "page": 7, "quote": "Proof. Intersect any compatible selection with the blocks in B. Heredity preserves compatibility,\nand the sum is unchanged because all other blocks of z vanish. Conversely, each selection on the\nright of (3.7) is a selection in the full maximum. This proves equality.\nFor a pair of vertices there are at most 2d distinct query blocks. First aggregate the coordinate\ncontributions of all occurrences of each repeated query. For each distinct query i, enumerate the at\nmost qd−1 ≤r assignments on its scope, and keep exactly those satisfying every source constraint\ninternal to that scope. This computes Λi and its slots. Next choose either no label or one of these\nlabels in each block. There are at most (1+r)2d choices. For each choice, test agreement on repeated\nvariables and scan the source list to test every constraint internal to the union of the chosen scopes.\nThis union has at most 2d(d −1) variables. The surviving choices are precisely the selections on the\nright of (3.7), and their absolute coordinate sums give the required maximum. Neither the labels\nnor the compatibility test requires satisfying any constraint outside the indicated scope union." } ], "guide": [] }, { "id": "proposition-6", "kind": "proposition", "label": "Proposition 3.3", "source": { "blockId": "p7-44df83b758", "page": 7, "quote": "Proposition 3.3 (Deterministic polynomial construction). For a fixed integer m ≥4, a fixed even\nsquare d satisfying Equation (2.1), and fixed source alphabets, the graph G can be output explicitly\nin deterministic time polynomial in the source bit length. Every edge test is exact." }, "statement": [ { "blockId": "p7-44df83b758", "page": 7, "quote": "Proposition 3.3 (Deterministic polynomial construction). For a fixed integer m ≥4, a fixed even\nsquare d satisfying Equation (2.1), and fixed source alphabets, the graph G can be output explicitly\nin deterministic time polynomial in the source bit length. Every edge test is exact." } ], "proof": [ { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, { "blockId": "p7-a2cbb096de", "page": 7, "quote": "d/(4m) and\n√" }, { "blockId": "p7-f81474c6c5", "page": 7, "quote": "d is an integer, the edge\ntest is precisely\n2mζ ≤p\n√" }, { "blockId": "p7-96c0742a89", "page": 7, "quote": "d." }, { "blockId": "p8-a1cd47b267", "page": 8, "quote": "Both sides are integers, so the test involves no approximation of a real number.\nChecking all pairs and writing every qualifying edge takes" }, { "blockId": "p8-efeb5ddc74", "page": 8, "quote": "O\n\n(1 + n2) poly(L + log n)\n" }, { "blockId": "p8-4c07ed3ec7", "page": 8, "quote": "bit operations, with constants depending only on the fixed parameters and alphabets. Formula (3.6)\nalso computes n by integer arithmetic with O(1 + log n) output bits. Thus the construction has\npolynomial bit complexity, including its explicit vertex and edge lists. It runs on every source\ninstance with these alphabets, regardless of whether the gap promise holds." } ], "guide": [] }, { "id": "lemma-7", "kind": "lemma", "label": "Lemma 3.4", "source": { "blockId": "p8-e6ad8608a9", "page": 8, "quote": "Lemma 3.4 (Midpoint-grid tail). If T1, . . . , Td are independent uniform elements of T , then" }, "statement": [ { "blockId": "p8-e6ad8608a9", "page": 8, "quote": "Lemma 3.4 (Midpoint-grid tail). If T1, . . . , Td are independent uniform elements of T , then" }, { "blockId": "p8-ea95170805", "page": 8, "quote": "" }, { "blockId": "p8-51d5efe1fe", "page": 8, "quote": "" }, { "blockId": "p8-28babd5945", "page": 8, "quote": "\n> 1" }, { "blockId": "p8-ce5e07173f", "page": 8, "quote": "Pr" }, { "blockId": "p8-f1fbe0ca46", "page": 8, "quote": "2\n\na + 1\n√" }, { "blockId": "p8-a23b41f071", "page": 8, "quote": "> 1" }, { "blockId": "p8-e4f07383fb", "page": 8, "quote": "\nd\nX" }, { "blockId": "p8-f74f24c930", "page": 8, "quote": "2 −\n√" }, { "blockId": "p8-b6a686e9a6", "page": 8, "quote": "2 −1" }, { "blockId": "p8-9b6f9115d5", "page": 8, "quote": "m.\n(3.8)" }, { "blockId": "p8-a3956b9f07", "page": 8, "quote": "d" }, { "blockId": "p8-8dcacefcbd", "page": 8, "quote": "j=1\nTj > t" } ], "proof": [ { "blockId": "p8-1ca49a3d7f", "page": 8, "quote": "Proof. Take independent uniform C1, . . . , Cd on [−1, 1], and round each to the midpoint of its\nsubinterval in the partition defining T . Boundary choices have probability zero. The rounded\nvariables have the required independent laws, and" }, { "blockId": "p8-727b08d172", "page": 8, "quote": "Put A = t + d/p = a\n√" }, { "blockId": "p8-600e9628f5", "page": 8, "quote": "d + d/p. We bound the central probability Pr[| P\nj Cj| ≤A].\nEach Cj may be generated as (Lj + Uj)/2, where Lj is a fair sign and Uj is uniform on [−1, 1],\nall independent. The two sign choices give uniform distributions on the two half intervals of [−1, 1],\nso this is indeed the uniform distribution on the full interval. Conditional on (U1, . . . , Ud), the event\n| P\nj Cj| ≤A places P\nj Lj in an interval of length" }, { "blockId": "p8-118861676d", "page": 8, "quote": "4A = 4a\n√" }, { "blockId": "p8-4d5f75491e", "page": 8, "quote": "d + 4d/p < 4a\n√" }, { "blockId": "p8-5acf1943ce", "page": 8, "quote": "d + 2." }, { "blockId": "p8-d0282f7caa", "page": 8, "quote": "The possible sign sums are spaced by two. An interval of length D contains at most D/2 + 1 values\nfrom such a lattice, so this interval contains fewer than 2a\n√" }, { "blockId": "p8-d9106e79cf", "page": 8, "quote": "d + 2 possible sign sums.\nWrite d = 2d0. The sign-sum probabilities are 4−d02d0\nv\n\n, for 0 ≤v ≤2d0. Consecutive ratios\n(2d0 −v)/(v + 1) show that the largest is the central one. It satisfies" }, { "blockId": "p8-e6253f14c4", "page": 8, "quote": "√" }, { "blockId": "p8-189d3f8929", "page": 8, "quote": "d0\nY" }, { "blockId": "p8-c4aadcc866", "page": 8, "quote": "2\n√" }, { "blockId": "p8-14b30f04b1", "page": 8, "quote": "\n=" }, { "blockId": "p8-2511573de9", "page": 8, "quote": "\n1 −1" }, { "blockId": "p8-30ca1ed264", "page": 8, "quote": "2u" }, { "blockId": "p8-fa4dbc7885", "page": 8, "quote": "4−d0\n2d0\nd0" }, { "blockId": "p8-9c912a64b0", "page": 8, "quote": "d\n." }, { "blockId": "p8-c54bdfcb05", "page": 8, "quote": "\n≤\n1\n√d0 + 1 <" }, { "blockId": "p8-feeb022966", "page": 8, "quote": "u=1" }, { "blockId": "p8-8468fa2a8d", "page": 8, "quote": "For the middle inequality, square the product and use" }, { "blockId": "p8-10ca91b149", "page": 8, "quote": "\n1 −1" }, { "blockId": "p8-93eeff13ac", "page": 8, "quote": "2u" }, { "blockId": "p8-e81194fff6", "page": 8, "quote": "2\n≤\nu\nu + 1\n(u ≥1);" }, { "blockId": "p9-c71449aab6", "page": 9, "quote": "indeed the right side minus the left side is (3u −1)/(4u2(u + 1)) > 0, and the resulting product\ntelescopes. The conditional central probability, and therefore its average, is bounded by" }, { "blockId": "p9-99a2f256bf", "page": 9, "quote": "" }, { "blockId": "p9-f3e92c2f4a", "page": 9, "quote": "" }, { "blockId": "p9-5e552a328b", "page": 9, "quote": "X" }, { "blockId": "p9-44b84aaa52", "page": 9, "quote": "Pr" }, { "blockId": "p9-18de59b25e", "page": 9, "quote": "2\n\n2a + 2\n√" }, { "blockId": "p9-07793bd3f9", "page": 9, "quote": "\n." }, { "blockId": "p9-073372b1af", "page": 9, "quote": "" }, { "blockId": "p9-98087f1300", "page": 9, "quote": "<\n√" }, { "blockId": "p9-684f17cbfb", "page": 9, "quote": "d" }, { "blockId": "p9-0e267b2d7b", "page": 9, "quote": "j\nCj" }, { "blockId": "p9-2e74a88638", "page": 9, "quote": "≤A" }, { "blockId": "p9-4748fcaa26", "page": 9, "quote": "Symmetry gives" }, { "blockId": "p9-1e99e2c25b", "page": 9, "quote": "" }, { "blockId": "p9-91736fe6c3", "page": 9, "quote": "" }, { "blockId": "p9-46041a7a1b", "page": 9, "quote": "" }, { "blockId": "p9-89d81e722c", "page": 9, "quote": "" }, { "blockId": "p9-daf6fd7094", "page": 9, "quote": "" }, { "blockId": "p9-d385f378ce", "page": 9, "quote": "" }, { "blockId": "p9-b74dcd8805", "page": 9, "quote": "X" }, { "blockId": "p9-7148aadd92", "page": 9, "quote": "Pr" }, { "blockId": "p9-83e5b3d3af", "page": 9, "quote": "2\n\na + 1\n√" }, { "blockId": "p9-dc1f8d853e", "page": 9, "quote": "\n." }, { "blockId": "p9-7c9977043d", "page": 9, "quote": "X" }, { "blockId": "p9-360e6cecf9", "page": 9, "quote": "= 1" }, { "blockId": "p9-48a3660190", "page": 9, "quote": "1 −Pr" }, { "blockId": "p9-af653ea944", "page": 9, "quote": "" }, { "blockId": "p9-87a910e71a", "page": 9, "quote": "" }, { "blockId": "p9-2aba014ba5", "page": 9, "quote": "> 1" }, { "blockId": "p9-2475fb3364", "page": 9, "quote": "2" }, { "blockId": "p9-2d508dc2ff", "page": 9, "quote": "2 −\n√" }, { "blockId": "p9-3d36ccf3dc", "page": 9, "quote": "d" }, { "blockId": "p9-86cd798522", "page": 9, "quote": "j\nCj > A" }, { "blockId": "p9-5412b72e44", "page": 9, "quote": "j\nCj" }, { "blockId": "p9-c319622ff8", "page": 9, "quote": "≤A" }, { "blockId": "p9-1a87c54021", "page": 9, "quote": "d > 4m = 1/a by Equation (2.1), and hence" }, { "blockId": "p9-6b0616f428", "page": 9, "quote": "On this event, rounding leaves P\nj Tj > t. Finally,\n√" }, { "blockId": "p9-a07da64554", "page": 9, "quote": "√" }, { "blockId": "p9-a3bdff380b", "page": 9, "quote": "2\n\na + 1\n√" }, { "blockId": "p9-1d590713dc", "page": 9, "quote": "\n< 2\n√" }, { "blockId": "p9-596ebf83d5", "page": 9, "quote": "2 a < 3a =\n3\n4m < 1" }, { "blockId": "p9-547dfc3974", "page": 9, "quote": "m." }, { "blockId": "p9-88c3936ad4", "page": 9, "quote": "d" }, { "blockId": "p9-e705b163a8", "page": 9, "quote": "This proves both inequalities in (3.8)." } ], "guide": [] }, { "id": "proposition-8", "kind": "proposition", "label": "Proposition 3.5", "source": { "blockId": "p9-0650d48b6b", "page": 9, "quote": "Proposition 3.5 (Completeness). If val(Φ) = 1, then G has an independent set of size greater than\n(1/2 −1/m)n. Consequently," }, "statement": [ { "blockId": "p9-0650d48b6b", "page": 9, "quote": "Proposition 3.5 (Completeness). If val(Φ) = 1, then G has an independent set of size greater than\n(1/2 −1/m)n. Consequently," }, { "blockId": "p9-faaf97dc8c", "page": 9, "quote": "τ(G) <\n1" }, { "blockId": "p9-c6723462c5", "page": 9, "quote": "\nn." }, { "blockId": "p9-9dcbb35417", "page": 9, "quote": "2 + 1" }, { "blockId": "p9-567d80c1f7", "page": 9, "quote": "m" } ], "proof": [ { "blockId": "p9-005d0d4dda", "page": 9, "quote": "Proof. Let A be a labeling satisfying every source constraint. Its restriction λA,i to P(i) belongs to\nΛi for every query i. The selection" }, { "blockId": "p9-1a9392c0bb", "page": 9, "quote": "IA = {(i, λA,i) : i ∈Q}" }, { "blockId": "p9-3bcdab94d7", "page": 9, "quote": "belongs to I: the labels agree on overlaps, and every constraint internal to their scope union is\nsatisfied by A. Keep precisely the vertices for which" }, { "blockId": "p9-186d7f2b8e", "page": 9, "quote": "d\nX" }, { "blockId": "p9-18496e9242", "page": 9, "quote": "j=1\nsj,slotij(c)(λA,ij(c)) > t.\n(3.9)" }, { "blockId": "p9-c90bc9285c", "page": 9, "quote": "For a uniform vertex, conditional on the seed tuple, the selected slot in each weight vector is fixed.\nIts entries are independent uniform elements of T , one from each independent sj. Coincident\nendpoint questions or slot numbers do not change this independence. By Theorem 3.4, more than a\n(1/2 −1/m) fraction of vertices are kept.\nFor two distinct kept vertices v, w, evaluating Sv + Sw on IA gives exactly the sum of their two\nexpressions in (3.9). This includes all contributions even when the same query appears in both\nvertices. The sum exceeds 2t, so (3.1) rules out the edge in (3.5). The kept vertices are therefore\nindependent. Their complement is a vertex cover of size less than (1/2 + 1/m)n." }, { "blockId": "p9-95e52082ad", "page": 9, "quote": "4\nA fixed function from an independent set" }, { "blockId": "p9-a83501fa40", "page": 9, "quote": "The soundness argument begins by turning a large independent set into one fixed Lipschitz function.\nWe need a variance lower bound on fresh random inputs and control of how much the function\nchanges when one position changes. Set" }, { "blockId": "p9-7f11e26df2", "page": 9, "quote": "b0 =\n1\n2m,\nν = b0a2" }, { "blockId": "p9-59ad20044b", "page": 9, "quote": "4\n=\n1\n27m3 .\n(4.1)" }, { "blockId": "p10-2f523fd0e7", "page": 10, "quote": "Thus the independent-set density we must rule out is 2b0. First discard the low-norm vertices. All\nvertices whose vectors have norm at most t form a clique: for any two of them the triangle inequality\ngives ∥Sv + Sw∥∗≤2t. An independent set therefore contains at most one such vertex. Since\nn ≥p > 1/b0, an independent set of size at least n/m = 2b0n contains more than b0n vertices of\nnorm greater than t. The following lemma turns these remaining vectors into the required function." } ], "guide": [] }, { "id": "lemma-9", "kind": "lemma", "label": "Lemma 4.1", "source": { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" }, "statement": [ { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" }, { "blockId": "p10-11188e04c8", "page": 10, "quote": "Pr\nv∈V (G)[Sv ∈A∗] ≥b0,\n∥x + y∥∗> 2t\n(x, y ∈A∗)," }, { "blockId": "p10-a993943676", "page": 10, "quote": "where the vertex is uniform. For a nonempty finite set B, write dist∗(z, B) = minb∈B ∥z −b∥∗. The\nfunction" }, { "blockId": "p10-df6a937533", "page": 10, "quote": "F(z) = 1" }, { "blockId": "p10-9d68e8efbd", "page": 10, "quote": "2\n\ndist∗(z, −A∗) −dist∗(z, A∗)\n\n(4.2)" }, { "blockId": "p10-a9f3ade5d7", "page": 10, "quote": "is odd and 1-Lipschitz for ∥·∥∗, and F(z) > t on A∗.\nLet E =\n[d]\n2\n\n. Draw independent uniform occurrence seeds c = (ce)e∈E ∈[M]E and independent\nweight blocks sj uniform on [−1, 1]r, independently of the seeds. Put" }, { "blockId": "p10-65a7a41f31", "page": 10, "quote": "" }, { "blockId": "p10-95587f0582", "page": 10, "quote": "" }, { "blockId": "p10-5f455e7325", "page": 10, "quote": "f(c, s) = F" }, { "blockId": "p10-254bfac670", "page": 10, "quote": ".\n(4.3)" }, { "blockId": "p10-6ad29e28b1", "page": 10, "quote": "\nd\nX" }, { "blockId": "p10-ea4997ae80", "page": 10, "quote": "j=1\nz(ij(c), sj)" }, { "blockId": "p10-97e4c85526", "page": 10, "quote": "This bounded function satisfies" }, { "blockId": "p10-6aa5afb122", "page": 10, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(4.4)" }, { "blockId": "p10-cdf2d5a8a1", "page": 10, "quote": "On the weight cubes it has the following three oscillation bounds:" }, { "blockId": "p10-557d000b43", "page": 10, "quote": "(i) Replacing one weight block sj, with all other data fixed, changes f by at most 2." }, { "blockId": "p10-e354d1c2e8", "page": 10, "quote": "(ii) Replacing one seed ce, with all other data fixed, changes f by at most 4." }, { "blockId": "p10-c0769b2092", "page": 10, "quote": "(iii) Fix k ∈[d]. Simultaneously replacing any seeds incident to k, while keeping all nonincident\nseeds, all weights, and the question received at every opposite position j ̸= k from {j, k}\nunchanged, changes f by at most 2." }, { "blockId": "p10-9d5d3433b6", "page": 10, "quote": "The set A∗, the function F, and its formula are fixed from an independent set of the entire graph\nbefore the fresh seeds above are drawn. A seed enters f only through its endpoint questions." } ], "proof": [ { "blockId": "p10-3eb0619124", "page": 10, "quote": "Proof. Take A∗to be the set of sum vectors of the stipulated high-norm vertices in the independent\nset. The number of those vertices gives the probability bound even if several vertices represent the\nsame vector. For distinct x, y ∈A∗, their representing vertices are distinct and nonadjacent, so\nEquation (3.5) gives ∥x + y∥∗> 2t. If x = y, the same inequality follows from ∥2x∥∗= 2 ∥x∥∗> 2t.\nDistance to a nonempty finite set is the minimum of the distances to its points. The triangle\ninequality makes each distance function 1-Lipschitz; the half-difference in (4.2) is therefore 1-\nLipschitz. Symmetry of the norm gives dist∗(−z, −A∗) = dist∗(z, A∗), proving oddness. For z ∈A∗,\nits distance to A∗is zero, while" }, { "blockId": "p10-76aac6ebec", "page": 10, "quote": "dist∗(z, −A∗) = min\nx∈A∗∥z + x∥∗> 2t." }, { "blockId": "p10-9db5c214f3", "page": 10, "quote": "Finiteness preserves the strict inequality, so F(z) > t." }, { "blockId": "p11-acf2eb48a1", "page": 11, "quote": "For fixed seeds, negating all weights preserves their joint law and negates f. This proves the\nfirst assertion in (4.4). Also F(0) = 0, and the increment bound ∥z(i, s)∥∗≤∥s∥∞implies |f| ≤d\non the sampling space.\nRound every continuous entry to the midpoint of its grid interval. The resulting entries are\nindependent uniform elements of T ; interval-boundary choices have probability zero. The change in\none increment has norm at most 1/p, by linearity in its weights and the increment bound. Thus\nrounding changes the argument of F, and hence its value, by at most d/p < t/2. The unchanged\nseeds together with the rounded weights have exactly the uniform vertex law. With probability\nat least b0, their sum belongs to A∗, and on that event the unrounded value of f exceeds t/2.\nConsequently\nEf2 ≥b0t2/4 = b0a2d/4 = νd." }, { "blockId": "p11-b7370a234c", "page": 11, "quote": "Every increment on a weight cube has norm at most one. Replacing one weight block changes\none increment by norm at most two, proving (i). A single seed affects only its two endpoint queries;\nreplacing it changes at most two increments, each by norm at most two, proving (ii). In (iii), each\nposition other than k keeps every question in its query, so only the increment at k may change. The\nsame bound of two proves the claim. All these arguments use the single fixed function F in (4.2); it\nis not chosen again for a sampled seed tuple." }, { "blockId": "p11-d0d0393bb5", "page": 11, "quote": "5\nA restriction preserving own-question variance" }, { "blockId": "p11-8c7eaa450a", "page": 11, "quote": "The function from the previous section has variance of order d, but the functions used to extract\nlabels must depend only on one position’s own questions and weights. We shall freeze all but\na batch of h positions and retain variance of order h in conditional averages with precisely this\ndependence. Questions at different positions are correlated because they share pair seeds; our\northogonal decomposition therefore uses the independent seeds and weights themselves.\nLet d ≥4 and r, M ≥1 be integers, and put E =\n[d]\n2\n\n. For each e ∈E, draw ce uniformly from\n[M], independently over e. At each endpoint j ∈e, this seed reveals a question Qe,j(ce), where\nQe,j is a fixed map to a finite set. There is no independence assumption on the two questions\nfrom the same seed. In the construction, for e = {j, k} with j < k, these maps are Qe,j(c) = xc\nand Qe,k(c) = yc. Independently of all seeds, draw sj uniformly from [−1, 1]r for each j ∈[d],\nindependently over j. All expectations and L2-norms in this section initially refer to the resulting\nproduct law on\nΩ= [M]E × ([−1, 1]r)d." }, { "blockId": "p11-229e7bb8ab", "page": 11, "quote": "We first specify the restrictions and conditional averages in the lemma. Given a bounded Borel\nmeasurable function f(c, s) and a set J ⊆[d], its frozen data are" }, { "blockId": "p11-95ea1c8256", "page": 11, "quote": "a =\n\n(ce)e/∈(J\n2), (sk)k/∈J\n\n.\n(5.1)" }, { "blockId": "p11-0fd1fee442", "page": 11, "quote": "Fixing these values leaves a function f∗of the hidden seeds (ce)e∈(J\n2) and the weights (sj)j∈J, with\ntheir original product law. For j ∈J, write" }, { "blockId": "p11-580f678b17", "page": 11, "quote": "uj =\n\nQ{j,k},j(c{j,k})\n" }, { "blockId": "p11-4bcab43585", "page": 11, "quote": "k∈J\\{j}," }, { "blockId": "p11-a7c5fe9a95", "page": 11, "quote": "with entries in increasing order of k. Define HJ,a\nj\n(u, s) by inserting sj = s in f∗and averaging the\nremaining variables conditional on uj = u. Explicitly, each incident hidden seed c{j,k} is drawn\nuniformly from the fiber of Q{j,k},j specified by the corresponding entry of u; these draws are\nindependent. The other hidden seeds and other weights retain their original independent laws. On" }, { "blockId": "p12-70559baa5b", "page": 12, "quote": "a question tuple of probability zero set HJ,a\nj\n= 0. This definition uses ordinary finite averages\nand cube integrals with the frozen values plugged in, so it also defines the averages when a has\ncontinuous coordinates. We abbreviate HJ,a\nj\nto Hj when the restriction is fixed." } ], "guide": [] }, { "id": "lemma-10", "kind": "lemma", "label": "Lemma 5.1", "source": { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" }, "statement": [ { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" }, { "blockId": "p12-e17a0fda8b", "page": 12, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(5.2)" }, { "blockId": "p12-2805c87b00", "page": 12, "quote": "Suppose also that the following oscillation bounds hold with all coordinates not mentioned kept fixed:" }, { "blockId": "p12-b9f098e02e", "page": 12, "quote": "(i) Changing one weight block sj changes f by at most 2." }, { "blockId": "p12-cd9ae429c6", "page": 12, "quote": "(ii) Changing one seed ce changes f by at most 4." }, { "blockId": "p12-140fd00286", "page": 12, "quote": "(iii) For any k ∈[d], simultaneously changing seeds incident to k, while preserving Q{k,j},j(c{k,j})\nfor every j ̸= k, changes f by at most 2." }, { "blockId": "p12-d963b5526f", "page": 12, "quote": "If" }, { "blockId": "p12-7a4e49fe92", "page": 12, "quote": "16h + 32h2 ≤νd" }, { "blockId": "p12-ad0f187d8f", "page": 12, "quote": "2 ,\n(5.3)" }, { "blockId": "p12-ea2c452c9c", "page": 12, "quote": "then there are a fixed h-element set J and fixed frozen data a such that" }, { "blockId": "p12-f2d71b1e41", "page": 12, "quote": "X" }, { "blockId": "p12-eb730192d5", "page": 12, "quote": "4 .\n(5.4)" }, { "blockId": "p12-20efa93216", "page": 12, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh" }, { "blockId": "p12-bbd8c74311", "page": 12, "quote": "The law in (5.4) is the product law of the remaining hidden seeds and batch weights described above.\nIn particular it is not conditioned on an event involving a draw of those remaining variables." } ], "proof": [ { "blockId": "p12-a337498528", "page": 12, "quote": "Proof. The conditional mean Hj averages all unfrozen weights except sj. It also averages every\nhidden seed on a pair avoiding j, and retains only j’s question on each hidden pair incident to\nj. These are the three ways in which averaging can lose information. We use the product-space\northogonal decomposition associated with the Efron–Stein method [7, Section 2], giving the needed\nprojection identities explicitly. We assign each component to one weight block on which it depends,\nand ask when a random batch preserves that component in its assigned conditional mean. The\noscillation bounds control the energy lost through each type of averaging." }, { "blockId": "p12-22f8b88cff", "page": 12, "quote": "Averaging projections and weight supports.\nLet W 0\nj average the weight block sj, retaining\nall other coordinates, and set W 1\nj = Id −W 0\nj . Let P 0\ne average the seed ce. For j ∈e, let Oe,j average\nthat seed conditional on its question Qe,j(ce), retaining all other coordinates. Each averaging is an\northogonal projection in L2(Ω): averaging twice gives the same function, and integrating over the\nretained data shows self-adjointness. Also" }, { "blockId": "p12-872013d74b", "page": 12, "quote": "ran(P 0\ne ) ⊆ran(Oe,j),\nP 0\ne Oe,j = Oe,jP 0\ne = P 0\ne ." }, { "blockId": "p12-c374d70b64", "page": 12, "quote": "Operators on distinct coordinates commute by the product law and Fubini’s theorem. The two\nendpoint projections Oe,j, Oe,k on a single seed need not commute; we will not assume that they do.\nFor S ⊆[d], put\nΠS =\nY" }, { "blockId": "p12-5043bdad9b", "page": 12, "quote": "j∈S\nW 1\nj\nY" }, { "blockId": "p12-62d9245ca1", "page": 12, "quote": "j /∈S\nW 0\nj ,\nfS = ΠSf." }, { "blockId": "p13-7db7caa7c4", "page": 13, "quote": "Expanding Q\nj(W 0\nj + W 1\nj ) gives f = P\nS fS. Distinct S’s give orthogonal components, because their\nprojections have a factor W 0\nj W 1\nj = 0 at some position. The first condition in (5.2) says that f∅= 0.\nFurthermore f −W 0\nj f = P\nS∋j fS, and hence" }, { "blockId": "p13-cb930c876c", "page": 13, "quote": "d\nX" }, { "blockId": "p13-50c6b220f1", "page": 13, "quote": "X" }, { "blockId": "p13-53aca89949", "page": 13, "quote": "j=1\n∥f −W 0\nj f∥2\n2 ≤4d.\n(5.5)" }, { "blockId": "p13-4ab0920958", "page": 13, "quote": "S̸=∅\n|S|∥fS∥2\n2 =" }, { "blockId": "p13-4a77849597", "page": 13, "quote": "Indeed, on every fiber obtained by fixing the other coordinates, the one-weight oscillation bound\nimplies |f −W 0\nj f| ≤2." }, { "blockId": "p13-eedd6923d1", "page": 13, "quote": "Refining the seed coordinates.\nFor every nonempty S, choose a designated position j(S) ∈S\nby a fixed rule, for example j(S) = min S. This choice depends only on the support S. With\nj = j(S), split each seed coordinate into the following projections:" }, { "blockId": "p13-1eb9806aa3", "page": 13, "quote": "j /∈e :\nP 0\ne ,\nId −P 0\ne ," }, { "blockId": "p13-76801848a6", "page": 13, "quote": "j ∈e :\nP 0\ne ,\nOe,j −P 0\ne ,\nId −Oe,j." }, { "blockId": "p13-ae279f868f", "page": 13, "quote": "The nested ranges just proved make the three projections in the second line pairwise orthogonal,\nand their sum is the identity. For a choice ω of one displayed projection at every seed, let fS,ω be\ntheir product applied to fS. For fixed S, these components give an orthogonal decomposition of\nfS. The seed projections commute with all weight projections, so every fS,ω remains in ran(ΠS).\nConsequently the components for different S’s are also orthogonal, even though the endpoint used\nto refine a seed can change with S. Thus\nX" }, { "blockId": "p13-3cfa9c14d3", "page": 13, "quote": "S,ω\n∥fS,ω∥2\n2 = ∥f∥2\n2,\n(5.6)" }, { "blockId": "p13-319e53e23f", "page": 13, "quote": "where henceforth these sums range over nonempty S and its allowed choices ω.\nLet TS,ω ⊆E be the seeds where the chosen projection is not P 0\ne . Let NS,ω ⊆[d] \\ {j(S)} be\nthe positions k for which the projection at e = {j(S), k} is Id −Oe,j(S). These are the incident\ncomponents not retained by the designated position’s own question. For each fixed e, applying\nId −P 0\ne keeps exactly the components with e ∈TS,ω. Orthogonality and the one-seed oscillation\nbound therefore give" }, { "blockId": "p13-c40cab8b36", "page": 13, "quote": "X" }, { "blockId": "p13-1830aafe32", "page": 13, "quote": "\n≤8d2.\n(5.7)" }, { "blockId": "p13-46470989f4", "page": 13, "quote": "S,ω\n|TS,ω|∥fS,ω∥2\n2 =\nX" }, { "blockId": "p13-5f0771b88d", "page": 13, "quote": "e∈E\n∥f −P 0\ne f∥2\n2 ≤16\nd\n2" }, { "blockId": "p13-45d4aca140", "page": 13, "quote": "Here |f −P 0\ne f| ≤4 follows by averaging values whose oscillation is at most 4.\nAt this stage the ordinary seed-energy bound is of order d2. That is sufficient for hidden pairs\navoiding the designated position, because both endpoints must enter the small batch. It is insufficient\nfor a hidden incident seed, which needs only one extra endpoint in the batch. The next estimate\nreduces the relevant incident energy to order d by resampling a whole star at once." }, { "blockId": "p13-d85cfd5384", "page": 13, "quote": "Charging the residual components to stars.\nFor each k ∈[d], define" }, { "blockId": "p13-89581310ca", "page": 13, "quote": "Dk =\nY" }, { "blockId": "p13-2caa193850", "page": 13, "quote": "j̸=k\nO{k,j},j." }, { "blockId": "p13-eb643b39c0", "page": 13, "quote": "The factors act on distinct seed coordinates, so this is an orthogonal projection. More explicitly,\nit retains all weights, all seeds not incident to k, and the opposite endpoint question at each" }, { "blockId": "p14-8ad8936f6c", "page": 14, "quote": "seed incident to k. Given these data, the incident seeds have independent conditional laws on\ntheir respective fibers; their product gives exactly the displayed operator. Figure 1 illustrates the\ninformation retained by this operation. Hypothesis (iii) bounds the oscillation of f on each such\nfiber by 2. It follows that\n∥f −Dkf∥2\n2 ≤4.\n(5.8)" }, { "blockId": "p14-acf38ee3fe", "page": 14, "quote": "position j′" }, { "blockId": "p14-21db4c168d", "page": 14, "quote": "question Q{k,j′},j′ fixed" }, { "blockId": "p14-e5e2d665b1", "page": 14, "quote": "position j\nquestion Q{k,j},j fixed" }, { "blockId": "p14-ef0a21e7d5", "page": 14, "quote": "c{k,j′}" }, { "blockId": "p14-b82f907fd7", "page": 14, "quote": "c{k,j}" }, { "blockId": "p14-f734cbf65f", "page": 14, "quote": "k" }, { "blockId": "p14-4210be5880", "page": 14, "quote": "every other position j′′" }, { "blockId": "p14-7075bcad44", "page": 14, "quote": "question Q{k,j′′},j′′ fixed" }, { "blockId": "p14-09264f0641", "page": 14, "quote": "Figure 1: The star operation centered at k. Each incident seed is resampled within the fiber of its\nquestion at the opposite endpoint. All nonincident seeds and all weights stay fixed. Thus every\nquery except the query at k stays fixed. The schematic depicts the conditional averaging operator\nDk, not additional graph edges." }, { "blockId": "p14-6f221287e3", "page": 14, "quote": "For a component designated at j, we charge invisible information on {j, k} to the star centered\nat k: that star retains j’s question on this pair. For every j ̸= k, the range of Dk is contained in\nthat of O{k,j},j. Distance to the larger closed subspace is no greater than distance to the smaller\none. Thus, for every v ∈L2," }, { "blockId": "p14-e6e5c65ba9", "page": 14, "quote": "∥(Id −O{k,j},j)v∥2\n2 ≤∥(Id −Dk)v∥2\n2." }, { "blockId": "p14-17d99adee0", "page": 14, "quote": "X" }, { "blockId": "p14-11c91b21e1", "page": 14, "quote": "For fixed S and k ̸= j(S), its seed decomposition shows that (Id −O{k,j(S)},j(S))fS is the sum of\nexactly the components with k ∈NS,ω. Consequently\nX" }, { "blockId": "p14-2de9842507", "page": 14, "quote": "S,ω\n|NS,ω|∥fS,ω∥2\n2 =\nX" }, { "blockId": "p14-7945191664", "page": 14, "quote": "S̸=∅" }, { "blockId": "p14-8b9100ab9b", "page": 14, "quote": "k̸=j(S)\n∥(Id −O{k,j(S)},j(S))fS∥2\n2" }, { "blockId": "p14-8531dc65e6", "page": 14, "quote": "d\nX" }, { "blockId": "p14-0104acb330", "page": 14, "quote": "X" }, { "blockId": "p14-b92b6db07f", "page": 14, "quote": "≤" }, { "blockId": "p14-19dfad80ba", "page": 14, "quote": "k=1" }, { "blockId": "p14-e6bd085493", "page": 14, "quote": "S̸=∅\n∥(Id −Dk)fS∥2\n2" }, { "blockId": "p14-472a841d98", "page": 14, "quote": "d\nX" }, { "blockId": "p14-2e18b1d1e1", "page": 14, "quote": "=" }, { "blockId": "p14-19092f176d", "page": 14, "quote": "k=1\n∥(Id −Dk)f∥2\n2 ≤4d.\n(5.9)" }, { "blockId": "p14-072202401e", "page": 14, "quote": "The inequality applies the preceding projection bound and adds the nonnegative terms with\nj(S) = k. The next equality uses the fact that Dk commutes with the weight projections, so the\nimages of distinct fS’s remain orthogonal. This argument uses neither products of different Dk’s\nnor commutation of opposite endpoint projections on one seed.\nThe three energy budgets are now available: weight interactions cost at most 4d, all nonconstant\nseeds at most 8d2, and incident information hidden from the designated question at most 4d. We\nnext check exactly which of these components a restriction preserves." }, { "blockId": "p15-56d3b89b17", "page": 15, "quote": "What the restriction keeps.\nFor fixed J and j ∈J, define the projection" }, { "blockId": "p15-b5277cc6cd", "page": 15, "quote": "P 0\ne\nY" }, { "blockId": "p15-0766ec7c51", "page": 15, "quote": "KJ,j = W 1\nj\nY" }, { "blockId": "p15-c1e6896573", "page": 15, "quote": "k∈J\\{j}\nO{j,k},j.\n(5.10)" }, { "blockId": "p15-0dcee1683c", "page": 15, "quote": "k∈J\\{j}\nW 0\nk\nY" }, { "blockId": "p15-f55def92af", "page": 15, "quote": "e∈(J\\{j}\n2 )" }, { "blockId": "p15-779c685497", "page": 15, "quote": "All seed factors here act on distinct coordinates. Before applying W 1\nj , these operators average\nexactly the unfrozen data other than the own questions uj and the own weights sj. They retain the\nfrozen data a as variables. The final centering W 1\nj subtracts the mean in sj, which is independent\nof the retained seed information. Thus" }, { "blockId": "p15-29d4682409", "page": 15, "quote": "KJ,jf = HJ,a\nj\n(uj, sj) −Es′\njHJ,a\nj\n(uj, s′\nj)" }, { "blockId": "p15-5ecf6999b6", "page": 15, "quote": "as functions on the original product space. If" }, { "blockId": "p15-6015a4e176", "page": 15, "quote": "V(J, a) =\nX" }, { "blockId": "p15-2c1d909a80", "page": 15, "quote": "j∈J\nEuj Varsj HJ,a\nj\n(uj, sj)," }, { "blockId": "p15-52c23815fd", "page": 15, "quote": "then integration over the initially random frozen data gives" }, { "blockId": "p15-f24b2adea2", "page": 15, "quote": "EaV(J, a) =\nX" }, { "blockId": "p15-ce5cb4cd85", "page": 15, "quote": "j∈J\n∥KJ,jf∥2\n2.\n(5.11)" }, { "blockId": "p15-5747044d54", "page": 15, "quote": "This identity uses only independent seed coordinates. Conditioning on the own questions restricts\neach incident hidden seed to its own fiber, as in the definition of Hj.\nConsider fS,ω and its designated position j = j(S). This component is preserved by the term\nKJ,j whenever" }, { "blockId": "p15-5e4719ae21", "page": 15, "quote": "\n= ∅.\n(5.12)" }, { "blockId": "p15-aba5150e1a", "page": 15, "quote": "S ∩J = {j},\nNS,ω ∩J = ∅,\nTS,ω ∩\nJ \\ {j}\n2" }, { "blockId": "p15-57d41c550d", "page": 15, "quote": "To verify that its squared norm can be counted in (5.11), first note that KJ,j commutes with every\nΠS. Thus its outputs from different weight supports are orthogonal, and ∥KJ,jf∥2\n2 = P\nS ∥KJ,jfS∥2\n2.\nThe weight factors in KJ,j keep fS precisely when S∩J = {j}. For those S whose designated position\nis this j, the seed factors are diagonal in the chosen seed decomposition: P 0\ne on a nonincident hidden\nseed keeps just its constant component, and Oe,j on an incident hidden seed keeps its constant and\nown-question components. The other seed coordinates are untouched. They therefore keep exactly\nthe components satisfying the last two conditions of (5.12), with no cancellation among them. We\nmay discard all other nonnegative squared norms in (5.11) and count each component solely in its\ndesignated term. It follows that" }, { "blockId": "p15-9d1daa6e4a", "page": 15, "quote": "EaV(J, a) ≥\nX" }, { "blockId": "p15-a71b6ad9bd", "page": 15, "quote": "S,ω\n1{conditions (5.12) hold}∥fS,ω∥2\n2.\n(5.13)" }, { "blockId": "p15-986481f9b9", "page": 15, "quote": "Averaging the restriction.\nChoose J uniformly among all h-element subsets of [d]. For a fixed\ncomponent, the designated position belongs to J with probability h/d. Conditional on this inclusion,\na specified other position belongs to J with probability" }, { "blockId": "p15-b56edca5df", "page": 15, "quote": "p1 = h −1" }, { "blockId": "p15-1f94cfc585", "page": 15, "quote": "d −1," }, { "blockId": "p15-6bd54e81b0", "page": 15, "quote": "and a specified pair avoiding the designated position lies in J with probability" }, { "blockId": "p15-c9aa1f4715", "page": 15, "quote": "p2 = (h −1)(h −2)" }, { "blockId": "p15-9134ce7d33", "page": 15, "quote": "(d −1)(d −2) ." }, { "blockId": "p16-b015ba1dc9", "page": 16, "quote": "The first two conditions in (5.12) fail only if one of S \\ {j(S)} or NS,ω is included. The third can fail\nonly at a pair in TS,ω avoiding j(S). A union bound therefore gives conditional survival probability\nat least\n1 −p1\n\n|S| −1 + |NS,ω|\n\n−p2|TS,ω|.\n(5.14)" }, { "blockId": "p16-4b32130281", "page": 16, "quote": "Charging all of TS,ω only decreases this lower bound. The bound remains valid when the displayed\nexpression is negative.\nBy (5.5), (5.9), and the orthogonal refinement of each fS,\nX" }, { "blockId": "p16-7a2e765530", "page": 16, "quote": "\n|S| −1 + |NS,ω|\n\n∥fS,ω∥2\n2 ≤8d." }, { "blockId": "p16-889a85ba2b", "page": 16, "quote": "S,ω" }, { "blockId": "p16-bd2c4edccb", "page": 16, "quote": "Combining (5.6), (5.7), (5.13), and (5.14) now yields" }, { "blockId": "p16-bf27fb903b", "page": 16, "quote": "EJ,aV(J, a) ≥h" }, { "blockId": "p16-cda59802b3", "page": 16, "quote": "d\n\n∥f∥2\n2 −8dp1 −8d2p2\n" }, { "blockId": "p16-a1e221561d", "page": 16, "quote": "≥h" }, { "blockId": "p16-0119551a53", "page": 16, "quote": "d\n\nνd −16h −32h2\n≥νh" }, { "blockId": "p16-5b95b452d5", "page": 16, "quote": "2 .\n(5.15)" }, { "blockId": "p16-741afdd376", "page": 16, "quote": "For the second inequality, d ≥4 gives p1 ≤2h/d and p2 ≤4h2/d2; the last inequality is (5.3).\nThe nonnegative measurable quantity V(J, a) is bounded, since f is bounded. Its mean in (5.15)\nimplies that a set of choices (J, a) of positive probability has V(J, a) ≥νh/4. Choose one such\nrestriction. The conditional averages were defined by explicit integration with its frozen coordinates\nplugged in. All hidden seeds and batch weights are then drawn with their original product law,\nestablishing (5.4) with the asserted interpretation." } ], "guide": [] }, { "id": "lemma-11", "kind": "lemma", "label": "Lemma 6.1", "source": { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, "statement": [ { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, { "blockId": "p17-979e3f5786", "page": 17, "quote": "Consequently,\n∥gj∥1 ≤1,\ngj,t′ = 0\nif t′ /∈slotij(Λij).\n(6.6)" }, { "blockId": "p17-9e62eecfd5", "page": 17, "quote": "The rule uses the fixed formula for F, the frozen increments, and the supplied batch queries and\nweights. It does not inspect which hidden seed occurrences produced those queries." } ], "proof": [ { "blockId": "p17-ac67f82849", "page": 17, "quote": "Proof. Let 1I ∈RP be the indicator vector of I. For ζ ∈A∗∪(−A∗), I ∈I, and ε ∈{−1, 1}, form\nthe finite collection of affine expressions" }, { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, { "blockId": "p18-6de128f74e", "page": 18, "quote": "For either C = A∗or C = −A∗," }, { "blockId": "p18-cf4ae44581", "page": 18, "quote": "" }, { "blockId": "p18-0d949687c8", "page": 18, "quote": "" }, { "blockId": "p18-2acb0c5f3c", "page": 18, "quote": "dist∗" }, { "blockId": "p18-ff074c45f1", "page": 18, "quote": "c0 +\nX" }, { "blockId": "p18-aa516f4123", "page": 18, "quote": "= min\nζ∈C\nmax\nI∈I, ε∈{−1,1} Lζ,I,ε(sJ)." }, { "blockId": "p18-9202fa965c", "page": 18, "quote": "j∈J\nz(ij, sj), C" }, { "blockId": "p18-e33882be56", "page": 18, "quote": "In particular, ψi is continuous and piecewise affine, with finitely many pieces.\nCompare all pairs of candidates in (6.7). If two candidates with different slope vectors in the\nunfrozen weights tie in value, set g = 0. Otherwise take the true gradient of the finite formula\nfor ψi. This latter choice is well-defined. Two candidates with the same slope either never tie or\nare identical affine functions. All other comparisons are strict at such a point and persist in a\nneighborhood; each minimum and maximum there selects one affine function up to identical copies.\nThe resulting formula is affine in that neighborhood. Fixing a deterministic order on candidates\nresolves any choice between identical copies without changing its gradient.\nEach discarded equality has a nonzero normal in the unfrozen coordinates, hence defines a\nproper affine hyperplane. No choice of frozen weights changes that fact, since frozen weights change\nthe offsets but not the slopes. Identical slopes caused by repeated endpoint questions or other\ncoincidences are allowed. Finite comparisons prove measurability, and the rule is defined even at\nquery tuples that cannot arise from the hidden seed law. Its formula uses the queries themselves\nand no additional seed information.\nAt a point not discarded, choose an active affine expression in each distance formula. Write their\nslopes as ε−vI−(i) for the distance to −A∗and ε+vI+(i) for the distance to A∗. Since F is half the\ndifference of these distances," }, { "blockId": "p18-b1c46cdf93", "page": 18, "quote": "g = 1" }, { "blockId": "p18-9e10b8c9b8", "page": 18, "quote": "2\n\nε−vI−(i) −ε+vI+(i)\n\n." }, { "blockId": "p18-6bc5f5b15a", "page": 18, "quote": "Both vectors in this average are signed compatible-incidence vectors, proving (6.5). At discarded\npoints the chosen value zero also belongs to the symmetric hull. Every signed incidence vector has\nblock ℓ1 norm at most one and zero entries in the unused slots, so convexity proves (6.6) at every\ntuple." } ], "guide": [] }, { "id": "lemma-12", "kind": "lemma", "label": "Lemma 6.2", "source": { "blockId": "p18-0ff91ffdb7", "page": 18, "quote": "Lemma 6.2 (Conditional cube variance). For each j ∈J and supported own tuple u, the function\nHj(u, ·) is 1-Lipschitz with respect to ∥· ∥∞. For almost every own s ∈[−1, 1]r," }, "statement": [ { "blockId": "p18-0ff91ffdb7", "page": 18, "quote": "Lemma 6.2 (Conditional cube variance). For each j ∈J and supported own tuple u, the function\nHj(u, ·) is 1-Lipschitz with respect to ∥· ∥∞. For almost every own s ∈[−1, 1]r," }, { "blockId": "p18-eae830cc0a", "page": 18, "quote": "∂s(t′)Hj(u, s) = ¯gj,t′(u, s),\nt′ ∈[r].\n(6.10)" }, { "blockId": "p18-3f3cdd765c", "page": 18, "quote": "Its variance satisfies the dimension-independent bound" }, { "blockId": "p18-87f7eefec4", "page": 18, "quote": "Vars Hj(u, s) ≤2Es∥¯gj(u, s)∥2\n2.\n(6.11)" }, { "blockId": "p19-62d154de5a", "page": 19, "quote": "Consequently, under the fixed restricted law,\nX" }, { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" } ], "proof": [ { "blockId": "p19-a8b14455ed", "page": 19, "quote": "Proof. Replacing the own block s by s′, with a seed configuration and other weights fixed, changes\nthe integrand in (6.4) by at most ∥s −s′∥∞. The same bound holds after averaging. In particular,\nevery scalar-coordinate difference quotient of the integrand is bounded in absolute value by one.\nFor each fixed hidden seed tuple, the full gradient identification in Theorem 6.1 holds outside\na null set in all batch weights. Fubini implies that for almost every own s it holds for almost all\nother weight blocks. The conditional seed law κj(u) has finite support and does not depend on s, so\nthe exceptional sets may be united over its support and over the finitely many weight coordinates.\nAt the remaining own values, the coordinate difference quotients converge almost surely to gj,t′.\nDominated convergence gives (6.10); the boundary of the cube has measure zero.\nEvery coordinate-line restriction of Hj(u, ·) is Lipschitz and hence absolutely continuous. By\nFubini applied to (6.10), for almost every fixing of the other own coordinates its derivative agrees\nalmost everywhere with the corresponding entry of ¯gj. The fundamental theorem of calculus\ntherefore expresses differences between any two values on such a line as the integral of that entry.\nLet w0 be the restriction of Hj(u, ·) to one of these lines. For x, y ∈[−1, 1], Cauchy–Schwarz\ngives" }, { "blockId": "p19-19df7f52ba", "page": 19, "quote": "|w0(x) −w0(y)|2 ≤|x −y|\nZ max(x,y)" }, { "blockId": "p19-e1c5f370cb", "page": 19, "quote": "min(x,y)\n|w′\n0(z)|2 dz ≤2\nZ 1" }, { "blockId": "p19-3940d513b5", "page": 19, "quote": "−1\n|w′\n0(z)|2 dz." }, { "blockId": "p19-ed42ada1e1", "page": 19, "quote": "For independent uniform X, Y ∈[−1, 1], this implies" }, { "blockId": "p19-1d80de8bc3", "page": 19, "quote": "Var(w0(X)) = 1" }, { "blockId": "p19-9eb7cdcaa6", "page": 19, "quote": "2E|w0(X) −w0(Y )|2 ≤\nZ 1" }, { "blockId": "p19-cf750ab39d", "page": 19, "quote": "−1\n|w′\n0(z)|2 dz = 2E|w′\n0(X)|2." }, { "blockId": "p19-4579d69f50", "page": 19, "quote": "The product decomposition of Efron and Stein [7, Section 2] bounds variance by the sum of the\naveraged one-coordinate conditional variances. To see this consequence directly, decompose the\nidentity at each scalar coordinate into its averaging projection and its orthogonal complement. The\nresulting orthogonal components are indexed by subsets of coordinates. Total variance counts every\nnonconstant squared component once, while the sum of one-coordinate conditional variances counts\nit once for each member of its nonempty index subset. Applying the line estimate, integrating over\nthe other coordinates, and summing proves (6.11). Averaging that bound over the own tuples and\nsumming over j, then using (6.3), proves (6.12)." }, { "blockId": "p19-67aa380edf", "page": 19, "quote": "A particular own weight value can make an entire slice of the other weights exceptional. The\nderivative identity is only an almost-everywhere assertion, and (6.12) is integrated over the restricted\nlaw. In contrast, (6.5), (6.6), and (6.9) hold at every input of their respective rules, including\nexceptional weights." } ], "guide": [] }, { "id": "proposition-13", "kind": "proposition", "label": "Proposition 6.3", "source": { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," }, "statement": [ { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," }, { "blockId": "p20-76458f0154", "page": 20, "quote": "EZ ≥βh" }, { "blockId": "p20-0aeab76362", "page": 20, "quote": "2 = 2.\n(6.15)" } ], "proof": [ { "blockId": "p20-c6f50a529f", "page": 20, "quote": "Proof. The mean block bound gives" }, { "blockId": "p20-cd69f7e969", "page": 20, "quote": "β" }, { "blockId": "p20-3b2b95e21c", "page": 20, "quote": "2 |Lj(u, s)| ≤∥¯gj(u, s)∥1 ≤1,\n|Lj(u, s)| ≤2/β = ℓ." }, { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, { "blockId": "p20-6e2af3d2bb", "page": 20, "quote": "X" }, { "blockId": "p20-782719d0a8", "page": 20, "quote": "2" }, { "blockId": "p20-35d79cfb92", "page": 20, "quote": "2 ." }, { "blockId": "p20-675b340463", "page": 20, "quote": "t′ /∈Tj\n|¯gj,t′|2 ≤β" }, { "blockId": "p20-45b8dcaa68", "page": 20, "quote": "t′ /∈Tj\n|¯gj,t′| ≤β" }, { "blockId": "p20-595a7da18e", "page": 20, "quote": "For the list slots, each mean entry has magnitude at most one, and conditional Jensen gives" }, { "blockId": "p20-fa0e5d3b86", "page": 20, "quote": "" }, { "blockId": "p20-3f02c9ecce", "page": 20, "quote": "" }, { "blockId": "p20-e202306f84", "page": 20, "quote": "X" }, { "blockId": "p20-d1b01a20f2", "page": 20, "quote": "X" }, { "blockId": "p20-1eeedf5812", "page": 20, "quote": "." }, { "blockId": "p20-d0d23e4323", "page": 20, "quote": "t′∈Tj\n|¯gj,t′|2 ≤\nX" }, { "blockId": "p20-c428dc9d9e", "page": 20, "quote": "t′∈Tj\n|¯gj,t′| ≤E" }, { "blockId": "p20-2064ebe155", "page": 20, "quote": "t′∈Tj\n|gj,t′|" }, { "blockId": "p20-00249ca770", "page": 20, "quote": "Gj" }, { "blockId": "p20-953714c1b0", "page": 20, "quote": "The last inequality uses the fact that Tj is determined by Gj. Taking expectations, summing, and\napplying (6.12) therefore yields" }, { "blockId": "p20-746b002e72", "page": 20, "quote": "X" }, { "blockId": "p20-6be5de16df", "page": 20, "quote": "E\nX" }, { "blockId": "p20-3b446a8fc5", "page": 20, "quote": "2 h = βh" }, { "blockId": "p20-221866509a", "page": 20, "quote": "2 .\n(6.16)" }, { "blockId": "p20-758f251dd1", "page": 20, "quote": "j∈J" }, { "blockId": "p20-068f4fb2a5", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≥βh −β" }, { "blockId": "p20-acacb43f6e", "page": 20, "quote": "For arbitrary fixed queries and arbitrary lists of valid labels, the sum of absolute values over\ntheir slot sets is a convex function of the full vector in RJ×[r]. On a signed generator ±vI(i) it equals\nthe number of positions hit by that selection I, since each block contains at most one selected slot.\nIts value on every point of the hull in (6.5) is consequently at most the maximum such hit count.\nAt every realized tuple, apply this deterministic statement to the realized lists and the full vector g:\nX" }, { "blockId": "p20-b0ff6bffd0", "page": 20, "quote": "X" }, { "blockId": "p20-010d9b9e12", "page": 20, "quote": "j∈J" }, { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" }, { "blockId": "p20-aeda64e05a", "page": 20, "quote": "No independence between the lists and the gradient, or between positions, is required. Combining\n(6.16) and (6.17) proves (6.15), with βh/2 = 2 from (6.1).\nFinally, (6.4) and (6.8) integrate over hidden seeds against the fixed law κj(u), and over other\nweights against their fixed cube laws. Their actual realized values are not inputs to the resulting\nmean function. Thus (6.13) has exactly the stated own-input dependence. No efficiency of this rule\nis needed for the soundness argument." }, { "blockId": "p21-25d511a8a2", "page": 21, "quote": "The kernels and list maps in this section remain fixed throughout the subsequent argument.\nConditioning later on other hidden seeds or on all weights changes the distribution of the inputs\nto these maps; it does not recompute their conditional averages. Locality, validity, and list-size\nbounds continue to hold for every fixed weight choice. The lower bound (6.15) concerns the original\nrestricted product law." } ], "guide": [] }, { "id": "lemma-14", "kind": "lemma", "label": "Lemma 7.1", "source": { "blockId": "p21-70de857a8d", "page": 21, "quote": "Lemma 7.1 (Decoding a pair). For every fixed J, a and fixed list maps satisfying Equation (7.2)," }, "statement": [ { "blockId": "p21-70de857a8d", "page": 21, "quote": "Lemma 7.1 (Decoding a pair). For every fixed J, a and fixed list maps satisfying Equation (7.2)," }, { "blockId": "p21-95dbadf67e", "page": 21, "quote": "\nℓ2σ = 3" }, { "blockId": "p21-b31ee8abfa", "page": 21, "quote": "2 −1" }, { "blockId": "p21-e300ec6c86", "page": 21, "quote": "2h < 3" }, { "blockId": "p21-e1ca65d7b0", "page": 21, "quote": "2.\n(7.4)" }, { "blockId": "p21-092429201f", "page": 21, "quote": "Pr(Ejk) ≤ℓ2σ\n(j, k ∈J, j < k),\nEZ ≤1 +\nh\n2" } ], "proof": [ { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." }, { "blockId": "p22-943351b541", "page": 22, "quote": "\nπc(Aρ,τ(xc)) = Aρ,τ(yc)\n\n≤σ.\n(7.5)" }, { "blockId": "p22-68e9aacef7", "page": 22, "quote": "This gives one symbol at every source variable. If a substituted tuple has probability zero, its\nempty list invokes the same fallback. If the variable already occurs in another coordinate of the\nquery, its value in a present label is still unique: a local label is an assignment on the set of distinct\nvariables in the scope. Thus repeated variables, empty valid-label sets, and missing ranks cause no\nambiguity. The assigned symbol on U depends only on x, and the assigned symbol on V only on y,\nbesides fixed data. It does not use the occurrence index or its projection map. The source promise\nconsequently gives\nPr\nc uniform in [M]" }, { "blockId": "p22-e0975c3db8", "page": 22, "quote": "For every actual remaining occurrence ce, the reconstructed tuples uxce\nj\n, uyce\nk\nare supported under\nthe original hidden-seed law: the full seed tuple comprising the fixed other seeds and this occurrence\nhas positive probability in that product law. They are exactly the own tuples at which the actual\nlists are evaluated.\nSuppose Ejk occurs and choose the ranks ρ, τ of labels witnessing it. The endpoint xce belongs\nto P(ij), and yce to P(ik). Compatibility requires the induced labeling to satisfy every source\nconstraint with both endpoints in the union of the selected scopes. In particular it satisfies the\nsampled occurrence ce. Its endpoint labels are exactly those used by Aρ,τ, so" }, { "blockId": "p22-7166649a70", "page": 22, "quote": "πce(Aρ,τ(xce)) = Aρ,τ(yce)." }, { "blockId": "p22-b0f00cc8f4", "page": 22, "quote": "This uses the sampled occurrence even when other occurrences have the same endpoints. Union\nbounding Equation (7.5) over the ℓ2 rank pairs proves the pair estimate for the fixed weights and\nother seeds. It is uniform in those fixed values, so averaging proves Pr(Ejk) ≤ℓ2σ. No conditioning\non compatible labels or successful ranks is used.\nFor every integer z ≥0, z ≤1 +\nz\n2\n\n, since the difference is (z −1)(z −2)/2 ≥0. Choose a\nselection attaining the maximum in Equation (7.3). Each of the pairs among its Z hit positions\nwitnesses the corresponding event Ejk. Consequently, pointwise," }, { "blockId": "p22-bfe8d15a27", "page": 22, "quote": "\n≤1 +\nX" }, { "blockId": "p22-59c4b5bf59", "page": 22, "quote": "Z ≤1 +\nZ\n2" }, { "blockId": "p22-e9d6cf0750", "page": 22, "quote": "{j,k}∈(J\n2)\n1Ejk.\n(7.6)" }, { "blockId": "p22-2f8a26b267", "page": 22, "quote": "Taking expectations, using the pair estimate, and substituting σ = 1/(h2ℓ2) gives Equation (7.4)." } ], "guide": [] }, { "id": "corollary-15", "kind": "corollary", "label": "Corollary 7.2", "source": { "blockId": "p22-9e3b3dd195", "page": 22, "quote": "Corollary 7.2 (Soundness). For the graph constructed in Section 3, assume (6.2). If val(Φ) ≤σ,\nevery independent set has size strictly less than n/m. Consequently τ(G) > (1 −1/m)n." }, "statement": [ { "blockId": "p22-9e3b3dd195", "page": 22, "quote": "Corollary 7.2 (Soundness). For the graph constructed in Section 3, assume (6.2). If val(Φ) ≤σ,\nevery independent set has size strictly less than n/m. Consequently τ(G) > (1 −1/m)n." } ], "proof": [ { "blockId": "p22-a3b65e66e6", "page": 22, "quote": "Proof. Suppose an independent set A has size at least n/m = 2b0n. By the low-norm clique\nobservation at the start of Section 4, at most one vertex of A has ∥Sv∥∗≤t. Since n ≥p > 1/b0,\nthe number of its high-norm vertices is at least" }, { "blockId": "p22-49f1fefab3", "page": 22, "quote": "2b0n −1 > b0n." }, { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, { "blockId": "p22-61d47165b6", "page": 22, "quote": "EZ ≥βh" }, { "blockId": "p22-3667d65700", "page": 22, "quote": "2 = 2." }, { "blockId": "p22-a7124c215a", "page": 22, "quote": "This contradicts Theorem 7.1, which gives EZ < 3/2 under the same remaining product law.\nThe lower bound just used averages over the hidden seeds and continuous batch weights for\nthose fixed J, a. The upper bound fixes further weights and seeds only to decode a single occurrence" }, { "blockId": "p23-5e6bed9716", "page": 23, "quote": "and is then averaged back. It does not require an energy lower bound, or a derivative identity, at\nevery such further fixing.\nWe have ruled out every independent set of size at least n/m. The complement of any vertex\ncover is independent, so every cover has size strictly greater than n −n/m, as claimed." } ], "guide": [ { "text": "Assume a large independent set exists in a source instance of value at most σ. The proof passes through a Lipschitz function, a variance-preserving restriction, and local lists before decoding.", "sources": [ { "blockId": "p22-a3b65e66e6", "page": 22, "quote": "Proof. Suppose an independent set A has size at least n/m = 2b0n. By the low-norm clique\nobservation at the start of Section 4, at most one vertex of A has ∥Sv∥∗≤t. Since n ≥p > 1/b0,\nthe number of its high-norm vertices is at least" }, { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" } ] }, { "text": "The prose cites Theorems 4.1, 5.1 and 6.3. The corresponding displayed headings are Lemma 4.1, Lemma 5.1 and Proposition 6.3. These are reference-label inconsistencies; they do not by themselves demonstrate a mathematical gap.", "sources": [ { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" }, { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" }, { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," } ] } ], "guideStatus": "Agent-authored reading guide; not independently verified." }, { "id": "external-16", "kind": "external", "label": "Reference [1]", "source": { "blockId": "p24-4e9bc96f99", "page": 24, "quote": "[1] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof\nverification and the hardness of approximation problems. Journal of the ACM, 45(3):501–555,\n1998. Preliminary version in FOCS 1992." }, "statement": [ { "blockId": "p24-4e9bc96f99", "page": 24, "quote": "[1] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof\nverification and the hardness of approximation problems. Journal of the ACM, 45(3):501–555,\n1998. Preliminary version in FOCS 1992." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-17", "kind": "external", "label": "Reference [2]", "source": { "blockId": "p24-22d541541a", "page": 24, "quote": "[2] Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of\nNP. Journal of the ACM, 45(1):70–122, 1998. Preliminary version in FOCS 1992." }, "statement": [ { "blockId": "p24-22d541541a", "page": 24, "quote": "[2] Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of\nNP. Journal of the ACM, 45(1):70–122, 1998. Preliminary version in FOCS 1992." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-18", "kind": "external", "label": "Reference [3]", "source": { "blockId": "p24-6636378992", "page": 24, "quote": "[3] Boaz Barak, Pravesh K. Kothari, and David Steurer. Small-set expansion in Shortcode graph\nand the 2-to-2 conjecture. In Avrim Blum, editor, 10th Innovations in Theoretical Computer\nScience Conference, volume 124 of Leibniz International Proceedings in Informatics, pages\n9:1–9:12. Schloss Dagstuhl–Leibniz-Zentrum f¨ur Informatik, 2019. https://doi.org/10.4230/\nLIPIcs.ITCS.2019.9." }, "statement": [ { "blockId": "p24-6636378992", "page": 24, "quote": "[3] Boaz Barak, Pravesh K. Kothari, and David Steurer. Small-set expansion in Shortcode graph\nand the 2-to-2 conjecture. In Avrim Blum, editor, 10th Innovations in Theoretical Computer\nScience Conference, volume 124 of Leibniz International Proceedings in Informatics, pages\n9:1–9:12. Schloss Dagstuhl–Leibniz-Zentrum f¨ur Informatik, 2019. https://doi.org/10.4230/\nLIPIcs.ITCS.2019.9." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-19", "kind": "external", "label": "Reference [4]", "source": { "blockId": "p24-48f73cd783", "page": 24, "quote": "[4] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the\n2-to-1 games conjecture?\nTheory of Computing, 21(11):1–50, 2025. https://doi.org/10.\n4086/toc.2025.v021a011. Preliminary version in STOC 2018." }, "statement": [ { "blockId": "p24-48f73cd783", "page": 24, "quote": "[4] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the\n2-to-1 games conjecture?\nTheory of Computing, 21(11):1–50, 2025. https://doi.org/10.\n4086/toc.2025.v021a011. Preliminary version in STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-20", "kind": "external", "label": "Reference [5]", "source": { "blockId": "p24-08770ca53d", "page": 24, "quote": "[5] Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals\nof Mathematics, 162(1):439–485, 2005. https://doi.org/10.4007/annals.2005.162.439." }, "statement": [ { "blockId": "p24-08770ca53d", "page": 24, "quote": "[5] Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals\nof Mathematics, 162(1):439–485, 2005. https://doi.org/10.4007/annals.2005.162.439." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-21", "kind": "external", "label": "Reference [6]", "source": { "blockId": "p24-7f2debf64e", "page": 24, "quote": "[6] Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proceedings of\nthe Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14, pages 624–633.\nAssociation for Computing Machinery, 2014. https://doi.org/10.1145/2591796.2591884.\nFull version: https://arxiv.org/abs/1305.1979v3, Theorem 6.2." }, "statement": [ { "blockId": "p24-7f2debf64e", "page": 24, "quote": "[6] Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proceedings of\nthe Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14, pages 624–633.\nAssociation for Computing Machinery, 2014. https://doi.org/10.1145/2591796.2591884.\nFull version: https://arxiv.org/abs/1305.1979v3, Theorem 6.2." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-22", "kind": "external", "label": "Reference [7]", "source": { "blockId": "p24-410974a00c", "page": 24, "quote": "[7] Bradley Efron and Charles Stein. The jackknife estimate of variance. The Annals of Statistics,\n9(3):586–596, 1981. https://doi.org/10.1214/aos/1176345462." }, "statement": [ { "blockId": "p24-410974a00c", "page": 24, "quote": "[7] Bradley Efron and Charles Stein. The jackknife estimate of variance. The Annals of Statistics,\n9(3):586–596, 1981. https://doi.org/10.1214/aos/1176345462." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-23", "kind": "external", "label": "Reference [8]", "source": { "blockId": "p24-ed654e0889", "page": 24, "quote": "[8] Uriel Feige, Shafi Goldwasser, L´aszl´o Lov´asz, Shmuel Safra, and Mario Szegedy. Interactive\nproofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268–292, 1996." }, "statement": [ { "blockId": "p24-ed654e0889", "page": 24, "quote": "[8] Uriel Feige, Shafi Goldwasser, L´aszl´o Lov´asz, Shmuel Safra, and Mario Szegedy. Interactive\nproofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268–292, 1996." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-24", "kind": "external", "label": "Reference [9]", "source": { "blockId": "p24-477728a37a", "page": 24, "quote": "[9] Ehud Friedgut. Boolean functions with low average sensitivity depend on few coordinates.\nCombinatorica, 18(1):27–35, 1998. https://doi.org/10.1007/PL00009809." }, "statement": [ { "blockId": "p24-477728a37a", "page": 24, "quote": "[9] Ehud Friedgut. Boolean functions with low average sensitivity depend on few coordinates.\nCombinatorica, 18(1):27–35, 1998. https://doi.org/10.1007/PL00009809." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-25", "kind": "external", "label": "Reference [10]", "source": { "blockId": "p24-6b96535962", "page": 24, "quote": "[10] Johan H˚astad. Some optimal inapproximability results. Journal of the ACM, 48(4):798–859,\nJuly 2001. https://doi.org/10.1145/502090.502098." }, "statement": [ { "blockId": "p24-6b96535962", "page": 24, "quote": "[10] Johan H˚astad. Some optimal inapproximability results. Journal of the ACM, 48(4):798–859,\nJuly 2001. https://doi.org/10.1145/502090.502098." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-26", "kind": "external", "label": "Reference [11]", "source": { "blockId": "p25-03de4d0889", "page": 25, "quote": "[11] George Karakostas. A better approximation ratio for the Vertex Cover problem. ACM Trans-\nactions on Algorithms, 5(4):41:1–41:8, 2009. https://doi.org/10.1145/1597036.1597045.\nPreliminary version in ICALP 2005." }, "statement": [ { "blockId": "p25-03de4d0889", "page": 25, "quote": "[11] George Karakostas. A better approximation ratio for the Vertex Cover problem. ACM Trans-\nactions on Algorithms, 5(4):41:1–41:8, 2009. https://doi.org/10.1145/1597036.1597045.\nPreliminary version in ICALP 2005." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-27", "kind": "external", "label": "Reference [12]", "source": { "blockId": "p25-fd6b49eaa7", "page": 25, "quote": "[12] Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and\nJames W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum\nPress, 1972. https://doi.org/10.1007/978-1-4684-2001-2_9." }, "statement": [ { "blockId": "p25-fd6b49eaa7", "page": 25, "quote": "[12] Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and\nJames W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum\nPress, 1972. https://doi.org/10.1007/978-1-4684-2001-2_9." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-28", "kind": "external", "label": "Reference [13]", "source": { "blockId": "p25-ceea3c618d", "page": 25, "quote": "[13] Subhash Khot.\nOn the power of unique 2-prover 1-round games.\nIn Proceedings of the\nThirty-Fourth Annual ACM Symposium on Theory of Computing, pages 767–775. ACM, 2002." }, "statement": [ { "blockId": "p25-ceea3c618d", "page": 25, "quote": "[13] Subhash Khot.\nOn the power of unique 2-prover 1-round games.\nIn Proceedings of the\nThirty-Fourth Annual ACM Symposium on Theory of Computing, pages 767–775. ACM, 2002." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-29", "kind": "external", "label": "Reference [14]", "source": { "blockId": "p25-dd48505af8", "page": 25, "quote": "[14] Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in Grassmann graph have\nnear-perfect expansion. Annals of Mathematics, 198(1):1–92, 2023. https://doi.org/10.\n4007/annals.2023.198.1.1. Preliminary version in FOCS 2018." }, "statement": [ { "blockId": "p25-dd48505af8", "page": 25, "quote": "[14] Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in Grassmann graph have\nnear-perfect expansion. Annals of Mathematics, 198(1):1–92, 2023. https://doi.org/10.\n4007/annals.2023.198.1.1. Preliminary version in FOCS 2018." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-30", "kind": "external", "label": "Reference [15]", "source": { "blockId": "p25-6d011ec792", "page": 25, "quote": "[15] Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games and Grassmann\ngraphs.\nTheory of Computing, 21(10):1–55, 2025.\nhttps://doi.org/10.4086/toc.2025.\nv021a010. Preliminary version in STOC 2017." }, "statement": [ { "blockId": "p25-6d011ec792", "page": 25, "quote": "[15] Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games and Grassmann\ngraphs.\nTheory of Computing, 21(10):1–55, 2025.\nhttps://doi.org/10.4086/toc.2025.\nv021a010. Preliminary version in STOC 2017." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-31", "kind": "external", "label": "Reference [16]", "source": { "blockId": "p25-c4d4ae699c", "page": 25, "quote": "[16] Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2 −ϵ.\nJournal of Computer and System Sciences, 74(3):335–349, May 2008. https://doi.org/10.\n1016/j.jcss.2007.06.019." }, "statement": [ { "blockId": "p25-c4d4ae699c", "page": 25, "quote": "[16] Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2 −ϵ.\nJournal of Computer and System Sciences, 74(3):335–349, May 2008. https://doi.org/10.\n1016/j.jcss.2007.06.019." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-32", "kind": "external", "label": "Reference [17]", "source": { "blockId": "p25-9a70a69653", "page": 25, "quote": "[17] Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763–803, 1998." }, "statement": [ { "blockId": "p25-9a70a69653", "page": 25, "quote": "[17] Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763–803, 1998." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "equation-33", "kind": "equation", "label": "Equation (2.1)", "source": { "blockId": "p4-09267b9534", "page": 4, "quote": "d > 4m.\n(2.1)" }, "statement": [ { "blockId": "p4-09267b9534", "page": 4, "quote": "d > 4m.\n(2.1)" } ], "proof": [], "guide": [] }, { "id": "equation-34", "kind": "equation", "label": "Equation (2.2)", "source": { "blockId": "p4-dc0d9cf7e1", "page": 4, "quote": "d.\n(2.2)" }, "statement": [ { "blockId": "p4-dc0d9cf7e1", "page": 4, "quote": "d.\n(2.2)" } ], "proof": [], "guide": [] }, { "id": "equation-35", "kind": "equation", "label": "Equation (3.1)", "source": { "blockId": "p6-918024cee6", "page": 6, "quote": ".\n(3.1)" }, "statement": [ { "blockId": "p6-918024cee6", "page": 6, "quote": ".\n(3.1)" } ], "proof": [], "guide": [] }, { "id": "equation-36", "kind": "equation", "label": "Equation (3.2)", "source": { "blockId": "p6-5b94c9fe95", "page": 6, "quote": "Lemma 3.1 (Compatibility norm). The family I is hereditary and contains every coordinate\nsingleton. Formula (3.1) defines a norm. For each fixed query i, the increment is linear in s and\nobeys\n∥z(i, s)∥∗≤∥s∥∞.\n(3.2)" }, "statement": [ { "blockId": "p6-5b94c9fe95", "page": 6, "quote": "Lemma 3.1 (Compatibility norm). The family I is hereditary and contains every coordinate\nsingleton. Formula (3.1) defines a norm. For each fixed query i, the increment is linear in s and\nobeys\n∥z(i, s)∥∗≤∥s∥∞.\n(3.2)" } ], "proof": [], "guide": [] }, { "id": "equation-37", "kind": "equation", "label": "Equation (3.3)", "source": { "blockId": "p6-d3c9737c8b", "page": 6, "quote": "p : u = −d, −d + 1, . . . , d\n\n,\np = 2d + 1.\n(3.3)" }, "statement": [ { "blockId": "p6-d3c9737c8b", "page": 6, "quote": "p : u = −d, −d + 1, . . . , d\n\n,\np = 2d + 1.\n(3.3)" } ], "proof": [], "guide": [] }, { "id": "equation-38", "kind": "equation", "label": "Equation (3.4)", "source": { "blockId": "p6-c69bde6d77", "page": 6, "quote": "j=1\nz(ij(c), sj).\n(3.4)" }, "statement": [ { "blockId": "p6-c69bde6d77", "page": 6, "quote": "j=1\nz(ij(c), sj).\n(3.4)" } ], "proof": [], "guide": [] }, { "id": "equation-39", "kind": "equation", "label": "Equation (3.5)", "source": { "blockId": "p7-5fce302f5a", "page": 7, "quote": "{v, w} ∈E(G)\n⇐⇒\n∥Sv + Sw∥∗≤2t.\n(3.5)" }, "statement": [ { "blockId": "p7-5fce302f5a", "page": 7, "quote": "{v, w} ∈E(G)\n⇐⇒\n∥Sv + Sw∥∗≤2t.\n(3.5)" } ], "proof": [], "guide": [] }, { "id": "equation-40", "kind": "equation", "label": "Equation (3.6)", "source": { "blockId": "p7-9086e7b2b1", "page": 7, "quote": "n = |V (G)| = M(d\n2)prd ≥p > 2m,\n(3.6)" }, "statement": [ { "blockId": "p7-9086e7b2b1", "page": 7, "quote": "n = |V (G)| = M(d\n2)prd ≥p > 2m,\n(3.6)" } ], "proof": [], "guide": [] }, { "id": "equation-41", "kind": "equation", "label": "Equation (3.7)", "source": { "blockId": "p7-e0da83d6df", "page": 7, "quote": ".\n(3.7)" }, "statement": [ { "blockId": "p7-e0da83d6df", "page": 7, "quote": ".\n(3.7)" } ], "proof": [], "guide": [] }, { "id": "equation-42", "kind": "equation", "label": "Equation (3.8)", "source": { "blockId": "p8-9b6f9115d5", "page": 8, "quote": "m.\n(3.8)" }, "statement": [ { "blockId": "p8-9b6f9115d5", "page": 8, "quote": "m.\n(3.8)" } ], "proof": [], "guide": [] }, { "id": "equation-43", "kind": "equation", "label": "Equation (3.9)", "source": { "blockId": "p9-18496e9242", "page": 9, "quote": "j=1\nsj,slotij(c)(λA,ij(c)) > t.\n(3.9)" }, "statement": [ { "blockId": "p9-18496e9242", "page": 9, "quote": "j=1\nsj,slotij(c)(λA,ij(c)) > t.\n(3.9)" } ], "proof": [], "guide": [] }, { "id": "equation-44", "kind": "equation", "label": "Equation (4.1)", "source": { "blockId": "p9-59ad20044b", "page": 9, "quote": "4\n=\n1\n27m3 .\n(4.1)" }, "statement": [ { "blockId": "p9-59ad20044b", "page": 9, "quote": "4\n=\n1\n27m3 .\n(4.1)" } ], "proof": [], "guide": [] }, { "id": "equation-45", "kind": "equation", "label": "Equation (4.2)", "source": { "blockId": "p10-9d68e8efbd", "page": 10, "quote": "2\n\ndist∗(z, −A∗) −dist∗(z, A∗)\n\n(4.2)" }, "statement": [ { "blockId": "p10-9d68e8efbd", "page": 10, "quote": "2\n\ndist∗(z, −A∗) −dist∗(z, A∗)\n\n(4.2)" } ], "proof": [], "guide": [] }, { "id": "equation-46", "kind": "equation", "label": "Equation (4.3)", "source": { "blockId": "p10-254bfac670", "page": 10, "quote": ".\n(4.3)" }, "statement": [ { "blockId": "p10-254bfac670", "page": 10, "quote": ".\n(4.3)" } ], "proof": [], "guide": [] }, { "id": "equation-47", "kind": "equation", "label": "Equation (4.4)", "source": { "blockId": "p10-6aa5afb122", "page": 10, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(4.4)" }, "statement": [ { "blockId": "p10-6aa5afb122", "page": 10, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(4.4)" } ], "proof": [], "guide": [] }, { "id": "equation-48", "kind": "equation", "label": "Equation (5.1)", "source": { "blockId": "p11-95ea1c8256", "page": 11, "quote": "a =\n\n(ce)e/∈(J\n2), (sk)k/∈J\n\n.\n(5.1)" }, "statement": [ { "blockId": "p11-95ea1c8256", "page": 11, "quote": "a =\n\n(ce)e/∈(J\n2), (sk)k/∈J\n\n.\n(5.1)" } ], "proof": [], "guide": [] }, { "id": "equation-49", "kind": "equation", "label": "Equation (5.2)", "source": { "blockId": "p12-e17a0fda8b", "page": 12, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(5.2)" }, "statement": [ { "blockId": "p12-e17a0fda8b", "page": 12, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(5.2)" } ], "proof": [], "guide": [] }, { "id": "equation-50", "kind": "equation", "label": "Equation (5.3)", "source": { "blockId": "p12-ad0f187d8f", "page": 12, "quote": "2 ,\n(5.3)" }, "statement": [ { "blockId": "p12-ad0f187d8f", "page": 12, "quote": "2 ,\n(5.3)" } ], "proof": [], "guide": [] }, { "id": "equation-51", "kind": "equation", "label": "Equation (5.4)", "source": { "blockId": "p12-eb730192d5", "page": 12, "quote": "4 .\n(5.4)" }, "statement": [ { "blockId": "p12-eb730192d5", "page": 12, "quote": "4 .\n(5.4)" } ], "proof": [], "guide": [] }, { "id": "equation-52", "kind": "equation", "label": "Equation (5.5)", "source": { "blockId": "p13-53aca89949", "page": 13, "quote": "j=1\n∥f −W 0\nj f∥2\n2 ≤4d.\n(5.5)" }, "statement": [ { "blockId": "p13-53aca89949", "page": 13, "quote": "j=1\n∥f −W 0\nj f∥2\n2 ≤4d.\n(5.5)" } ], "proof": [], "guide": [] }, { "id": "equation-53", "kind": "equation", "label": "Equation (5.6)", "source": { "blockId": "p13-3cfa9c14d3", "page": 13, "quote": "S,ω\n∥fS,ω∥2\n2 = ∥f∥2\n2,\n(5.6)" }, "statement": [ { "blockId": "p13-3cfa9c14d3", "page": 13, "quote": "S,ω\n∥fS,ω∥2\n2 = ∥f∥2\n2,\n(5.6)" } ], "proof": [], "guide": [] }, { "id": "equation-54", "kind": "equation", "label": "Equation (5.7)", "source": { "blockId": "p13-1830aafe32", "page": 13, "quote": "\n≤8d2.\n(5.7)" }, "statement": [ { "blockId": "p13-1830aafe32", "page": 13, "quote": "\n≤8d2.\n(5.7)" } ], "proof": [], "guide": [] }, { "id": "equation-55", "kind": "equation", "label": "Equation (5.8)", "source": { "blockId": "p14-8ad8936f6c", "page": 14, "quote": "seed incident to k. Given these data, the incident seeds have independent conditional laws on\ntheir respective fibers; their product gives exactly the displayed operator. Figure 1 illustrates the\ninformation retained by this operation. Hypothesis (iii) bounds the oscillation of f on each such\nfiber by 2. It follows that\n∥f −Dkf∥2\n2 ≤4.\n(5.8)" }, "statement": [ { "blockId": "p14-8ad8936f6c", "page": 14, "quote": "seed incident to k. Given these data, the incident seeds have independent conditional laws on\ntheir respective fibers; their product gives exactly the displayed operator. Figure 1 illustrates the\ninformation retained by this operation. Hypothesis (iii) bounds the oscillation of f on each such\nfiber by 2. It follows that\n∥f −Dkf∥2\n2 ≤4.\n(5.8)" } ], "proof": [], "guide": [] }, { "id": "equation-56", "kind": "equation", "label": "Equation (5.9)", "source": { "blockId": "p14-19092f176d", "page": 14, "quote": "k=1\n∥(Id −Dk)f∥2\n2 ≤4d.\n(5.9)" }, "statement": [ { "blockId": "p14-19092f176d", "page": 14, "quote": "k=1\n∥(Id −Dk)f∥2\n2 ≤4d.\n(5.9)" } ], "proof": [], "guide": [] }, { "id": "equation-57", "kind": "equation", "label": "Equation (5.10)", "source": { "blockId": "p15-c1e6896573", "page": 15, "quote": "k∈J\\{j}\nO{j,k},j.\n(5.10)" }, "statement": [ { "blockId": "p15-c1e6896573", "page": 15, "quote": "k∈J\\{j}\nO{j,k},j.\n(5.10)" } ], "proof": [], "guide": [] }, { "id": "equation-58", "kind": "equation", "label": "Equation (5.11)", "source": { "blockId": "p15-ce5cb4cd85", "page": 15, "quote": "j∈J\n∥KJ,jf∥2\n2.\n(5.11)" }, "statement": [ { "blockId": "p15-ce5cb4cd85", "page": 15, "quote": "j∈J\n∥KJ,jf∥2\n2.\n(5.11)" } ], "proof": [], "guide": [] }, { "id": "equation-59", "kind": "equation", "label": "Equation (5.12)", "source": { "blockId": "p15-5e4719ae21", "page": 15, "quote": "\n= ∅.\n(5.12)" }, "statement": [ { "blockId": "p15-5e4719ae21", "page": 15, "quote": "\n= ∅.\n(5.12)" } ], "proof": [], "guide": [] }, { "id": "equation-60", "kind": "equation", "label": "Equation (5.13)", "source": { "blockId": "p15-a71b6ad9bd", "page": 15, "quote": "S,ω\n1{conditions (5.12) hold}∥fS,ω∥2\n2.\n(5.13)" }, "statement": [ { "blockId": "p15-a71b6ad9bd", "page": 15, "quote": "S,ω\n1{conditions (5.12) hold}∥fS,ω∥2\n2.\n(5.13)" } ], "proof": [], "guide": [] }, { "id": "equation-61", "kind": "equation", "label": "Equation (5.14)", "source": { "blockId": "p16-b015ba1dc9", "page": 16, "quote": "The first two conditions in (5.12) fail only if one of S \\ {j(S)} or NS,ω is included. The third can fail\nonly at a pair in TS,ω avoiding j(S). A union bound therefore gives conditional survival probability\nat least\n1 −p1\n\n|S| −1 + |NS,ω|\n\n−p2|TS,ω|.\n(5.14)" }, "statement": [ { "blockId": "p16-b015ba1dc9", "page": 16, "quote": "The first two conditions in (5.12) fail only if one of S \\ {j(S)} or NS,ω is included. The third can fail\nonly at a pair in TS,ω avoiding j(S). A union bound therefore gives conditional survival probability\nat least\n1 −p1\n\n|S| −1 + |NS,ω|\n\n−p2|TS,ω|.\n(5.14)" } ], "proof": [], "guide": [] }, { "id": "equation-62", "kind": "equation", "label": "Equation (5.15)", "source": { "blockId": "p16-5b95b452d5", "page": 16, "quote": "2 .\n(5.15)" }, "statement": [ { "blockId": "p16-5b95b452d5", "page": 16, "quote": "2 .\n(5.15)" } ], "proof": [], "guide": [] }, { "id": "equation-63", "kind": "equation", "label": "Equation (6.1)", "source": { "blockId": "p16-a0d26bb753", "page": 16, "quote": "β ,\nh = 2ℓ.\n(6.1)" }, "statement": [ { "blockId": "p16-a0d26bb753", "page": 16, "quote": "β ,\nh = 2ℓ.\n(6.1)" } ], "proof": [], "guide": [] }, { "id": "equation-64", "kind": "equation", "label": "Equation (6.2)", "source": { "blockId": "p16-dc57d17785", "page": 16, "quote": "2 .\n(6.2)" }, "statement": [ { "blockId": "p16-dc57d17785", "page": 16, "quote": "2 .\n(6.2)" } ], "proof": [], "guide": [] }, { "id": "equation-65", "kind": "equation", "label": "Equation (6.3)", "source": { "blockId": "p16-6ae5a923ab", "page": 16, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh/4.\n(6.3)" }, "statement": [ { "blockId": "p16-6ae5a923ab", "page": 16, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh/4.\n(6.3)" } ], "proof": [], "guide": [] }, { "id": "equation-66", "kind": "equation", "label": "Equation (6.4)", "source": { "blockId": "p17-2dbe9d3401", "page": 17, "quote": "Hj(u, s) = EcJ∼κj(u), (sk)k∈J\\{j}f∗(cJ, sJ)\n\nsj=s,\n(6.4)" }, "statement": [ { "blockId": "p17-2dbe9d3401", "page": 17, "quote": "Hj(u, s) = EcJ∼κj(u), (sk)k∈J\\{j}f∗(cJ, sJ)\n\nsj=s,\n(6.4)" } ], "proof": [], "guide": [] }, { "id": "equation-67", "kind": "equation", "label": "Equation (6.5)", "source": { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, "statement": [ { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" } ], "proof": [], "guide": [] }, { "id": "equation-68", "kind": "equation", "label": "Equation (6.6)", "source": { "blockId": "p17-979e3f5786", "page": 17, "quote": "Consequently,\n∥gj∥1 ≤1,\ngj,t′ = 0\nif t′ /∈slotij(Λij).\n(6.6)" }, "statement": [ { "blockId": "p17-979e3f5786", "page": 17, "quote": "Consequently,\n∥gj∥1 ≤1,\ngj,t′ = 0\nif t′ /∈slotij(Λij).\n(6.6)" } ], "proof": [], "guide": [] }, { "id": "equation-69", "kind": "equation", "label": "Equation (6.7)", "source": { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, "statement": [ { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" } ], "proof": [], "guide": [] }, { "id": "equation-70", "kind": "equation", "label": "Equation (6.8)", "source": { "blockId": "p18-7cd2b491d3", "page": 18, "quote": "¯gj(u, s) = EcJ∼κj(u), (sk)k∈J\\{j}gj(cJ, sJ)\n\nsj=s.\n(6.8)" }, "statement": [ { "blockId": "p18-7cd2b491d3", "page": 18, "quote": "¯gj(u, s) = EcJ∼κj(u), (sk)k∈J\\{j}gj(cJ, sJ)\n\nsj=s.\n(6.8)" } ], "proof": [], "guide": [] }, { "id": "equation-71", "kind": "equation", "label": "Equation (6.9)", "source": { "blockId": "p18-7ac4aecf1c", "page": 18, "quote": "∥¯gj(u, s)∥1 ≤1,\n¯gj,t′(u, s) = 0\non unused slots of ij(u).\n(6.9)" }, "statement": [ { "blockId": "p18-7ac4aecf1c", "page": 18, "quote": "∥¯gj(u, s)∥1 ≤1,\n¯gj,t′(u, s) = 0\non unused slots of ij(u).\n(6.9)" } ], "proof": [], "guide": [] }, { "id": "equation-72", "kind": "equation", "label": "Equation (6.10)", "source": { "blockId": "p18-eae830cc0a", "page": 18, "quote": "∂s(t′)Hj(u, s) = ¯gj,t′(u, s),\nt′ ∈[r].\n(6.10)" }, "statement": [ { "blockId": "p18-eae830cc0a", "page": 18, "quote": "∂s(t′)Hj(u, s) = ¯gj,t′(u, s),\nt′ ∈[r].\n(6.10)" } ], "proof": [], "guide": [] }, { "id": "equation-73", "kind": "equation", "label": "Equation (6.11)", "source": { "blockId": "p18-87f7eefec4", "page": 18, "quote": "Vars Hj(u, s) ≤2Es∥¯gj(u, s)∥2\n2.\n(6.11)" }, "statement": [ { "blockId": "p18-87f7eefec4", "page": 18, "quote": "Vars Hj(u, s) ≤2Es∥¯gj(u, s)∥2\n2.\n(6.11)" } ], "proof": [], "guide": [] }, { "id": "equation-74", "kind": "equation", "label": "Equation (6.12)", "source": { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" }, "statement": [ { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" } ], "proof": [], "guide": [] }, { "id": "equation-75", "kind": "equation", "label": "Equation (6.13)", "source": { "blockId": "p19-a43f673884", "page": 19, "quote": "\n.\n(6.13)" }, "statement": [ { "blockId": "p19-a43f673884", "page": 19, "quote": "\n.\n(6.13)" } ], "proof": [], "guide": [] }, { "id": "equation-76", "kind": "equation", "label": "Equation (6.14)", "source": { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, "statement": [ { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" } ], "proof": [], "guide": [] }, { "id": "equation-77", "kind": "equation", "label": "Equation (6.15)", "source": { "blockId": "p20-0aeab76362", "page": 20, "quote": "2 = 2.\n(6.15)" }, "statement": [ { "blockId": "p20-0aeab76362", "page": 20, "quote": "2 = 2.\n(6.15)" } ], "proof": [], "guide": [] }, { "id": "equation-78", "kind": "equation", "label": "Equation (6.16)", "source": { "blockId": "p20-221866509a", "page": 20, "quote": "2 .\n(6.16)" }, "statement": [ { "blockId": "p20-221866509a", "page": 20, "quote": "2 .\n(6.16)" } ], "proof": [], "guide": [] }, { "id": "equation-79", "kind": "equation", "label": "Equation (6.17)", "source": { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" }, "statement": [ { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" } ], "proof": [], "guide": [] }, { "id": "equation-80", "kind": "equation", "label": "Equation (7.1)", "source": { "blockId": "p21-04a624cdd6", "page": 21, "quote": "Set\nσ =\n1\nh2ℓ2 ,\n(7.1)" }, "statement": [ { "blockId": "p21-04a624cdd6", "page": 21, "quote": "Set\nσ =\n1\nh2ℓ2 ,\n(7.1)" } ], "proof": [], "guide": [] }, { "id": "equation-81", "kind": "equation", "label": "Equation (7.2)", "source": { "blockId": "p21-a4f76b578f", "page": 21, "quote": "Lj = Lj(uj, sj) ⊆Λij,\n|Lj| ≤ℓ.\n(7.2)" }, "statement": [ { "blockId": "p21-a4f76b578f", "page": 21, "quote": "Lj = Lj(uj, sj) ⊆Λij,\n|Lj| ≤ℓ.\n(7.2)" } ], "proof": [], "guide": [] }, { "id": "equation-82", "kind": "equation", "label": "Equation (7.3)", "source": { "blockId": "p21-9da6541b11", "page": 21, "quote": "Z = max\nI∈I #{j ∈J : some λ ∈Lj has (ij, λ) ∈I}.\n(7.3)" }, "statement": [ { "blockId": "p21-9da6541b11", "page": 21, "quote": "Z = max\nI∈I #{j ∈J : some λ ∈Lj has (ij, λ) ∈I}.\n(7.3)" } ], "proof": [], "guide": [] }, { "id": "equation-83", "kind": "equation", "label": "Equation (7.4)", "source": { "blockId": "p21-e1ca65d7b0", "page": 21, "quote": "2.\n(7.4)" }, "statement": [ { "blockId": "p21-e1ca65d7b0", "page": 21, "quote": "2.\n(7.4)" } ], "proof": [], "guide": [] }, { "id": "equation-84", "kind": "equation", "label": "Equation (7.5)", "source": { "blockId": "p22-943351b541", "page": 22, "quote": "\nπc(Aρ,τ(xc)) = Aρ,τ(yc)\n\n≤σ.\n(7.5)" }, "statement": [ { "blockId": "p22-943351b541", "page": 22, "quote": "\nπc(Aρ,τ(xc)) = Aρ,τ(yc)\n\n≤σ.\n(7.5)" } ], "proof": [], "guide": [] }, { "id": "equation-85", "kind": "equation", "label": "Equation (7.6)", "source": { "blockId": "p22-e9d6cf0750", "page": 22, "quote": "{j,k}∈(J\n2)\n1Ejk.\n(7.6)" }, "statement": [ { "blockId": "p22-e9d6cf0750", "page": 22, "quote": "{j,k}∈(J\n2)\n1Ejk.\n(7.6)" } ], "proof": [], "guide": [] }, { "id": "equation-86", "kind": "equation", "label": "Equation (8.1)", "source": { "blockId": "p23-13ee73bc9e", "page": 23, "quote": "(8.1)" }, "statement": [ { "blockId": "p23-13ee73bc9e", "page": 23, "quote": "(8.1)" } ], "proof": [], "guide": [] }, { "id": "equation-87", "kind": "equation", "label": "Equation (8.2)", "source": { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" }, "statement": [ { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" } ], "proof": [], "guide": [] }, { "id": "external-88", "kind": "external", "label": "[10] Theorem 8.1", "source": { "blockId": "p2-5fcd56bf85", "page": 2, "quote": "The connection between proof checking and approximation hardness was developed by Feige,\nGoldwasser, Lov´asz, Safra, and Szegedy [8]. Their graph encodes accepting local proof views, with\nconsistency of the answers governing adjacency. The PCP breakthrough of 1992, developed by\nArora and Safra and completed in constant-query form by Arora, Lund, Motwani, Sudan, and\nSzegedy [2, 1], yielded constant approximation gaps for problems including Vertex Cover. Raz’s\nparallel repetition theorem [17] reduces the soundness of projection games to any desired fixed\npositive constant while preserving perfect completeness. Together these results supply the Label\nCover promise stated precisely in Theorem 2.1, the only external hardness input used here.\nFor Vertex Cover, H˚astad established hardness for every fixed factor below 7/6 [10, Theorem 8.1].\nDinur and Safra raised this to every fixed factor below 10\n√" }, "statement": [ { "blockId": "p2-5fcd56bf85", "page": 2, "quote": "The connection between proof checking and approximation hardness was developed by Feige,\nGoldwasser, Lov´asz, Safra, and Szegedy [8]. Their graph encodes accepting local proof views, with\nconsistency of the answers governing adjacency. The PCP breakthrough of 1992, developed by\nArora and Safra and completed in constant-query form by Arora, Lund, Motwani, Sudan, and\nSzegedy [2, 1], yielded constant approximation gaps for problems including Vertex Cover. Raz’s\nparallel repetition theorem [17] reduces the soundness of projection games to any desired fixed\npositive constant while preserving perfect completeness. Together these results supply the Label\nCover promise stated precisely in Theorem 2.1, the only external hardness input used here.\nFor Vertex Cover, H˚astad established hardness for every fixed factor below 7/6 [10, Theorem 8.1].\nDinur and Safra raised this to every fixed factor below 10\n√" }, { "blockId": "p24-6b96535962", "page": 24, "quote": "[10] Johan H˚astad. Some optimal inapproximability results. Journal of the ACM, 48(4):798–859,\nJuly 2001. https://doi.org/10.1145/502090.502098." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-89", "kind": "external", "label": "[5] Theorem 1.1", "source": { "blockId": "p2-706b6597ff", "page": 2, "quote": "5 −21, approximately 1.36068 [5,\nTheorem 1.1]. Their analysis uses biased set families and Boolean-function structure to control\nindependent-set density. Khot’s Unique Games Conjecture [13] proposed a stronger source of\napproximation gaps. Assuming this conjecture, Khot and Regev obtained hardness for every fixed\nfactor below two [16]. Their proof uses a biased long code and an equivalent, stronger formulation of\nthe source promise that supports short lists of candidate labels. Friedgut’s approximation theorem\nfor Boolean functions of low average sensitivity [9] supplies bounded sets of relevant coordinates in\nthat analysis.\nThe 2017–2018 developments in the 2-to-2 Games program gave an unconditional advance. The\nGrassmann-graph reduction of Khot, Minzer, and Safra [15], the work of Dinur, Khot, Kindler,\nMinzer, and Safra [4], and the contribution of Barak, Kothari, and Steurer [3] culminated in the\nexpansion theorem of Khot, Minzer, and Safra [14]. This completed the Games Theorem with\nimperfect completeness and yielded unconditional Vertex Cover hardness for every fixed factor below\n√" }, "statement": [ { "blockId": "p2-706b6597ff", "page": 2, "quote": "5 −21, approximately 1.36068 [5,\nTheorem 1.1]. Their analysis uses biased set families and Boolean-function structure to control\nindependent-set density. Khot’s Unique Games Conjecture [13] proposed a stronger source of\napproximation gaps. Assuming this conjecture, Khot and Regev obtained hardness for every fixed\nfactor below two [16]. Their proof uses a biased long code and an equivalent, stronger formulation of\nthe source promise that supports short lists of candidate labels. Friedgut’s approximation theorem\nfor Boolean functions of low average sensitivity [9] supplies bounded sets of relevant coordinates in\nthat analysis.\nThe 2017–2018 developments in the 2-to-2 Games program gave an unconditional advance. The\nGrassmann-graph reduction of Khot, Minzer, and Safra [15], the work of Dinur, Khot, Kindler,\nMinzer, and Safra [4], and the contribution of Barak, Kothari, and Steurer [3] culminated in the\nexpansion theorem of Khot, Minzer, and Safra [14]. This completed the Games Theorem with\nimperfect completeness and yielded unconditional Vertex Cover hardness for every fixed factor below\n√" }, { "blockId": "p24-08770ca53d", "page": 24, "quote": "[5] Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals\nof Mathematics, 162(1):439–485, 2005. https://doi.org/10.4007/annals.2005.162.439." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-90", "kind": "external", "label": "[6] Theorem 6.2", "source": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "statement": [ { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, { "blockId": "p24-7f2debf64e", "page": 24, "quote": "[6] Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proceedings of\nthe Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14, pages 624–633.\nAssociation for Computing Machinery, 2014. https://doi.org/10.1145/2591796.2591884.\nFull version: https://arxiv.org/abs/1305.1979v3, Theorem 6.2." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-91", "kind": "external", "label": "[4] Definition 1.1", "source": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "statement": [ { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, { "blockId": "p24-48f73cd783", "page": 24, "quote": "[4] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the\n2-to-1 games conjecture?\nTheory of Computing, 21(11):1–50, 2025. https://doi.org/10.\n4086/toc.2025.v021a011. Preliminary version in STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-92", "kind": "external", "label": "[4] Theorem 1.3", "source": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "statement": [ { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, { "blockId": "p24-48f73cd783", "page": 24, "quote": "[4] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the\n2-to-1 games conjecture?\nTheory of Computing, 21(11):1–50, 2025. https://doi.org/10.\n4086/toc.2025.v021a011. Preliminary version in STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "curated-93", "kind": "definition", "label": "Vertex cover and τ(G)", "source": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "statement": [ { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." } ], "proof": [], "guide": [] }, { "id": "curated-94", "kind": "definition", "label": "Source value val(Φ)", "source": { "blockId": "p4-c1403a8a36", "page": 4, "quote": "The value val(Φ) is the maximum fraction of satisfied occurrences. List entries count with their\nmultiplicity. Endpoints and map tables are given explicitly in binary; unused variable declarations\ncan be removed. We require no regularity or expansion condition on the constraint graph." }, "statement": [ { "blockId": "p4-c1403a8a36", "page": 4, "quote": "The value val(Φ) is the maximum fraction of satisfied occurrences. List entries count with their\nmultiplicity. Endpoints and map tables are given explicitly in binary; unused variable declarations\ncan be removed. We require no regularity or expansion condition on the constraint graph." } ], "proof": [], "guide": [] } ], "edges": [ { "id": "edge-1", "from": "equation-36", "to": "lemma-4", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p6-5b94c9fe95", "page": 6, "quote": "Lemma 3.1 (Compatibility norm). The family I is hereditary and contains every coordinate\nsingleton. Formula (3.1) defines a norm. For each fixed query i, the increment is linear in s and\nobeys\n∥z(i, s)∥∗≤∥s∥∞.\n(3.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-2", "from": "equation-41", "to": "lemma-5", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p7-e0da83d6df", "page": 7, "quote": ".\n(3.7)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-3", "from": "equation-42", "to": "lemma-7", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p8-9b6f9115d5", "page": 8, "quote": "m.\n(3.8)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-4", "from": "equation-43", "to": "proposition-8", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p9-18496e9242", "page": 9, "quote": "j=1\nsj,slotij(c)(λA,ij(c)) > t.\n(3.9)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-5", "from": "equation-44", "to": "proposition-8", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p9-59ad20044b", "page": 9, "quote": "4\n=\n1\n27m3 .\n(4.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-6", "from": "equation-45", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-9d68e8efbd", "page": 10, "quote": "2\n\ndist∗(z, −A∗) −dist∗(z, A∗)\n\n(4.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-7", "from": "equation-46", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-254bfac670", "page": 10, "quote": ".\n(4.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-8", "from": "equation-47", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-6aa5afb122", "page": 10, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(4.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-9", "from": "equation-48", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p11-95ea1c8256", "page": 11, "quote": "a =\n\n(ce)e/∈(J\n2), (sk)k/∈J\n\n.\n(5.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-10", "from": "equation-49", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p12-e17a0fda8b", "page": 12, "quote": "Esf(c, s) = 0\nfor every c,\nEf2 ≥νd.\n(5.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-11", "from": "equation-50", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p12-ad0f187d8f", "page": 12, "quote": "2 ,\n(5.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-12", "from": "equation-51", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p12-eb730192d5", "page": 12, "quote": "4 .\n(5.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-13", "from": "equation-52", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p13-53aca89949", "page": 13, "quote": "j=1\n∥f −W 0\nj f∥2\n2 ≤4d.\n(5.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-14", "from": "equation-53", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p13-3cfa9c14d3", "page": 13, "quote": "S,ω\n∥fS,ω∥2\n2 = ∥f∥2\n2,\n(5.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-15", "from": "equation-54", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p13-1830aafe32", "page": 13, "quote": "\n≤8d2.\n(5.7)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-16", "from": "equation-55", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p14-8ad8936f6c", "page": 14, "quote": "seed incident to k. Given these data, the incident seeds have independent conditional laws on\ntheir respective fibers; their product gives exactly the displayed operator. Figure 1 illustrates the\ninformation retained by this operation. Hypothesis (iii) bounds the oscillation of f on each such\nfiber by 2. It follows that\n∥f −Dkf∥2\n2 ≤4.\n(5.8)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-17", "from": "equation-56", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p14-19092f176d", "page": 14, "quote": "k=1\n∥(Id −Dk)f∥2\n2 ≤4d.\n(5.9)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-18", "from": "equation-57", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-c1e6896573", "page": 15, "quote": "k∈J\\{j}\nO{j,k},j.\n(5.10)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-19", "from": "equation-58", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-ce5cb4cd85", "page": 15, "quote": "j∈J\n∥KJ,jf∥2\n2.\n(5.11)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-20", "from": "equation-59", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-5e4719ae21", "page": 15, "quote": "\n= ∅.\n(5.12)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-21", "from": "equation-60", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-a71b6ad9bd", "page": 15, "quote": "S,ω\n1{conditions (5.12) hold}∥fS,ω∥2\n2.\n(5.13)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-22", "from": "equation-61", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p16-b015ba1dc9", "page": 16, "quote": "The first two conditions in (5.12) fail only if one of S \\ {j(S)} or NS,ω is included. The third can fail\nonly at a pair in TS,ω avoiding j(S). A union bound therefore gives conditional survival probability\nat least\n1 −p1\n\n|S| −1 + |NS,ω|\n\n−p2|TS,ω|.\n(5.14)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-23", "from": "equation-62", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p16-5b95b452d5", "page": 16, "quote": "2 .\n(5.15)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-24", "from": "equation-67", "to": "lemma-11", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-25", "from": "equation-68", "to": "lemma-11", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p17-979e3f5786", "page": 17, "quote": "Consequently,\n∥gj∥1 ≤1,\ngj,t′ = 0\nif t′ /∈slotij(Λij).\n(6.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-26", "from": "equation-69", "to": "lemma-11", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-27", "from": "equation-72", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p18-eae830cc0a", "page": 18, "quote": "∂s(t′)Hj(u, s) = ¯gj,t′(u, s),\nt′ ∈[r].\n(6.10)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-28", "from": "equation-73", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p18-87f7eefec4", "page": 18, "quote": "Vars Hj(u, s) ≤2Es∥¯gj(u, s)∥2\n2.\n(6.11)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-29", "from": "equation-74", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-30", "from": "equation-77", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p20-0aeab76362", "page": 20, "quote": "2 = 2.\n(6.15)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-31", "from": "equation-78", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p20-221866509a", "page": 20, "quote": "2 .\n(6.16)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-32", "from": "equation-79", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-33", "from": "equation-83", "to": "lemma-14", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p21-e1ca65d7b0", "page": 21, "quote": "2.\n(7.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-34", "from": "equation-84", "to": "lemma-14", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p22-943351b541", "page": 22, "quote": "\nπc(Aρ,τ(xc)) = Aρ,τ(yc)\n\n≤σ.\n(7.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-35", "from": "equation-85", "to": "lemma-14", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p22-e9d6cf0750", "page": 22, "quote": "{j,k}∈(J\n2)\n1Ejk.\n(7.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-36", "from": "equation-87", "to": "theorem-1", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-37", "from": "equation-87", "to": "corollary-2", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-38", "from": "external-17", "to": "theorem-3", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-39", "from": "external-16", "to": "theorem-3", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-40", "from": "external-90", "to": "theorem-3", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-41", "from": "external-91", "to": "theorem-3", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-42", "from": "external-92", "to": "theorem-3", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-43", "from": "equation-33", "to": "proposition-6", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p7-44df83b758", "page": 7, "quote": "Proposition 3.3 (Deterministic polynomial construction). For a fixed integer m ≥4, a fixed even\nsquare d satisfying Equation (2.1), and fixed source alphabets, the graph G can be output explicitly\nin deterministic time polynomial in the source bit length. Every edge test is exact." }, "note": "" }, { "id": "edge-44", "from": "lemma-5", "to": "proposition-6", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, "note": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch." }, { "id": "edge-45", "from": "equation-34", "to": "proposition-6", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, "note": "" }, { "id": "edge-46", "from": "equation-33", "to": "lemma-7", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p9-1a87c54021", "page": 9, "quote": "d > 4m = 1/a by Equation (2.1), and hence" }, "note": "" }, { "id": "edge-47", "from": "lemma-7", "to": "proposition-8", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p9-c90bc9285c", "page": 9, "quote": "For a uniform vertex, conditional on the seed tuple, the selected slot in each weight vector is fixed.\nIts entries are independent uniform elements of T , one from each independent sj. Coincident\nendpoint questions or slot numbers do not change this independence. By Theorem 3.4, more than a\n(1/2 −1/m) fraction of vertices are kept.\nFor two distinct kept vertices v, w, evaluating Sv + Sw on IA gives exactly the sum of their two\nexpressions in (3.9). This includes all contributions even when the same query appears in both\nvertices. The sum exceeds 2t, so (3.1) rules out the edge in (3.5). The kept vertices are therefore\nindependent. Their complement is a vertex cover of size less than (1/2 + 1/m)n." }, "note": "Text says Theorem 3.4; candidate heading is Lemma 3.4. Check the mismatch." }, { "id": "edge-48", "from": "equation-39", "to": "lemma-9", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p10-3eb0619124", "page": 10, "quote": "Proof. Take A∗to be the set of sum vectors of the stipulated high-norm vertices in the independent\nset. The number of those vertices gives the probability bound even if several vertices represent the\nsame vector. For distinct x, y ∈A∗, their representing vertices are distinct and nonadjacent, so\nEquation (3.5) gives ∥x + y∥∗> 2t. If x = y, the same inequality follows from ∥2x∥∗= 2 ∥x∥∗> 2t.\nDistance to a nonempty finite set is the minimum of the distances to its points. The triangle\ninequality makes each distance function 1-Lipschitz; the half-difference in (4.2) is therefore 1-\nLipschitz. Symmetry of the norm gives dist∗(−z, −A∗) = dist∗(z, A∗), proving oddness. For z ∈A∗,\nits distance to A∗is zero, while" }, "note": "" }, { "id": "edge-49", "from": "external-22", "to": "lemma-10", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p12-a337498528", "page": 12, "quote": "Proof. The conditional mean Hj averages all unfrozen weights except sj. It also averages every\nhidden seed on a pair avoiding j, and retains only j’s question on each hidden pair incident to\nj. These are the three ways in which averaging can lose information. We use the product-space\northogonal decomposition associated with the Efron–Stein method [7, Section 2], giving the needed\nprojection identities explicitly. We assign each component to one weight block on which it depends,\nand ask when a random batch preserves that component in its assigned conditional mean. The\noscillation bounds control the energy lost through each type of averaging." }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-50", "from": "lemma-11", "to": "lemma-12", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p19-a8b14455ed", "page": 19, "quote": "Proof. Replacing the own block s by s′, with a seed configuration and other weights fixed, changes\nthe integrand in (6.4) by at most ∥s −s′∥∞. The same bound holds after averaging. In particular,\nevery scalar-coordinate difference quotient of the integrand is bounded in absolute value by one.\nFor each fixed hidden seed tuple, the full gradient identification in Theorem 6.1 holds outside\na null set in all batch weights. Fubini implies that for almost every own s it holds for almost all\nother weight blocks. The conditional seed law κj(u) has finite support and does not depend on s, so\nthe exceptional sets may be united over its support and over the finitely many weight coordinates.\nAt the remaining own values, the coordinate difference quotients converge almost surely to gj,t′.\nDominated convergence gives (6.10); the boundary of the cube has measure zero.\nEvery coordinate-line restriction of Hj(u, ·) is Lipschitz and hence absolutely continuous. By\nFubini applied to (6.10), for almost every fixing of the other own coordinates its derivative agrees\nalmost everywhere with the corresponding entry of ¯gj. The fundamental theorem of calculus\ntherefore expresses differences between any two values on such a line as the integral of that entry.\nLet w0 be the restriction of Hj(u, ·) to one of these lines. For x, y ∈[−1, 1], Cauchy–Schwarz\ngives" }, "note": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch." }, { "id": "edge-51", "from": "external-22", "to": "lemma-12", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p19-4579d69f50", "page": 19, "quote": "The product decomposition of Efron and Stein [7, Section 2] bounds variance by the sum of the\naveraged one-coordinate conditional variances. To see this consequence directly, decompose the\nidentity at each scalar coordinate into its averaging projection and its orthogonal complement. The\nresulting orthogonal components are indexed by subsets of coordinates. Total variance counts every\nnonconstant squared component once, while the sum of one-coordinate conditional variances counts\nit once for each member of its nonempty index subset. Applying the line estimate, integrating over\nthe other coordinates, and summing proves (6.11). Averaging that bound over the own tuples and\nsumming over j, then using (6.3), proves (6.12)." }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-52", "from": "equation-81", "to": "lemma-14", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p21-70de857a8d", "page": 21, "quote": "Lemma 7.1 (Decoding a pair). For every fixed J, a and fixed list maps satisfying Equation (7.2)," }, "note": "" }, { "id": "edge-53", "from": "proposition-13", "to": "lemma-14", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." }, "note": "Text says Theorem 6.3; candidate heading is Proposition 6.3. Check the mismatch." }, { "id": "edge-54", "from": "equation-84", "to": "lemma-14", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p22-b0f00cc8f4", "page": 22, "quote": "This uses the sampled occurrence even when other occurrences have the same endpoints. Union\nbounding Equation (7.5) over the ℓ2 rank pairs proves the pair estimate for the fixed weights and\nother seeds. It is uniform in those fixed values, so averaging proves Pr(Ejk) ≤ℓ2σ. No conditioning\non compatible labels or successful ranks is used.\nFor every integer z ≥0, z ≤1 +\nz\n2\n\n, since the difference is (z −1)(z −2)/2 ≥0. Choose a\nselection attaining the maximum in Equation (7.3). Each of the pairs among its Z hit positions\nwitnesses the corresponding event Ejk. Consequently, pointwise," }, "note": "" }, { "id": "edge-55", "from": "equation-82", "to": "lemma-14", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p22-b0f00cc8f4", "page": 22, "quote": "This uses the sampled occurrence even when other occurrences have the same endpoints. Union\nbounding Equation (7.5) over the ℓ2 rank pairs proves the pair estimate for the fixed weights and\nother seeds. It is uniform in those fixed values, so averaging proves Pr(Ejk) ≤ℓ2σ. No conditioning\non compatible labels or successful ranks is used.\nFor every integer z ≥0, z ≤1 +\nz\n2\n\n, since the difference is (z −1)(z −2)/2 ≥0. Choose a\nselection attaining the maximum in Equation (7.3). Each of the pairs among its Z hit positions\nwitnesses the corresponding event Ejk. Consequently, pointwise," }, "note": "" }, { "id": "edge-56", "from": "equation-83", "to": "lemma-14", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p22-2f8a26b267", "page": 22, "quote": "Taking expectations, using the pair estimate, and substituting σ = 1/(h2ℓ2) gives Equation (7.4)." }, "note": "" }, { "id": "edge-57", "from": "lemma-9", "to": "corollary-15", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "note": "Text says Theorem 4.1; candidate heading is Lemma 4.1. Check the mismatch." }, { "id": "edge-58", "from": "lemma-10", "to": "corollary-15", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "note": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch." }, { "id": "edge-59", "from": "proposition-13", "to": "corollary-15", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "note": "Text says Theorem 6.3; candidate heading is Proposition 6.3. Check the mismatch." }, { "id": "edge-60", "from": "equation-77", "to": "corollary-15", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "note": "" }, { "id": "edge-61", "from": "lemma-14", "to": "corollary-15", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p22-a7124c215a", "page": 22, "quote": "This contradicts Theorem 7.1, which gives EZ < 3/2 under the same remaining product law.\nThe lower bound just used averages over the hidden seeds and continuous batch weights for\nthose fixed J, a. The upper bound fixes further weights and seeds only to decode a single occurrence" }, "note": "Text says Theorem 7.1; candidate heading is Lemma 7.1. Check the mismatch." }, { "id": "edge-62", "from": "theorem-3", "to": "theorem-1", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-63", "from": "theorem-3", "to": "corollary-2", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-64", "from": "proposition-6", "to": "theorem-1", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Text says Theorem 3.3; candidate heading is Proposition 3.3. Check the mismatch. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-65", "from": "proposition-6", "to": "corollary-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Text says Theorem 3.3; candidate heading is Proposition 3.3. Check the mismatch. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-66", "from": "proposition-8", "to": "theorem-1", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Text says Theorem 3.5; candidate heading is Proposition 3.5. Check the mismatch. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-67", "from": "proposition-8", "to": "corollary-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Text says Theorem 3.5; candidate heading is Proposition 3.5. Check the mismatch. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-68", "from": "corollary-15", "to": "theorem-1", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Text says Theorem 7.2; candidate heading is Corollary 7.2. Check the mismatch. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-69", "from": "corollary-15", "to": "corollary-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Text says Theorem 7.2; candidate heading is Corollary 7.2. Check the mismatch. Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-70", "from": "equation-86", "to": "theorem-1", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-71", "from": "equation-86", "to": "corollary-2", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Joint proof of Theorem 1.1, Corollary 1.2; the reference is not assigned independently to each conclusion." }, { "id": "edge-72", "from": "curated-93", "to": "theorem-1", "relation": "defines", "basis": "agent-inferred", "evidence": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "note": "Agent-proposed dependency from the cited passage; not proof verification." }, { "id": "edge-73", "from": "curated-94", "to": "theorem-3", "relation": "defines", "basis": "agent-inferred", "evidence": { "blockId": "p4-15866c4f93", "page": 4, "quote": "Theorem 2.1 (Perfect-completeness Label Cover hardness). For every fixed rational σ ∈(0, 1),\nthere are nonempty constant alphabets and a deterministic polynomial-time reduction from 3SAT to\ninstances of the form above such that satisfiable formulas give val(Φ) = 1, and unsatisfiable formulas\ngive val(Φ) ≤σ." }, "note": "Agent-proposed dependency from the cited passage; not proof verification." }, { "id": "edge-74", "from": "proposition-6", "to": "theorem-1", "relation": "supports", "basis": "agent-inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Agent-proposed dependency from the cited passage; not proof verification." }, { "id": "edge-75", "from": "proposition-8", "to": "theorem-1", "relation": "supports", "basis": "agent-inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Agent-proposed dependency from the cited passage; not proof verification." }, { "id": "edge-76", "from": "corollary-15", "to": "theorem-1", "relation": "supports", "basis": "agent-inferred", "evidence": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "note": "Agent-proposed dependency from the cited passage; not proof verification." } ], "notation": [ { "symbol": "m", "meaning": "Fixed integer controlling the cover gap.", "status": "source-located", "scope": "m ≥ 4 in Theorem 1.1 and the final parameter choice.", "source": { "blockId": "p1-9c682e0dc0", "page": 1, "quote": "Theorem 1.1 (Explicit cover gap). For every fixed integer m ≥4, there is a deterministic polynomial-\ntime reduction from 3SAT to finite simple undirected unweighted graphs with the following guarantees:" }, "occurrences": [ { "blockId": "p1-9c682e0dc0", "page": 1, "quote": "Theorem 1.1 (Explicit cover gap). For every fixed integer m ≥4, there is a deterministic polynomial-\ntime reduction from 3SAT to finite simple undirected unweighted graphs with the following guarantees:" } ], "occurrenceCount": 1 }, { "symbol": "σ", "meaning": "Soundness threshold in the source hardness promise.", "status": "source-located", "scope": "Fixed rational number in (0,1), Theorem 2.1.", "source": { "blockId": "p4-15866c4f93", "page": 4, "quote": "Theorem 2.1 (Perfect-completeness Label Cover hardness). For every fixed rational σ ∈(0, 1),\nthere are nonempty constant alphabets and a deterministic polynomial-time reduction from 3SAT to\ninstances of the form above such that satisfiable formulas give val(Φ) = 1, and unsatisfiable formulas\ngive val(Φ) ≤σ." }, "occurrences": [ { "blockId": "p4-15866c4f93", "page": 4, "quote": "Theorem 2.1 (Perfect-completeness Label Cover hardness). For every fixed rational σ ∈(0, 1),\nthere are nonempty constant alphabets and a deterministic polynomial-time reduction from 3SAT to\ninstances of the form above such that satisfiable formulas give val(Φ) = 1, and unsatisfiable formulas\ngive val(Φ) ≤σ." } ], "occurrenceCount": 1 }, { "symbol": "val(Φ)", "meaning": "Maximum fraction of satisfied constraint occurrences.", "status": "source-located", "scope": "The source instance; multiplicities count.", "source": { "blockId": "p4-c1403a8a36", "page": 4, "quote": "The value val(Φ) is the maximum fraction of satisfied occurrences. List entries count with their\nmultiplicity. Endpoints and map tables are given explicitly in binary; unused variable declarations\ncan be removed. We require no regularity or expansion condition on the constraint graph." }, "occurrences": [ { "blockId": "p4-c1403a8a36", "page": 4, "quote": "The value val(Φ) is the maximum fraction of satisfied occurrences. List entries count with their\nmultiplicity. Endpoints and map tables are given explicitly in binary; unused variable declarations\ncan be removed. We require no regularity or expansion condition on the constraint graph." } ], "occurrenceCount": 1 }, { "symbol": "n", "meaning": "Number of vertices |V(G)|.", "status": "source-located", "scope": "The constructed graph in the main approximation argument.", "source": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "occurrences": [ { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." } ], "occurrenceCount": 1 }, { "symbol": "τ(G)", "meaning": "Minimum number of vertices in a vertex cover of G.", "status": "source-located", "scope": "Finite simple undirected graph G, introduced in Section 1.", "source": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "occurrences": [ { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." } ], "occurrenceCount": 1 }, { "symbol": "A", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "occurrences": [ { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, { "blockId": "p2-86f65bca52", "page": 2, "quote": "2 and arbitrarily\nsmall soundness density. The gap in Theorem 1.1 instead has completeness density approaching\n1/2, which is what gives the factor-two threshold.\nOur argument retains the objective of extracting short local label lists from a large independent set,\nbut obtains the lists from Lipschitz derivatives. The central step preserves variance while restricting\nthe information available to each list. We adapt the product-space orthogonal decomposition of the\nEfron–Stein method [7, Section 2], combining it with a resampling estimate that controls the loss\ncaused by this information restriction. A variance inequality on a product of intervals then supplies\ngradient energy, from which short candidate-label lists can be extracted without an alphabet-size\nfactor. These ingredients are proved in full in Sections 5 and 6; the final decoding uses the ordinary\nLabel Cover promise above." }, { "blockId": "p2-6cf80fbffe", "page": 2, "quote": "A Label Cover instance is a bipartite constraint system: every edge asks that a specified map send\nthe label on its left endpoint to the label on its right endpoint. Its value is the largest fraction of\nconstraints satisfied by one assignment of labels to all variables. Our source promise separates value\none from arbitrarily small fixed value. The alphabets may grow as that fixed value decreases; our\nanalytic estimates will not depend on their sizes. The exact source theorem is stated in Theorem 2.1.\nImagine d positions, with an independent source constraint assigned to each unordered pair. At\na pair j < k, position j receives the constraint’s left endpoint and position k its right endpoint.\nA position’s query is its type together with these d −1 endpoint questions. It does not reveal\nconstraint indices or opposite endpoints. A local label assigns symbols to the variables in one query\nand satisfies every source constraint internal to that scope. A compatible family chooses at most\none label per query; its members agree on overlaps and satisfy every constraint internal to their\ncombined scopes." }, { "blockId": "p3-887993e02b", "page": 3, "quote": "Give one vector coordinate to each query–label pair. A compatible family acts on a vector by\nsumming the coordinates it selects; the largest absolute value of these sums defines a norm, ∥·∥∗. At\neach position, a vector of grid weights is supported on that position’s query block. A graph vertex\nspecifies all pair constraints and all position weights. Let Sv be the sum of its position vectors. For\na positive threshold t, distinct vertices v, w are adjacent precisely when ∥Sv + Sw∥∗≤2t.\nA satisfying source assignment restricts to compatible labels for every query. Its linear functional\nexceeds t on an independent set: on the sum of two such vertices it exceeds 2t. A tail estimate for a\nsum of symmetric weights shows that this set occupies nearly half the graph. Section 3 gives the\nconstruction, its deterministic implementation, and this completeness argument." }, { "blockId": "p4-25c6142b08", "page": 4, "quote": "Write [k] = {1, . . . , k}. A Label Cover instance Φ consists of two disjoint finite variable sets U, V ,\ntwo nonempty finite alphabets ΣU, ΣV , and an explicit nonempty list of M constraint occurrences.\nOccurrence c ∈[M] has endpoints xc ∈U, yc ∈V , and a total map πc : ΣU →ΣV . A labeling A\nassigns each variable a symbol from its side’s alphabet. It satisfies occurrence c when" }, { "blockId": "p4-7d2b5e1281", "page": 4, "quote": "πc(A(xc)) = A(yc)." }, { "blockId": "p4-1554d76840", "page": 4, "quote": "This is the standard consequence of the probabilistically checkable proof (PCP) theorem [2, 1]\nand parallel repetition [17]. A quantitative formulation appears in the full version of Dinur and\nSteurer [6, Theorem 6.2]. The precise unweighted, total-projection formulation is recorded in [4,\nDefinitions 1.1–1.2 and Theorem 1.3]. There the constraints are even D-to-one for a suitable constant\nD, a property we do not use. The constants, including the alphabets and D, may depend on σ.\nThis theorem is the only external hardness input to our proof." }, { "blockId": "p5-86047a999b", "page": 5, "quote": "A query will involve at most d −1 variables, and r bounds its number of local labels. For fixed m, d\nand alphabets, every parameter introduced here is constant with respect to the source input size." }, { "blockId": "p5-65da3010f3", "page": 5, "quote": "Fix a source instance and the construction parameters from Section 2. Each vertex will specify\nsource constraints on the pairs of d positions and weights on the labels available at each position. We\nfirst define when local labels fit together and use their coordinate sums to define a norm. A satisfying\nsource labeling will then provide a linear functional whose values above t select an independent set." }, { "blockId": "p5-c91061e26c", "page": 5, "quote": "Let E =\n[d]\n2\n\nbe the set of unordered pairs of distinct positions. A query of type j ∈[d] has the form" }, { "blockId": "p5-84ffa95f13", "page": 5, "quote": "The tuple is ordered by increasing k. Let Q be the set of all such queries of all types, and let\nP(i) ⊆U ∪V be the set of distinct variables appearing in i. The side of each variable is part of its\nidentity, since U and V are disjoint. Repeated entries in the tuple contribute only one variable to\nP(i); thus |P(i)| ≤d −1.\nA label λ for i assigns a symbol from the appropriate alphabet to each variable of P(i) and\nsatisfies every source constraint occurrence whose two endpoints belong to P(i). Write Λi for the\nset of these labels. This set may be empty, for example when constraints internal to the scope are\ninconsistent. There are at most q|P(i)| ≤r labels. Fix orders on variables and alphabet symbols,\nand list the valid assignments in lexicographic order to define an injection" }, { "blockId": "p5-919440e5bc", "page": 5, "quote": "A compatible selection is a subset I ⊆P with at most one label per query, such that the selected\nlabels agree on overlaps and their induced labeling satisfies every source constraint whose two\nendpoints lie in\nP(I) =\n[" }, { "blockId": "p6-9352d79c2f", "page": 6, "quote": "A seed tuple is an indexed family c = (ce)e∈E ∈[M]E. For e = {j, k} with j < k, seed ce gives\nendpoint xce to position j and endpoint yce to position k. It thereby specifies queries ij(c) through" }, { "blockId": "p6-934a43ac0f", "page": 6, "quote": "A query retains its type and its ordered endpoint questions. It does not retain the sampled occurrence\nindices, the sampled projection maps, or the opposite endpoint in any pair. Different seed tuples\ncan give the same queries. Queries of different position types are nevertheless distinct, even if\nsome endpoint values coincide. The source instance used to define labels and slots is fixed; the\nseed-dependent data in a query are only its endpoint questions.\nUse the symmetric grid" }, { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, { "blockId": "p8-0a79cefe7b", "page": 8, "quote": "A satisfying source assignment will give a linear functional whose value exceeds t on an independent\nset. To show that this set occupies nearly half the vertices, we first bound the corresponding tail of\na sum of independent uniform grid variables." }, { "blockId": "p8-727b08d172", "page": 8, "quote": "Put A = t + d/p = a\n√" }, { "blockId": "p8-600e9628f5", "page": 8, "quote": "d + d/p. We bound the central probability Pr[| P\nj Cj| ≤A].\nEach Cj may be generated as (Lj + Uj)/2, where Lj is a fair sign and Uj is uniform on [−1, 1],\nall independent. The two sign choices give uniform distributions on the two half intervals of [−1, 1],\nso this is indeed the uniform distribution on the full interval. Conditional on (U1, . . . , Ud), the event\n| P\nj Cj| ≤A places P\nj Lj in an interval of length" }, { "blockId": "p8-118861676d", "page": 8, "quote": "4A = 4a\n√" }, { "blockId": "p9-2e74a88638", "page": 9, "quote": "≤A" } ], "occurrenceCount": 48 }, { "symbol": "E", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-c91061e26c", "page": 5, "quote": "Let E =\n[d]\n2\n\nbe the set of unordered pairs of distinct positions. A query of type j ∈[d] has the form" }, "occurrences": [ { "blockId": "p5-c91061e26c", "page": 5, "quote": "Let E =\n[d]\n2\n\nbe the set of unordered pairs of distinct positions. A query of type j ∈[d] has the form" }, { "blockId": "p6-9352d79c2f", "page": 6, "quote": "A seed tuple is an indexed family c = (ce)e∈E ∈[M]E. For e = {j, k} with j < k, seed ce gives\nendpoint xce to position j and endpoint yce to position k. It thereby specifies queries ij(c) through" }, { "blockId": "p6-efa5c4b66e", "page": 6, "quote": "These are the midpoints of the p equal consecutive subintervals of [−1, 1]. The vertex set consists of\nall pairs\nv = (c, s),\nc ∈[M]E,\ns = (s1, . . . , sd) ∈(T r)d." }, { "blockId": "p7-5fce302f5a", "page": 7, "quote": "{v, w} ∈E(G)\n⇐⇒\n∥Sv + Sw∥∗≤2t.\n(3.5)" }, { "blockId": "p10-a9f3ade5d7", "page": 10, "quote": "is odd and 1-Lipschitz for ∥·∥∗, and F(z) > t on A∗.\nLet E =\n[d]\n2\n\n. Draw independent uniform occurrence seeds c = (ce)e∈E ∈[M]E and independent\nweight blocks sj uniform on [−1, 1]r, independently of the seeds. Put" }, { "blockId": "p11-8c7eaa450a", "page": 11, "quote": "The function from the previous section has variance of order d, but the functions used to extract\nlabels must depend only on one position’s own questions and weights. We shall freeze all but\na batch of h positions and retain variance of order h in conditional averages with precisely this\ndependence. Questions at different positions are correlated because they share pair seeds; our\northogonal decomposition therefore uses the independent seeds and weights themselves.\nLet d ≥4 and r, M ≥1 be integers, and put E =\n[d]\n2\n\n. For each e ∈E, draw ce uniformly from\n[M], independently over e. At each endpoint j ∈e, this seed reveals a question Qe,j(ce), where\nQe,j is a fixed map to a finite set. There is no independence assumption on the two questions\nfrom the same seed. In the construction, for e = {j, k} with j < k, these maps are Qe,j(c) = xc\nand Qe,k(c) = yc. Independently of all seeds, draw sj uniformly from [−1, 1]r for each j ∈[d],\nindependently over j. All expectations and L2-norms in this section initially refer to the resulting\nproduct law on\nΩ= [M]E × ([−1, 1]r)d." }, { "blockId": "p13-319e53e23f", "page": 13, "quote": "where henceforth these sums range over nonempty S and its allowed choices ω.\nLet TS,ω ⊆E be the seeds where the chosen projection is not P 0\ne . Let NS,ω ⊆[d] \\ {j(S)} be\nthe positions k for which the projection at e = {j(S), k} is Id −Oe,j(S). These are the incident\ncomponents not retained by the designated position’s own question. For each fixed e, applying\nId −P 0\ne keeps exactly the components with e ∈TS,ω. Orthogonality and the one-seed oscillation\nbound therefore give" }, { "blockId": "p13-5f0771b88d", "page": 13, "quote": "e∈E\n∥f −P 0\ne f∥2\n2 ≤16\nd\n2" }, { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" }, { "blockId": "p19-9eb7cdcaa6", "page": 19, "quote": "2E|w0(X) −w0(Y )|2 ≤\nZ 1" }, { "blockId": "p19-cf750ab39d", "page": 19, "quote": "−1\n|w′\n0(z)|2 dz = 2E|w′\n0(X)|2." }, { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, { "blockId": "p20-c428dc9d9e", "page": 20, "quote": "t′∈Tj\n|¯gj,t′| ≤E" }, { "blockId": "p20-6be5de16df", "page": 20, "quote": "E\nX" } ], "occurrenceCount": 14 }, { "symbol": "Ejk", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p21-7b7e7e18ec", "page": 21, "quote": "Each map receives only the indicated own questions and weights; on an unsupported own-question\ntuple it returns the empty list. Order each list by its slots. The lists in Theorem 6.3 have exactly\nthese properties.\nFor distinct j, k ∈J, let Ejk be the event that some I ∈I hits both lists: it contains (ij, λj)\nand (ik, λk) for labels λj ∈Lj, λk ∈Lk. Define" }, "occurrences": [ { "blockId": "p21-7b7e7e18ec", "page": 21, "quote": "Each map receives only the indicated own questions and weights; on an unsupported own-question\ntuple it returns the empty list. Order each list by its slots. The lists in Theorem 6.3 have exactly\nthese properties.\nFor distinct j, k ∈J, let Ejk be the event that some I ∈I hits both lists: it contains (ij, λj)\nand (ik, λk) for labels λj ∈Lj, λk ∈Lk. Define" }, { "blockId": "p21-092429201f", "page": 21, "quote": "Pr(Ejk) ≤ℓ2σ\n(j, k ∈J, j < k),\nEZ ≤1 +\nh\n2" }, { "blockId": "p22-e0975c3db8", "page": 22, "quote": "For every actual remaining occurrence ce, the reconstructed tuples uxce\nj\n, uyce\nk\nare supported under\nthe original hidden-seed law: the full seed tuple comprising the fixed other seeds and this occurrence\nhas positive probability in that product law. They are exactly the own tuples at which the actual\nlists are evaluated.\nSuppose Ejk occurs and choose the ranks ρ, τ of labels witnessing it. The endpoint xce belongs\nto P(ij), and yce to P(ik). Compatibility requires the induced labeling to satisfy every source\nconstraint with both endpoints in the union of the selected scopes. In particular it satisfies the\nsampled occurrence ce. Its endpoint labels are exactly those used by Aρ,τ, so" }, { "blockId": "p22-b0f00cc8f4", "page": 22, "quote": "This uses the sampled occurrence even when other occurrences have the same endpoints. Union\nbounding Equation (7.5) over the ℓ2 rank pairs proves the pair estimate for the fixed weights and\nother seeds. It is uniform in those fixed values, so averaging proves Pr(Ejk) ≤ℓ2σ. No conditioning\non compatible labels or successful ranks is used.\nFor every integer z ≥0, z ≤1 +\nz\n2\n\n, since the difference is (z −1)(z −2)/2 ≥0. Choose a\nselection attaining the maximum in Equation (7.3). Each of the pairs among its Z hit positions\nwitnesses the corresponding event Ejk. Consequently, pointwise," }, { "blockId": "p22-e9d6cf0750", "page": 22, "quote": "{j,k}∈(J\n2)\n1Ejk.\n(7.6)" } ], "occurrenceCount": 5, "latex": "E_{jk}", "latexBasis": "pdf-script-position" }, { "symbol": "I", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-51373afcc5", "page": 5, "quote": "Let I be the family of compatible selections, including the empty selection. Agreement on overlaps\nalone is not enough: constraints joining variables from different selected scopes must also be\nsatisfied. Thus labels on any two selected queries satisfy every source constraint between their\nscopes. Compatibility imposes no condition on a constraint with an endpoint outside P(I); the\ninduced labeling need not extend to a satisfying labeling of the entire instance." }, "occurrences": [ { "blockId": "p5-919440e5bc", "page": 5, "quote": "A compatible selection is a subset I ⊆P with at most one label per query, such that the selected\nlabels agree on overlaps and their induced labeling satisfies every source constraint whose two\nendpoints lie in\nP(I) =\n[" }, { "blockId": "p5-aca0ba4dde", "page": 5, "quote": "(i,λ)∈I\nP(i)." }, { "blockId": "p5-51373afcc5", "page": 5, "quote": "Let I be the family of compatible selections, including the empty selection. Agreement on overlaps\nalone is not enough: constraints joining variables from different selected scopes must also be\nsatisfied. Thus labels on any two selected queries satisfy every source constraint between their\nscopes. Compatibility imposes no condition on a constraint with an endpoint outside P(I); the\ninduced labeling need not extend to a satisfying labeling of the entire instance." }, { "blockId": "p6-63e66a997c", "page": 6, "quote": "∥z∥∗= max\nI∈I" }, { "blockId": "p6-99c6986cec", "page": 6, "quote": "(i,λ)∈I\nzi,λ" }, { "blockId": "p6-5b94c9fe95", "page": 6, "quote": "Lemma 3.1 (Compatibility norm). The family I is hereditary and contains every coordinate\nsingleton. Formula (3.1) defines a norm. For each fixed query i, the increment is linear in s and\nobeys\n∥z(i, s)∥∗≤∥s∥∞.\n(3.2)" }, { "blockId": "p6-4581aaf54a", "page": 6, "quote": "Proof. Removing labels preserves agreement and can only shrink the scope union. Every constraint\ninternal to the smaller union was therefore satisfied before removal. This proves heredity. Every\nsingleton belongs to I by the definition of a valid query label. The maximum of absolute linear forms\nis absolutely homogeneous and satisfies the triangle inequality; the singleton forms ensure that it\nvanishes only at zero. If the coordinate set is empty, this is the unique norm on the zero-dimensional\nspace. Finally, a compatible selection contains at most one coordinate of the i-block, proving (3.2).\nLinearity follows from the coordinate formula." }, { "blockId": "p7-65cbc81ba7", "page": 7, "quote": "(i,λ)∈I\nzi,λ" }, { "blockId": "p7-182f70c256", "page": 7, "quote": "∥z∥∗=\nmax\nI∈I:\n(i,λ)∈I =⇒i∈B" }, { "blockId": "p9-3bcdab94d7", "page": 9, "quote": "belongs to I: the labels agree on overlaps, and every constraint internal to their scope union is\nsatisfied by A. Keep precisely the vertices for which" }, { "blockId": "p17-c91c50c010", "page": 17, "quote": "For I ∈I, let vI(i) ∈RJ×[r] have entry one at (j, slotij(λ)) if (ij, λ) ∈I, and zero elsewhere. It has\nat most one nonzero entry in each block j. We will represent the gradient using these incidence\nvectors. At exceptional weights where a derivative need not exist, the same representation will hold\nfor a prescribed value of the gradient rule." }, { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, { "blockId": "p17-ac67f82849", "page": 17, "quote": "Proof. Let 1I ∈RP be the indicator vector of I. For ζ ∈A∗∪(−A∗), I ∈I, and ε ∈{−1, 1}, form\nthe finite collection of affine expressions" }, { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, { "blockId": "p18-aa516f4123", "page": 18, "quote": "= min\nζ∈C\nmax\nI∈I, ε∈{−1,1} Lζ,I,ε(sJ)." }, { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, { "blockId": "p20-acacb43f6e", "page": 20, "quote": "For arbitrary fixed queries and arbitrary lists of valid labels, the sum of absolute values over\ntheir slot sets is a convex function of the full vector in RJ×[r]. On a signed generator ±vI(i) it equals\nthe number of positions hit by that selection I, since each block contains at most one selected slot.\nIts value on every point of the hull in (6.5) is consequently at most the maximum such hit count.\nAt every realized tuple, apply this deterministic statement to the realized lists and the full vector g:\nX" }, { "blockId": "p21-7b7e7e18ec", "page": 21, "quote": "Each map receives only the indicated own questions and weights; on an unsupported own-question\ntuple it returns the empty list. Order each list by its slots. The lists in Theorem 6.3 have exactly\nthese properties.\nFor distinct j, k ∈J, let Ejk be the event that some I ∈I hits both lists: it contains (ij, λj)\nand (ik, λk) for labels λj ∈Lj, λk ∈Lk. Define" }, { "blockId": "p21-9da6541b11", "page": 21, "quote": "Z = max\nI∈I #{j ∈J : some λ ∈Lj has (ij, λ) ∈I}.\n(7.3)" } ], "occurrenceCount": 19 }, { "symbol": "J", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, "occurrences": [ { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, { "blockId": "p11-229e7bb8ab", "page": 11, "quote": "We first specify the restrictions and conditional averages in the lemma. Given a bounded Borel\nmeasurable function f(c, s) and a set J ⊆[d], its frozen data are" }, { "blockId": "p11-95ea1c8256", "page": 11, "quote": "a =\n\n(ce)e/∈(J\n2), (sk)k/∈J\n\n.\n(5.1)" }, { "blockId": "p11-0fd1fee442", "page": 11, "quote": "Fixing these values leaves a function f∗of the hidden seeds (ce)e∈(J\n2) and the weights (sj)j∈J, with\ntheir original product law. For j ∈J, write" }, { "blockId": "p11-4bcab43585", "page": 11, "quote": "k∈J\\{j}," }, { "blockId": "p12-ea2c452c9c", "page": 12, "quote": "then there are a fixed h-element set J and fixed frozen data a such that" }, { "blockId": "p12-20efa93216", "page": 12, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh" }, { "blockId": "p15-56d3b89b17", "page": 15, "quote": "What the restriction keeps.\nFor fixed J and j ∈J, define the projection" }, { "blockId": "p15-c1e6896573", "page": 15, "quote": "k∈J\\{j}\nO{j,k},j.\n(5.10)" }, { "blockId": "p15-0dcee1683c", "page": 15, "quote": "k∈J\\{j}\nW 0\nk\nY" }, { "blockId": "p15-f55def92af", "page": 15, "quote": "e∈(J\\{j}\n2 )" }, { "blockId": "p15-6015a4e176", "page": 15, "quote": "V(J, a) =\nX" }, { "blockId": "p15-2c1d909a80", "page": 15, "quote": "j∈J\nEuj Varsj HJ,a\nj\n(uj, sj)," }, { "blockId": "p15-f24b2adea2", "page": 15, "quote": "EaV(J, a) =\nX" }, { "blockId": "p15-ce5cb4cd85", "page": 15, "quote": "j∈J\n∥KJ,jf∥2\n2.\n(5.11)" }, { "blockId": "p15-aba5150e1a", "page": 15, "quote": "S ∩J = {j},\nNS,ω ∩J = ∅,\nTS,ω ∩\nJ \\ {j}\n2" }, { "blockId": "p15-57d41c550d", "page": 15, "quote": "To verify that its squared norm can be counted in (5.11), first note that KJ,j commutes with every\nΠS. Thus its outputs from different weight supports are orthogonal, and ∥KJ,jf∥2\n2 = P\nS ∥KJ,jfS∥2\n2.\nThe weight factors in KJ,j keep fS precisely when S∩J = {j}. For those S whose designated position\nis this j, the seed factors are diagonal in the chosen seed decomposition: P 0\ne on a nonincident hidden\nseed keeps just its constant component, and Oe,j on an incident hidden seed keeps its constant and\nown-question components. The other seed coordinates are untouched. They therefore keep exactly\nthe components satisfying the last two conditions of (5.12), with no cancellation among them. We\nmay discard all other nonnegative squared norms in (5.11) and count each component solely in its\ndesignated term. It follows that" }, { "blockId": "p15-9d1daa6e4a", "page": 15, "quote": "EaV(J, a) ≥\nX" }, { "blockId": "p15-986481f9b9", "page": 15, "quote": "Averaging the restriction.\nChoose J uniformly among all h-element subsets of [d]. For a fixed\ncomponent, the designated position belongs to J with probability h/d. Conditional on this inclusion,\na specified other position belongs to J with probability" }, { "blockId": "p15-6bd54e81b0", "page": 15, "quote": "and a specified pair avoiding the designated position lies in J with probability" } ], "occurrenceCount": 47 }, { "symbol": "L", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, "occurrences": [ { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, { "blockId": "p8-efeb5ddc74", "page": 8, "quote": "O\n\n(1 + n2) poly(L + log n)\n" }, { "blockId": "p11-8c7eaa450a", "page": 11, "quote": "The function from the previous section has variance of order d, but the functions used to extract\nlabels must depend only on one position’s own questions and weights. We shall freeze all but\na batch of h positions and retain variance of order h in conditional averages with precisely this\ndependence. Questions at different positions are correlated because they share pair seeds; our\northogonal decomposition therefore uses the independent seeds and weights themselves.\nLet d ≥4 and r, M ≥1 be integers, and put E =\n[d]\n2\n\n. For each e ∈E, draw ce uniformly from\n[M], independently over e. At each endpoint j ∈e, this seed reveals a question Qe,j(ce), where\nQe,j is a fixed map to a finite set. There is no independence assumption on the two questions\nfrom the same seed. In the construction, for e = {j, k} with j < k, these maps are Qe,j(c) = xc\nand Qe,k(c) = yc. Independently of all seeds, draw sj uniformly from [−1, 1]r for each j ∈[d],\nindependently over j. All expectations and L2-norms in this section initially refer to the resulting\nproduct law on\nΩ= [M]E × ([−1, 1]r)d." }, { "blockId": "p12-22f8b88cff", "page": 12, "quote": "Averaging projections and weight supports.\nLet W 0\nj average the weight block sj, retaining\nall other coordinates, and set W 1\nj = Id −W 0\nj . Let P 0\ne average the seed ce. For j ∈e, let Oe,j average\nthat seed conditional on its question Qe,j(ce), retaining all other coordinates. Each averaging is an\northogonal projection in L2(Ω): averaging twice gives the same function, and integrating over the\nretained data shows self-adjointness. Also" }, { "blockId": "p14-6f221287e3", "page": 14, "quote": "For a component designated at j, we charge invisible information on {j, k} to the star centered\nat k: that star retains j’s question on this pair. For every j ̸= k, the range of Dk is contained in\nthat of O{k,j},j. Distance to the larger closed subspace is no greater than distance to the smaller\none. Thus, for every v ∈L2," }, { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, { "blockId": "p18-aa516f4123", "page": 18, "quote": "= min\nζ∈C\nmax\nI∈I, ε∈{−1,1} Lζ,I,ε(sJ)." } ], "occurrenceCount": 7 }, { "symbol": "Lj", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, "occurrences": [ { "blockId": "p8-600e9628f5", "page": 8, "quote": "d + d/p. We bound the central probability Pr[| P\nj Cj| ≤A].\nEach Cj may be generated as (Lj + Uj)/2, where Lj is a fair sign and Uj is uniform on [−1, 1],\nall independent. The two sign choices give uniform distributions on the two half intervals of [−1, 1],\nso this is indeed the uniform distribution on the full interval. Conditional on (U1, . . . , Ud), the event\n| P\nj Cj| ≤A places P\nj Lj in an interval of length" }, { "blockId": "p19-e116ff9258", "page": 19, "quote": "Lj(u, s) =\n\nλ ∈Λij(u) :\n¯gj,slotij(u)(λ)(u, s)\n ≥β" }, { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," }, { "blockId": "p20-3b2b95e21c", "page": 20, "quote": "2 |Lj(u, s)| ≤∥¯gj(u, s)∥1 ≤1,\n|Lj(u, s)| ≤2/β = ℓ." }, { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, { "blockId": "p20-068f4fb2a5", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≥βh −β" }, { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" }, { "blockId": "p21-a4f76b578f", "page": 21, "quote": "Lj = Lj(uj, sj) ⊆Λij,\n|Lj| ≤ℓ.\n(7.2)" }, { "blockId": "p21-7b7e7e18ec", "page": 21, "quote": "Each map receives only the indicated own questions and weights; on an unsupported own-question\ntuple it returns the empty list. Order each list by its slots. The lists in Theorem 6.3 have exactly\nthese properties.\nFor distinct j, k ∈J, let Ejk be the event that some I ∈I hits both lists: it contains (ij, λj)\nand (ik, λk) for labels λj ∈Lj, λk ∈Lk. Define" }, { "blockId": "p21-9da6541b11", "page": 21, "quote": "Z = max\nI∈I #{j ∈J : some λ ∈Lj has (ij, λ) ∈I}.\n(7.3)" }, { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." } ], "occurrenceCount": 12, "latex": "L_{j}", "latexBasis": "pdf-script-position" }, { "symbol": "Q", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-84ffa95f13", "page": 5, "quote": "The tuple is ordered by increasing k. Let Q be the set of all such queries of all types, and let\nP(i) ⊆U ∪V be the set of distinct variables appearing in i. The side of each variable is part of its\nidentity, since U and V are disjoint. Repeated entries in the tuple contribute only one variable to\nP(i); thus |P(i)| ≤d −1.\nA label λ for i assigns a symbol from the appropriate alphabet to each variable of P(i) and\nsatisfies every source constraint occurrence whose two endpoints belong to P(i). Write Λi for the\nset of these labels. This set may be empty, for example when constraints internal to the scope are\ninconsistent. There are at most q|P(i)| ≤r labels. Fix orders on variables and alphabet symbols,\nand list the valid assignments in lexicographic order to define an injection" }, "occurrences": [ { "blockId": "p5-84ffa95f13", "page": 5, "quote": "The tuple is ordered by increasing k. Let Q be the set of all such queries of all types, and let\nP(i) ⊆U ∪V be the set of distinct variables appearing in i. The side of each variable is part of its\nidentity, since U and V are disjoint. Repeated entries in the tuple contribute only one variable to\nP(i); thus |P(i)| ≤d −1.\nA label λ for i assigns a symbol from the appropriate alphabet to each variable of P(i) and\nsatisfies every source constraint occurrence whose two endpoints belong to P(i). Write Λi for the\nset of these labels. This set may be empty, for example when constraints internal to the scope are\ninconsistent. There are at most q|P(i)| ≤r labels. Fix orders on variables and alphabet symbols,\nand list the valid assignments in lexicographic order to define an injection" }, { "blockId": "p5-c0dae8c4e1", "page": 5, "quote": "P = {(i, λ) : i ∈Q, λ ∈Λi}." }, { "blockId": "p7-bd0f6459db", "page": 7, "quote": "Lemma 3.2 (Local evaluation of the norm). Let B ⊆Q be a set of query blocks supporting z ∈RP.\nThen" }, { "blockId": "p9-1a9392c0bb", "page": 9, "quote": "IA = {(i, λA,i) : i ∈Q}" }, { "blockId": "p11-580f678b17", "page": 11, "quote": "uj =\n\nQ{j,k},j(c{j,k})\n" }, { "blockId": "p11-a7c5fe9a95", "page": 11, "quote": "with entries in increasing order of k. Define HJ,a\nj\n(u, s) by inserting sj = s in f∗and averaging the\nremaining variables conditional on uj = u. Explicitly, each incident hidden seed c{j,k} is drawn\nuniformly from the fiber of Q{j,k},j specified by the corresponding entry of u; these draws are\nindependent. The other hidden seeds and other weights retain their original independent laws. On" }, { "blockId": "p12-140fd00286", "page": 12, "quote": "(iii) For any k ∈[d], simultaneously changing seeds incident to k, while preserving Q{k,j},j(c{k,j})\nfor every j ̸= k, changes f by at most 2." }, { "blockId": "p13-7db7caa7c4", "page": 13, "quote": "Expanding Q\nj(W 0\nj + W 1\nj ) gives f = P\nS fS. Distinct S’s give orthogonal components, because their\nprojections have a factor W 0\nj W 1\nj = 0 at some position. The first condition in (5.2) says that f∅= 0.\nFurthermore f −W 0\nj f = P\nS∋j fS, and hence" }, { "blockId": "p14-21db4c168d", "page": 14, "quote": "question Q{k,j′},j′ fixed" }, { "blockId": "p14-e5e2d665b1", "page": 14, "quote": "position j\nquestion Q{k,j},j fixed" }, { "blockId": "p14-7075bcad44", "page": 14, "quote": "question Q{k,j′′},j′′ fixed" } ], "occurrenceCount": 11 }, { "symbol": "Sv", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p3-887993e02b", "page": 3, "quote": "Give one vector coordinate to each query–label pair. A compatible family acts on a vector by\nsumming the coordinates it selects; the largest absolute value of these sums defines a norm, ∥·∥∗. At\neach position, a vector of grid weights is supported on that position’s query block. A graph vertex\nspecifies all pair constraints and all position weights. Let Sv be the sum of its position vectors. For\na positive threshold t, distinct vertices v, w are adjacent precisely when ∥Sv + Sw∥∗≤2t.\nA satisfying source assignment restricts to compatible labels for every query. Its linear functional\nexceeds t on an independent set: on the sum of two such vertices it exceeds 2t. A tail estimate for a\nsum of symmetric weights shows that this set occupies nearly half the graph. Section 3 gives the\nconstruction, its deterministic implementation, and this completeness argument." }, "occurrences": [ { "blockId": "p3-887993e02b", "page": 3, "quote": "Give one vector coordinate to each query–label pair. A compatible family acts on a vector by\nsumming the coordinates it selects; the largest absolute value of these sums defines a norm, ∥·∥∗. At\neach position, a vector of grid weights is supported on that position’s query block. A graph vertex\nspecifies all pair constraints and all position weights. Let Sv be the sum of its position vectors. For\na positive threshold t, distinct vertices v, w are adjacent precisely when ∥Sv + Sw∥∗≤2t.\nA satisfying source assignment restricts to compatible labels for every query. Its linear functional\nexceeds t on an independent set: on the sum of two such vertices it exceeds 2t. A tail estimate for a\nsum of symmetric weights shows that this set occupies nearly half the graph. Section 3 gives the\nconstruction, its deterministic implementation, and this completeness argument." }, { "blockId": "p6-76822b6c61", "page": 6, "quote": "Sv =" }, { "blockId": "p7-5fce302f5a", "page": 7, "quote": "{v, w} ∈E(G)\n⇐⇒\n∥Sv + Sw∥∗≤2t.\n(3.5)" }, { "blockId": "p7-fddd405aad", "page": 7, "quote": "For z = Sv + Sw, this maximum can be evaluated by at most (1 + r)2d choices, each checked by a\nscan of the source constraints." }, { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, { "blockId": "p9-c90bc9285c", "page": 9, "quote": "For a uniform vertex, conditional on the seed tuple, the selected slot in each weight vector is fixed.\nIts entries are independent uniform elements of T , one from each independent sj. Coincident\nendpoint questions or slot numbers do not change this independence. By Theorem 3.4, more than a\n(1/2 −1/m) fraction of vertices are kept.\nFor two distinct kept vertices v, w, evaluating Sv + Sw on IA gives exactly the sum of their two\nexpressions in (3.9). This includes all contributions even when the same query appears in both\nvertices. The sum exceeds 2t, so (3.1) rules out the edge in (3.5). The kept vertices are therefore\nindependent. Their complement is a vertex cover of size less than (1/2 + 1/m)n." }, { "blockId": "p10-2f523fd0e7", "page": 10, "quote": "Thus the independent-set density we must rule out is 2b0. First discard the low-norm vertices. All\nvertices whose vectors have norm at most t form a clique: for any two of them the triangle inequality\ngives ∥Sv + Sw∥∗≤2t. An independent set therefore contains at most one such vertex. Since\nn ≥p > 1/b0, an independent set of size at least n/m = 2b0n contains more than b0n vertices of\nnorm greater than t. The following lemma turns these remaining vectors into the required function." }, { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" }, { "blockId": "p10-11188e04c8", "page": 10, "quote": "Pr\nv∈V (G)[Sv ∈A∗] ≥b0,\n∥x + y∥∗> 2t\n(x, y ∈A∗)," }, { "blockId": "p22-a3b65e66e6", "page": 22, "quote": "Proof. Suppose an independent set A has size at least n/m = 2b0n. By the low-norm clique\nobservation at the start of Section 4, at most one vertex of A has ∥Sv∥∗≤t. Since n ≥p > 1/b0,\nthe number of its high-norm vertices is at least" } ], "occurrenceCount": 10, "latex": "S_{v}", "latexBasis": "pdf-script-position" }, { "symbol": "Tj", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, "occurrences": [ { "blockId": "p8-8dcacefcbd", "page": 8, "quote": "j=1\nTj > t" }, { "blockId": "p8-a1ffa28ba5", "page": 8, "quote": "j\nTj −\nX" }, { "blockId": "p9-6b0616f428", "page": 9, "quote": "On this event, rounding leaves P\nj Tj > t. Finally,\n√" }, { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, { "blockId": "p20-675b340463", "page": 20, "quote": "t′ /∈Tj\n|¯gj,t′|2 ≤β" }, { "blockId": "p20-45b8dcaa68", "page": 20, "quote": "t′ /∈Tj\n|¯gj,t′| ≤β" }, { "blockId": "p20-d0d23e4323", "page": 20, "quote": "t′∈Tj\n|¯gj,t′|2 ≤\nX" }, { "blockId": "p20-c428dc9d9e", "page": 20, "quote": "t′∈Tj\n|¯gj,t′| ≤E" }, { "blockId": "p20-2064ebe155", "page": 20, "quote": "t′∈Tj\n|gj,t′|" }, { "blockId": "p20-953714c1b0", "page": 20, "quote": "The last inequality uses the fact that Tj is determined by Gj. Taking expectations, summing, and\napplying (6.12) therefore yields" } ], "occurrenceCount": 10, "latex": "T_{j}", "latexBasis": "pdf-script-position" }, { "symbol": "Z", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p3-e75cdd5634", "page": 3, "quote": "From variance to compatible lists.\nFor fixed batch queries, differentiate F composed with the\nsum of the position vectors. The full gradient lies in the convex hull of signed incidence vectors of\ncompatible label selections (Theorem 6.1). It carries a compatible representation, but can depend\non all batch inputs. Conversely, the derivatives of the own-question means have the required\ninformation restriction, but need not together lie in that same hull.\nThe cube variance bound in Theorem 6.2 gives enough squared energy in the own-question\ngradients. Each block has ℓ1 norm at most one, so thresholding its entries produces a short list\nwithout an alphabet-size factor. Conditional Jensen then transfers the retained list mass back to\nthe single full gradient before its compatible-incidence bound is used. If Z is the largest number of\nlists hit by one compatible selection, this gives EZ ≥2 (Theorem 6.3). The pair-decoding argument\nabove, summed over pairs of positions, gives EZ < 3/2 under source soundness (Theorem 7.1), a\ncontradiction." }, "occurrences": [ { "blockId": "p3-e75cdd5634", "page": 3, "quote": "From variance to compatible lists.\nFor fixed batch queries, differentiate F composed with the\nsum of the position vectors. The full gradient lies in the convex hull of signed incidence vectors of\ncompatible label selections (Theorem 6.1). It carries a compatible representation, but can depend\non all batch inputs. Conversely, the derivatives of the own-question means have the required\ninformation restriction, but need not together lie in that same hull.\nThe cube variance bound in Theorem 6.2 gives enough squared energy in the own-question\ngradients. Each block has ℓ1 norm at most one, so thresholding its entries produces a short list\nwithout an alphabet-size factor. Conditional Jensen then transfers the retained list mass back to\nthe single full gradient before its compatible-incidence bound is used. If Z is the largest number of\nlists hit by one compatible selection, this gives EZ ≥2 (Theorem 6.3). The pair-decoding argument\nabove, summed over pairs of positions, gives EZ < 3/2 under source soundness (Theorem 7.1), a\ncontradiction." }, { "blockId": "p19-19df7f52ba", "page": 19, "quote": "|w0(x) −w0(y)|2 ≤|x −y|\nZ max(x,y)" }, { "blockId": "p19-e1c5f370cb", "page": 19, "quote": "min(x,y)\n|w′\n0(z)|2 dz ≤2\nZ 1" }, { "blockId": "p19-9eb7cdcaa6", "page": 19, "quote": "2E|w0(X) −w0(Y )|2 ≤\nZ 1" }, { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" }, { "blockId": "p21-9da6541b11", "page": 21, "quote": "Z = max\nI∈I #{j ∈J : some λ ∈Lj has (ij, λ) ∈I}.\n(7.3)" }, { "blockId": "p22-b0f00cc8f4", "page": 22, "quote": "This uses the sampled occurrence even when other occurrences have the same endpoints. Union\nbounding Equation (7.5) over the ℓ2 rank pairs proves the pair estimate for the fixed weights and\nother seeds. It is uniform in those fixed values, so averaging proves Pr(Ejk) ≤ℓ2σ. No conditioning\non compatible labels or successful ranks is used.\nFor every integer z ≥0, z ≤1 +\nz\n2\n\n, since the difference is (z −1)(z −2)/2 ≥0. Choose a\nselection attaining the maximum in Equation (7.3). Each of the pairs among its Z hit positions\nwitnesses the corresponding event Ejk. Consequently, pointwise," }, { "blockId": "p22-59c4b5bf59", "page": 22, "quote": "Z ≤1 +\nZ\n2" } ], "occurrenceCount": 9 }, { "symbol": "d", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, "occurrences": [ { "blockId": "p2-6cf80fbffe", "page": 2, "quote": "A Label Cover instance is a bipartite constraint system: every edge asks that a specified map send\nthe label on its left endpoint to the label on its right endpoint. Its value is the largest fraction of\nconstraints satisfied by one assignment of labels to all variables. Our source promise separates value\none from arbitrarily small fixed value. The alphabets may grow as that fixed value decreases; our\nanalytic estimates will not depend on their sizes. The exact source theorem is stated in Theorem 2.1.\nImagine d positions, with an independent source constraint assigned to each unordered pair. At\na pair j < k, position j receives the constraint’s left endpoint and position k its right endpoint.\nA position’s query is its type together with these d −1 endpoint questions. It does not reveal\nconstraint indices or opposite endpoints. A local label assigns symbols to the variables in one query\nand satisfies every source constraint internal to that scope. A compatible family chooses at most\none label per query; its members agree on overlaps and satisfy every constraint internal to their\ncombined scopes." }, { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, { "blockId": "p4-507d2b975a", "page": 4, "quote": "Fix the requested gap integer m ≥4. We first construct and analyze graphs for arbitrary fixed\nsource alphabets and an even square integer d satisfying" }, { "blockId": "p4-912156e740", "page": 4, "quote": "d ≥4,\n√" }, { "blockId": "p4-09267b9534", "page": 4, "quote": "d > 4m.\n(2.1)" }, { "blockId": "p4-b3b991007e", "page": 4, "quote": "The number d will count the positions in the construction. The soundness argument will impose\nfurther lower bounds on d and determine the required source soundness. In Section 8 we choose all\nthese constants as functions of m, before invoking Theorem 2.1 to obtain the alphabets.\nFor the graph construction put" }, { "blockId": "p4-dc0d9cf7e1", "page": 4, "quote": "d.\n(2.2)" }, { "blockId": "p4-9a2dc1e7d9", "page": 4, "quote": "a =\n1\n4m,\np = 2d + 1,\nt = a\n√" }, { "blockId": "p4-10ecaa0721", "page": 4, "quote": "d\np < 1" }, { "blockId": "p5-4439bbf76c", "page": 5, "quote": "d are integers, so t is rational. For a source instance over fixed alphabets, set" }, { "blockId": "p5-86047a999b", "page": 5, "quote": "A query will involve at most d −1 variables, and r bounds its number of local labels. For fixed m, d\nand alphabets, every parameter introduced here is constant with respect to the source input size." }, { "blockId": "p5-65da3010f3", "page": 5, "quote": "Fix a source instance and the construction parameters from Section 2. Each vertex will specify\nsource constraints on the pairs of d positions and weights on the labels available at each position. We\nfirst define when local labels fit together and use their coordinate sums to define a norm. A satisfying\nsource labeling will then provide a linear functional whose values above t select an independent set." }, { "blockId": "p5-c91061e26c", "page": 5, "quote": "Let E =\n[d]\n2\n\nbe the set of unordered pairs of distinct positions. A query of type j ∈[d] has the form" }, { "blockId": "p5-80dd8529d7", "page": 5, "quote": "i =\n\nj, (qjk)k∈[d]\\{j}\n\n,\nqjk ∈" }, { "blockId": "p5-84ffa95f13", "page": 5, "quote": "The tuple is ordered by increasing k. Let Q be the set of all such queries of all types, and let\nP(i) ⊆U ∪V be the set of distinct variables appearing in i. The side of each variable is part of its\nidentity, since U and V are disjoint. Repeated entries in the tuple contribute only one variable to\nP(i); thus |P(i)| ≤d −1.\nA label λ for i assigns a symbol from the appropriate alphabet to each variable of P(i) and\nsatisfies every source constraint occurrence whose two endpoints belong to P(i). Write Λi for the\nset of these labels. This set may be empty, for example when constraints internal to the scope are\ninconsistent. There are at most q|P(i)| ≤r labels. Fix orders on variables and alphabet symbols,\nand list the valid assignments in lexicographic order to define an injection" }, { "blockId": "p6-d3c9737c8b", "page": 6, "quote": "p : u = −d, −d + 1, . . . , d\n\n,\np = 2d + 1.\n(3.3)" }, { "blockId": "p6-efa5c4b66e", "page": 6, "quote": "These are the midpoints of the p equal consecutive subintervals of [−1, 1]. The vertex set consists of\nall pairs\nv = (c, s),\nc ∈[M]E,\ns = (s1, . . . , sd) ∈(T r)d." }, { "blockId": "p6-1bbd2b8815", "page": 6, "quote": "d\nX" }, { "blockId": "p7-9086e7b2b1", "page": 7, "quote": "n = |V (G)| = M(d\n2)prd ≥p > 2m,\n(3.6)" }, { "blockId": "p7-fddd405aad", "page": 7, "quote": "For z = Sv + Sw, this maximum can be evaluated by at most (1 + r)2d choices, each checked by a\nscan of the source constraints." } ], "occurrenceCount": 82 }, { "symbol": "e", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p10-a9f3ade5d7", "page": 10, "quote": "is odd and 1-Lipschitz for ∥·∥∗, and F(z) > t on A∗.\nLet E =\n[d]\n2\n\n. Draw independent uniform occurrence seeds c = (ce)e∈E ∈[M]E and independent\nweight blocks sj uniform on [−1, 1]r, independently of the seeds. Put" }, "occurrences": [ { "blockId": "p6-9352d79c2f", "page": 6, "quote": "A seed tuple is an indexed family c = (ce)e∈E ∈[M]E. For e = {j, k} with j < k, seed ce gives\nendpoint xce to position j and endpoint yce to position k. It thereby specifies queries ij(c) through" }, { "blockId": "p10-a9f3ade5d7", "page": 10, "quote": "is odd and 1-Lipschitz for ∥·∥∗, and F(z) > t on A∗.\nLet E =\n[d]\n2\n\n. Draw independent uniform occurrence seeds c = (ce)e∈E ∈[M]E and independent\nweight blocks sj uniform on [−1, 1]r, independently of the seeds. Put" }, { "blockId": "p11-8c7eaa450a", "page": 11, "quote": "The function from the previous section has variance of order d, but the functions used to extract\nlabels must depend only on one position’s own questions and weights. We shall freeze all but\na batch of h positions and retain variance of order h in conditional averages with precisely this\ndependence. Questions at different positions are correlated because they share pair seeds; our\northogonal decomposition therefore uses the independent seeds and weights themselves.\nLet d ≥4 and r, M ≥1 be integers, and put E =\n[d]\n2\n\n. For each e ∈E, draw ce uniformly from\n[M], independently over e. At each endpoint j ∈e, this seed reveals a question Qe,j(ce), where\nQe,j is a fixed map to a finite set. There is no independence assumption on the two questions\nfrom the same seed. In the construction, for e = {j, k} with j < k, these maps are Qe,j(c) = xc\nand Qe,k(c) = yc. Independently of all seeds, draw sj uniformly from [−1, 1]r for each j ∈[d],\nindependently over j. All expectations and L2-norms in this section initially refer to the resulting\nproduct law on\nΩ= [M]E × ([−1, 1]r)d." }, { "blockId": "p11-95ea1c8256", "page": 11, "quote": "a =\n\n(ce)e/∈(J\n2), (sk)k/∈J\n\n.\n(5.1)" }, { "blockId": "p11-0fd1fee442", "page": 11, "quote": "Fixing these values leaves a function f∗of the hidden seeds (ce)e∈(J\n2) and the weights (sj)j∈J, with\ntheir original product law. For j ∈J, write" }, { "blockId": "p12-22f8b88cff", "page": 12, "quote": "Averaging projections and weight supports.\nLet W 0\nj average the weight block sj, retaining\nall other coordinates, and set W 1\nj = Id −W 0\nj . Let P 0\ne average the seed ce. For j ∈e, let Oe,j average\nthat seed conditional on its question Qe,j(ce), retaining all other coordinates. Each averaging is an\northogonal projection in L2(Ω): averaging twice gives the same function, and integrating over the\nretained data shows self-adjointness. Also" }, { "blockId": "p12-872013d74b", "page": 12, "quote": "ran(P 0\ne ) ⊆ran(Oe,j),\nP 0\ne Oe,j = Oe,jP 0\ne = P 0\ne ." }, { "blockId": "p13-1eb9806aa3", "page": 13, "quote": "j /∈e :\nP 0\ne ,\nId −P 0\ne ," }, { "blockId": "p13-76801848a6", "page": 13, "quote": "j ∈e :\nP 0\ne ,\nOe,j −P 0\ne ,\nId −Oe,j." }, { "blockId": "p13-319e53e23f", "page": 13, "quote": "where henceforth these sums range over nonempty S and its allowed choices ω.\nLet TS,ω ⊆E be the seeds where the chosen projection is not P 0\ne . Let NS,ω ⊆[d] \\ {j(S)} be\nthe positions k for which the projection at e = {j(S), k} is Id −Oe,j(S). These are the incident\ncomponents not retained by the designated position’s own question. For each fixed e, applying\nId −P 0\ne keeps exactly the components with e ∈TS,ω. Orthogonality and the one-seed oscillation\nbound therefore give" }, { "blockId": "p13-5f0771b88d", "page": 13, "quote": "e∈E\n∥f −P 0\ne f∥2\n2 ≤16\nd\n2" }, { "blockId": "p13-45d4aca140", "page": 13, "quote": "Here |f −P 0\ne f| ≤4 follows by averaging values whose oscillation is at most 4.\nAt this stage the ordinary seed-energy bound is of order d2. That is sufficient for hidden pairs\navoiding the designated position, because both endpoints must enter the small batch. It is insufficient\nfor a hidden incident seed, which needs only one extra endpoint in the batch. The next estimate\nreduces the relevant incident energy to order d by resampling a whole star at once." }, { "blockId": "p15-b5277cc6cd", "page": 15, "quote": "P 0\ne\nY" }, { "blockId": "p15-f55def92af", "page": 15, "quote": "e∈(J\\{j}\n2 )" }, { "blockId": "p15-57d41c550d", "page": 15, "quote": "To verify that its squared norm can be counted in (5.11), first note that KJ,j commutes with every\nΠS. Thus its outputs from different weight supports are orthogonal, and ∥KJ,jf∥2\n2 = P\nS ∥KJ,jfS∥2\n2.\nThe weight factors in KJ,j keep fS precisely when S∩J = {j}. For those S whose designated position\nis this j, the seed factors are diagonal in the chosen seed decomposition: P 0\ne on a nonincident hidden\nseed keeps just its constant component, and Oe,j on an incident hidden seed keeps its constant and\nown-question components. The other seed coordinates are untouched. They therefore keep exactly\nthe components satisfying the last two conditions of (5.12), with no cancellation among them. We\nmay discard all other nonnegative squared norms in (5.11) and count each component solely in its\ndesignated term. It follows that" }, { "blockId": "p16-3ddfbe0fc9", "page": 16, "quote": "The joint parameter choice in Section 8 will ensure these conditions. Applying Theorem 5.1 to the\nfunction in Theorem 4.1 supplies a fixed set J ⊂[d] of size h and fixed data a consisting of the\nweights outside J and every seed outside EJ =\nJ\n2\n\n. Write f∗(cJ, sJ) for f with these values plugged\nin. In the rest of this section the probability law is the product of independent uniform hidden seeds\n(ce)e∈EJ and independent uniform weight blocks (sj)j∈J on [−1, 1]r. The choice of the fixed data\ndoes not condition these remaining draws on any event of success.\nFor j ∈J, write uj = (qjk)k∈J\\{j}, in increasing order of k, for the tuple of questions received\nby j from its incident hidden seeds. Together with a, it specifies the whole query ij; denote the\ncompleted query by ij(uj). The conditional averages supplied by the restriction lemma satisfy\nX" }, { "blockId": "p21-ae56abe745", "page": 21, "quote": "and assume throughout this section that val(Φ) ≤σ. The source promise bounds every labeling\nof U ∪V , so the decoders below may use arbitrary fixed instance-dependent data. Their essential\nrestriction is that the label returned at a variable does not also depend on the sampled constraint\noccurrence.\nFix a set J ⊆[d] of size h and frozen data a as in Theorem 5.1. The remaining seeds ce, e ∈\nJ\n2\n\n,\nare independent uniform elements of [M]; the weight blocks sj, j ∈J, are independent uniforms on\n[−1, 1]r, independent of the seeds. Let uj be position j’s tuple of questions from its incident hidden\nseeds. Together with a, this tuple determines its query ij.\nConsider any fixed total measurable maps" }, { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." } ], "occurrenceCount": 18 }, { "symbol": "g", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, "occurrences": [ { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" }, { "blockId": "p18-e33882be56", "page": 18, "quote": "In particular, ψi is continuous and piecewise affine, with finitely many pieces.\nCompare all pairs of candidates in (6.7). If two candidates with different slope vectors in the\nunfrozen weights tie in value, set g = 0. Otherwise take the true gradient of the finite formula\nfor ψi. This latter choice is well-defined. Two candidates with the same slope either never tie or\nare identical affine functions. All other comparisons are strict at such a point and persist in a\nneighborhood; each minimum and maximum there selects one affine function up to identical copies.\nThe resulting formula is affine in that neighborhood. Fixing a deterministic order on candidates\nresolves any choice between identical copies without changing its gradient.\nEach discarded equality has a nonzero normal in the unfrozen coordinates, hence defines a\nproper affine hyperplane. No choice of frozen weights changes that fact, since frozen weights change\nthe offsets but not the slopes. Identical slopes caused by repeated endpoint questions or other\ncoincidences are allowed. Finite comparisons prove measurability, and the rule is defined even at\nquery tuples that cannot arise from the hidden seed law. Its formula uses the queries themselves\nand no additional seed information.\nAt a point not discarded, choose an active affine expression in each distance formula. Write their\nslopes as ε−vI−(i) for the distance to −A∗and ε+vI+(i) for the distance to A∗. Since F is half the\ndifference of these distances," }, { "blockId": "p18-b1c46cdf93", "page": 18, "quote": "g = 1" }, { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, { "blockId": "p20-acacb43f6e", "page": 20, "quote": "For arbitrary fixed queries and arbitrary lists of valid labels, the sum of absolute values over\ntheir slot sets is a convex function of the full vector in RJ×[r]. On a signed generator ±vI(i) it equals\nthe number of positions hit by that selection I, since each block contains at most one selected slot.\nIts value on every point of the hull in (6.5) is consequently at most the maximum such hit count.\nAt every realized tuple, apply this deterministic statement to the realized lists and the full vector g:\nX" } ], "occurrenceCount": 5 }, { "symbol": "h", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, "occurrences": [ { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, { "blockId": "p11-8c7eaa450a", "page": 11, "quote": "The function from the previous section has variance of order d, but the functions used to extract\nlabels must depend only on one position’s own questions and weights. We shall freeze all but\na batch of h positions and retain variance of order h in conditional averages with precisely this\ndependence. Questions at different positions are correlated because they share pair seeds; our\northogonal decomposition therefore uses the independent seeds and weights themselves.\nLet d ≥4 and r, M ≥1 be integers, and put E =\n[d]\n2\n\n. For each e ∈E, draw ce uniformly from\n[M], independently over e. At each endpoint j ∈e, this seed reveals a question Qe,j(ce), where\nQe,j is a fixed map to a finite set. There is no independence assumption on the two questions\nfrom the same seed. In the construction, for e = {j, k} with j < k, these maps are Qe,j(c) = xc\nand Qe,k(c) = yc. Independently of all seeds, draw sj uniformly from [−1, 1]r for each j ∈[d],\nindependently over j. All expectations and L2-norms in this section initially refer to the resulting\nproduct law on\nΩ= [M]E × ([−1, 1]r)d." }, { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" }, { "blockId": "p12-7a4e49fe92", "page": 12, "quote": "16h + 32h2 ≤νd" }, { "blockId": "p12-ea2c452c9c", "page": 12, "quote": "then there are a fixed h-element set J and fixed frozen data a such that" }, { "blockId": "p12-20efa93216", "page": 12, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh" }, { "blockId": "p15-986481f9b9", "page": 15, "quote": "Averaging the restriction.\nChoose J uniformly among all h-element subsets of [d]. For a fixed\ncomponent, the designated position belongs to J with probability h/d. Conditional on this inclusion,\na specified other position belongs to J with probability" }, { "blockId": "p15-b56edca5df", "page": 15, "quote": "p1 = h −1" }, { "blockId": "p15-c9aa1f4715", "page": 15, "quote": "p2 = (h −1)(h −2)" }, { "blockId": "p16-bf27fb903b", "page": 16, "quote": "EJ,aV(J, a) ≥h" }, { "blockId": "p16-a1e221561d", "page": 16, "quote": "≥h" }, { "blockId": "p16-0119551a53", "page": 16, "quote": "d\n\nνd −16h −32h2\n≥νh" }, { "blockId": "p16-741afdd376", "page": 16, "quote": "For the second inequality, d ≥4 gives p1 ≤2h/d and p2 ≤4h2/d2; the last inequality is (5.3).\nThe nonnegative measurable quantity V(J, a) is bounded, since f is bounded. Its mean in (5.15)\nimplies that a set of choices (J, a) of positive probability has V(J, a) ≥νh/4. Choose one such\nrestriction. The conditional averages were defined by explicit integration with its frozen coordinates\nplugged in. All hidden seeds and batch weights are then drawn with their original product law,\nestablishing (5.4) with the asserted interpretation." }, { "blockId": "p16-a0d26bb753", "page": 16, "quote": "β ,\nh = 2ℓ.\n(6.1)" }, { "blockId": "p16-064cae7e41", "page": 16, "quote": "Since ν = 1/(27m3), both ℓand h are positive integers depending only on m. Assume from now on\nthat the number d of graph positions also satisfies" }, { "blockId": "p16-d2531d9d0f", "page": 16, "quote": "h < d,\n16h + 32h2 ≤νd" }, { "blockId": "p16-3ddfbe0fc9", "page": 16, "quote": "The joint parameter choice in Section 8 will ensure these conditions. Applying Theorem 5.1 to the\nfunction in Theorem 4.1 supplies a fixed set J ⊂[d] of size h and fixed data a consisting of the\nweights outside J and every seed outside EJ =\nJ\n2\n\n. Write f∗(cJ, sJ) for f with these values plugged\nin. In the rest of this section the probability law is the product of independent uniform hidden seeds\n(ce)e∈EJ and independent uniform weight blocks (sj)j∈J on [−1, 1]r. The choice of the fixed data\ndoes not condition these remaining draws on any event of success.\nFor j ∈J, write uj = (qjk)k∈J\\{j}, in increasing order of k, for the tuple of questions received\nby j from its incident hidden seeds. Together with a, it specifies the whole query ij; denote the\ncompleted query by ij(uj). The conditional averages supplied by the restriction lemma satisfy\nX" }, { "blockId": "p16-6ae5a923ab", "page": 16, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh/4.\n(6.3)" }, { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" }, { "blockId": "p19-abc62cd133", "page": 19, "quote": "The conditional means now retain total squared gradient energy at least βh, and each block has ℓ1" } ], "occurrenceCount": 37 }, { "symbol": "uj", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p11-a7c5fe9a95", "page": 11, "quote": "with entries in increasing order of k. Define HJ,a\nj\n(u, s) by inserting sj = s in f∗and averaging the\nremaining variables conditional on uj = u. Explicitly, each incident hidden seed c{j,k} is drawn\nuniformly from the fiber of Q{j,k},j specified by the corresponding entry of u; these draws are\nindependent. The other hidden seeds and other weights retain their original independent laws. On" }, "occurrences": [ { "blockId": "p11-580f678b17", "page": 11, "quote": "uj =\n\nQ{j,k},j(c{j,k})\n" }, { "blockId": "p11-a7c5fe9a95", "page": 11, "quote": "with entries in increasing order of k. Define HJ,a\nj\n(u, s) by inserting sj = s in f∗and averaging the\nremaining variables conditional on uj = u. Explicitly, each incident hidden seed c{j,k} is drawn\nuniformly from the fiber of Q{j,k},j specified by the corresponding entry of u; these draws are\nindependent. The other hidden seeds and other weights retain their original independent laws. On" }, { "blockId": "p12-20efa93216", "page": 12, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh" }, { "blockId": "p15-779c685497", "page": 15, "quote": "All seed factors here act on distinct coordinates. Before applying W 1\nj , these operators average\nexactly the unfrozen data other than the own questions uj and the own weights sj. They retain the\nfrozen data a as variables. The final centering W 1\nj subtracts the mean in sj, which is independent\nof the retained seed information. Thus" }, { "blockId": "p15-29d4682409", "page": 15, "quote": "KJ,jf = HJ,a\nj\n(uj, sj) −Es′\njHJ,a\nj\n(uj, s′\nj)" }, { "blockId": "p15-2c1d909a80", "page": 15, "quote": "j∈J\nEuj Varsj HJ,a\nj\n(uj, sj)," }, { "blockId": "p16-3ddfbe0fc9", "page": 16, "quote": "The joint parameter choice in Section 8 will ensure these conditions. Applying Theorem 5.1 to the\nfunction in Theorem 4.1 supplies a fixed set J ⊂[d] of size h and fixed data a consisting of the\nweights outside J and every seed outside EJ =\nJ\n2\n\n. Write f∗(cJ, sJ) for f with these values plugged\nin. In the rest of this section the probability law is the product of independent uniform hidden seeds\n(ce)e∈EJ and independent uniform weight blocks (sj)j∈J on [−1, 1]r. The choice of the fixed data\ndoes not condition these remaining draws on any event of success.\nFor j ∈J, write uj = (qjk)k∈J\\{j}, in increasing order of k, for the tuple of questions received\nby j from its incident hidden seeds. Together with a, it specifies the whole query ij; denote the\ncompleted query by ij(uj). The conditional averages supplied by the restriction lemma satisfy\nX" }, { "blockId": "p16-6ae5a923ab", "page": 16, "quote": "j∈J\nEuj Varsj Hj(uj, sj) ≥νh/4.\n(6.3)" }, { "blockId": "p17-ce323cea6d", "page": 17, "quote": "Here Hj(u, s) is the mean of f∗given the own questions uj = u and the own weight block sj = s.\nOur aim is to turn this variance into own-input lists of size at most ℓwhose maximum compatible\nhit count has expectation at least two.\nTo differentiate these means, we use the explicit integral from Section 5. For a tuple u in supp uj,\nlet κj(u) be the conditional distribution of all hidden seeds given uj = u. On an incident pair {j, k},\nthis is uniform on the occurrence indices producing the specified question at j; on a nonincident\nhidden pair it is uniform on [M]. All these conditional seed coordinates are independent. This\nfollows from independence of the original seeds and from the fact that each condition fixes a function\nof just one seed. Thus" }, { "blockId": "p19-a8692ca128", "page": 19, "quote": "j∈J\nE∥¯gj(uj, sj)∥2\n2 ≥νh/8 = βh.\n(6.12)" }, { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," }, { "blockId": "p20-d850944fa1", "page": 20, "quote": "It also gives zero dummy entries, so thresholding returns only valid labels.\nFinite threshold\ncomparisons of measurable functions, with an empty-list fallback, prove totality and measurability.\nWe first transfer the retained mass of the own conditional means to the full gradient rule g, and\nthen apply its pointwise hull bound. For an actual own input, let Tj ⊂[r] be the set of slots of Lj,\nand write Gj for the information (uj, sj). The conditional mean in (6.8) is a version of E[gj | Gj],\ndefined by the displayed kernel even at own values of probability zero. For slots outside Tj, (6.9)\nand the threshold give\nX" }, { "blockId": "p21-ae56abe745", "page": 21, "quote": "and assume throughout this section that val(Φ) ≤σ. The source promise bounds every labeling\nof U ∪V , so the decoders below may use arbitrary fixed instance-dependent data. Their essential\nrestriction is that the label returned at a variable does not also depend on the sampled constraint\noccurrence.\nFix a set J ⊆[d] of size h and frozen data a as in Theorem 5.1. The remaining seeds ce, e ∈\nJ\n2\n\n,\nare independent uniform elements of [M]; the weight blocks sj, j ∈J, are independent uniforms on\n[−1, 1]r, independent of the seeds. Let uj be position j’s tuple of questions from its incident hidden\nseeds. Together with a, this tuple determines its query ij.\nConsider any fixed total measurable maps" }, { "blockId": "p21-a4f76b578f", "page": 21, "quote": "Lj = Lj(uj, sj) ⊆Λij,\n|Lj| ≤ℓ.\n(7.2)" }, { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." } ], "occurrenceCount": 16, "latex": "u_{j}", "latexBasis": "pdf-script-position" }, { "symbol": "w0", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p19-a8b14455ed", "page": 19, "quote": "Proof. Replacing the own block s by s′, with a seed configuration and other weights fixed, changes\nthe integrand in (6.4) by at most ∥s −s′∥∞. The same bound holds after averaging. In particular,\nevery scalar-coordinate difference quotient of the integrand is bounded in absolute value by one.\nFor each fixed hidden seed tuple, the full gradient identification in Theorem 6.1 holds outside\na null set in all batch weights. Fubini implies that for almost every own s it holds for almost all\nother weight blocks. The conditional seed law κj(u) has finite support and does not depend on s, so\nthe exceptional sets may be united over its support and over the finitely many weight coordinates.\nAt the remaining own values, the coordinate difference quotients converge almost surely to gj,t′.\nDominated convergence gives (6.10); the boundary of the cube has measure zero.\nEvery coordinate-line restriction of Hj(u, ·) is Lipschitz and hence absolutely continuous. By\nFubini applied to (6.10), for almost every fixing of the other own coordinates its derivative agrees\nalmost everywhere with the corresponding entry of ¯gj. The fundamental theorem of calculus\ntherefore expresses differences between any two values on such a line as the integral of that entry.\nLet w0 be the restriction of Hj(u, ·) to one of these lines. For x, y ∈[−1, 1], Cauchy–Schwarz\ngives" }, "occurrences": [ { "blockId": "p19-a8b14455ed", "page": 19, "quote": "Proof. Replacing the own block s by s′, with a seed configuration and other weights fixed, changes\nthe integrand in (6.4) by at most ∥s −s′∥∞. The same bound holds after averaging. In particular,\nevery scalar-coordinate difference quotient of the integrand is bounded in absolute value by one.\nFor each fixed hidden seed tuple, the full gradient identification in Theorem 6.1 holds outside\na null set in all batch weights. Fubini implies that for almost every own s it holds for almost all\nother weight blocks. The conditional seed law κj(u) has finite support and does not depend on s, so\nthe exceptional sets may be united over its support and over the finitely many weight coordinates.\nAt the remaining own values, the coordinate difference quotients converge almost surely to gj,t′.\nDominated convergence gives (6.10); the boundary of the cube has measure zero.\nEvery coordinate-line restriction of Hj(u, ·) is Lipschitz and hence absolutely continuous. By\nFubini applied to (6.10), for almost every fixing of the other own coordinates its derivative agrees\nalmost everywhere with the corresponding entry of ¯gj. The fundamental theorem of calculus\ntherefore expresses differences between any two values on such a line as the integral of that entry.\nLet w0 be the restriction of Hj(u, ·) to one of these lines. For x, y ∈[−1, 1], Cauchy–Schwarz\ngives" }, { "blockId": "p19-19df7f52ba", "page": 19, "quote": "|w0(x) −w0(y)|2 ≤|x −y|\nZ max(x,y)" }, { "blockId": "p19-1d80de8bc3", "page": 19, "quote": "Var(w0(X)) = 1" }, { "blockId": "p19-9eb7cdcaa6", "page": 19, "quote": "2E|w0(X) −w0(Y )|2 ≤\nZ 1" } ], "occurrenceCount": 4, "latex": "w_{0}", "latexBasis": "pdf-script-position" }, { "symbol": "Θ", "meaning": "No explicit definition located by the extractor.", "status": "not-located", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-01ff3f3332", "page": 1, "quote": "Vertex Cover appeared as Node Cover among the problems Karp proved NP-complete in 1972 [12].\nExact hardness still leaves room for approximation, and the elementary maximal-matching algorithm\nachieves factor two. Subsequent algorithms improved this ratio by an amount depending on the\ngraph order. Karakostas obtained 2−Θ(1/√log n) using semidefinite programming [11]. This saving\ntends to zero as n grows. The threshold addressed here concerns a fixed saving below two, so it is\ncompatible with such algorithmic improvements." }, "occurrences": [ { "blockId": "p1-01ff3f3332", "page": 1, "quote": "Vertex Cover appeared as Node Cover among the problems Karp proved NP-complete in 1972 [12].\nExact hardness still leaves room for approximation, and the elementary maximal-matching algorithm\nachieves factor two. Subsequent algorithms improved this ratio by an amount depending on the\ngraph order. Karakostas obtained 2−Θ(1/√log n) using semidefinite programming [11]. This saving\ntends to zero as n grows. The threshold addressed here concerns a fixed saving below two, so it is\ncompatible with such algorithmic improvements." } ], "occurrenceCount": 1 }, { "symbol": "Φ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p4-25c6142b08", "page": 4, "quote": "Write [k] = {1, . . . , k}. A Label Cover instance Φ consists of two disjoint finite variable sets U, V ,\ntwo nonempty finite alphabets ΣU, ΣV , and an explicit nonempty list of M constraint occurrences.\nOccurrence c ∈[M] has endpoints xc ∈U, yc ∈V , and a total map πc : ΣU →ΣV . A labeling A\nassigns each variable a symbol from its side’s alphabet. It satisfies occurrence c when" }, "occurrences": [ { "blockId": "p4-25c6142b08", "page": 4, "quote": "Write [k] = {1, . . . , k}. A Label Cover instance Φ consists of two disjoint finite variable sets U, V ,\ntwo nonempty finite alphabets ΣU, ΣV , and an explicit nonempty list of M constraint occurrences.\nOccurrence c ∈[M] has endpoints xc ∈U, yc ∈V , and a total map πc : ΣU →ΣV . A labeling A\nassigns each variable a symbol from its side’s alphabet. It satisfies occurrence c when" }, { "blockId": "p4-c1403a8a36", "page": 4, "quote": "The value val(Φ) is the maximum fraction of satisfied occurrences. List entries count with their\nmultiplicity. Endpoints and map tables are given explicitly in binary; unused variable declarations\ncan be removed. We require no regularity or expansion condition on the constraint graph." }, { "blockId": "p4-15866c4f93", "page": 4, "quote": "Theorem 2.1 (Perfect-completeness Label Cover hardness). For every fixed rational σ ∈(0, 1),\nthere are nonempty constant alphabets and a deterministic polynomial-time reduction from 3SAT to\ninstances of the form above such that satisfiable formulas give val(Φ) = 1, and unsatisfiable formulas\ngive val(Φ) ≤σ." }, { "blockId": "p9-0650d48b6b", "page": 9, "quote": "Proposition 3.5 (Completeness). If val(Φ) = 1, then G has an independent set of size greater than\n(1/2 −1/m)n. Consequently," }, { "blockId": "p17-cb8e82d3e3", "page": 17, "quote": "where the other weight blocks have their original independent uniform laws. The seed law in this\nformula does not depend on s. We use zero as a fallback for Hj on unsupported own question tuples.\nThe fixed data throughout are Φ, F, J, and a." }, { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," }, { "blockId": "p21-ae56abe745", "page": 21, "quote": "and assume throughout this section that val(Φ) ≤σ. The source promise bounds every labeling\nof U ∪V , so the decoders below may use arbitrary fixed instance-dependent data. Their essential\nrestriction is that the label returned at a variable does not also depend on the sampled constraint\noccurrence.\nFix a set J ⊆[d] of size h and frozen data a as in Theorem 5.1. The remaining seeds ce, e ∈\nJ\n2\n\n,\nare independent uniform elements of [M]; the weight blocks sj, j ∈J, are independent uniforms on\n[−1, 1]r, independent of the seeds. Let uj be position j’s tuple of questions from its incident hidden\nseeds. Together with a, this tuple determines its query ij.\nConsider any fixed total measurable maps" }, { "blockId": "p22-9e3b3dd195", "page": 22, "quote": "Corollary 7.2 (Soundness). For the graph constructed in Section 3, assume (6.2). If val(Φ) ≤σ,\nevery independent set has size strictly less than n/m. Consequently τ(G) > (1 −1/m)n." } ], "occurrenceCount": 8 }, { "symbol": "α", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "occurrences": [ { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." }, { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, { "blockId": "p23-a579cda93f", "page": 23, "quote": "α < Rm :=\n1 −1/m\n1/2 + 1/m = 2 −\n6\nm + 2.\n(8.2)" }, { "blockId": "p23-ed115d16da", "page": 23, "quote": "Suppose an algorithm always returns a vertex cover of size at most ατ(G). On a completeness\ninstance its output has size strictly less than" }, { "blockId": "p23-48253c9453", "page": 23, "quote": "α\n1" }, { "blockId": "p24-38dac4ff9d", "page": 24, "quote": "uses only integer arithmetic: multiply the cardinality by m and compare with (m −1)n. The real\nnumber α is used only to choose the fixed integer m; it is not an input to the reduction. This proves\nTheorem 1.2." } ], "occurrenceCount": 7 }, { "symbol": "β", "meaning": "No explicit definition located by the extractor.", "status": "not-located", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p16-1533308f69", "page": 16, "quote": "β = ν" }, "occurrences": [ { "blockId": "p16-1533308f69", "page": 16, "quote": "β = ν" }, { "blockId": "p16-a0d26bb753", "page": 16, "quote": "β ,\nh = 2ℓ.\n(6.1)" }, { "blockId": "p19-e116ff9258", "page": 19, "quote": "Lj(u, s) =\n\nλ ∈Λij(u) :\n¯gj,slotij(u)(λ)(u, s)\n ≥β" }, { "blockId": "p20-cd69f7e969", "page": 20, "quote": "β" }, { "blockId": "p20-3b2b95e21c", "page": 20, "quote": "2 |Lj(u, s)| ≤∥¯gj(u, s)∥1 ≤1,\n|Lj(u, s)| ≤2/β = ℓ." }, { "blockId": "p20-675b340463", "page": 20, "quote": "t′ /∈Tj\n|¯gj,t′|2 ≤β" }, { "blockId": "p20-45b8dcaa68", "page": 20, "quote": "t′ /∈Tj\n|¯gj,t′| ≤β" }, { "blockId": "p20-068f4fb2a5", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≥βh −β" }, { "blockId": "p23-42af8c9eb4", "page": 23, "quote": "β =\n1\n210m3 ,\nℓ= 211m3,\nh = 212m3," } ], "occurrenceCount": 9 }, { "symbol": "ε", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p17-ac67f82849", "page": 17, "quote": "Proof. Let 1I ∈RP be the indicator vector of I. For ζ ∈A∗∪(−A∗), I ∈I, and ε ∈{−1, 1}, form\nthe finite collection of affine expressions" }, "occurrences": [ { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." }, { "blockId": "p17-ac67f82849", "page": 17, "quote": "Proof. Let 1I ∈RP be the indicator vector of I. For ζ ∈A∗∪(−A∗), I ∈I, and ε ∈{−1, 1}, form\nthe finite collection of affine expressions" }, { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, { "blockId": "p18-aa516f4123", "page": 18, "quote": "= min\nζ∈C\nmax\nI∈I, ε∈{−1,1} Lζ,I,ε(sJ)." }, { "blockId": "p18-e33882be56", "page": 18, "quote": "In particular, ψi is continuous and piecewise affine, with finitely many pieces.\nCompare all pairs of candidates in (6.7). If two candidates with different slope vectors in the\nunfrozen weights tie in value, set g = 0. Otherwise take the true gradient of the finite formula\nfor ψi. This latter choice is well-defined. Two candidates with the same slope either never tie or\nare identical affine functions. All other comparisons are strict at such a point and persist in a\nneighborhood; each minimum and maximum there selects one affine function up to identical copies.\nThe resulting formula is affine in that neighborhood. Fixing a deterministic order on candidates\nresolves any choice between identical copies without changing its gradient.\nEach discarded equality has a nonzero normal in the unfrozen coordinates, hence defines a\nproper affine hyperplane. No choice of frozen weights changes that fact, since frozen weights change\nthe offsets but not the slopes. Identical slopes caused by repeated endpoint questions or other\ncoincidences are allowed. Finite comparisons prove measurability, and the rule is defined even at\nquery tuples that cannot arise from the hidden seed law. Its formula uses the queries themselves\nand no additional seed information.\nAt a point not discarded, choose an active affine expression in each distance formula. Write their\nslopes as ε−vI−(i) for the distance to −A∗and ε+vI+(i) for the distance to A∗. Since F is half the\ndifference of these distances," }, { "blockId": "p18-9e10b8c9b8", "page": 18, "quote": "2\n\nε−vI−(i) −ε+vI+(i)\n\n." } ], "occurrenceCount": 6 }, { "symbol": "ζ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, "occurrences": [ { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, { "blockId": "p17-ac67f82849", "page": 17, "quote": "Proof. Let 1I ∈RP be the indicator vector of I. For ζ ∈A∗∪(−A∗), I ∈I, and ε ∈{−1, 1}, form\nthe finite collection of affine expressions" }, { "blockId": "p17-eb8bb78f33", "page": 17, "quote": "Lζ,I,ε(sJ) = ε⟨1I, c0 −ζ⟩+ ε⟨vI(i), sJ⟩.\n(6.7)" }, { "blockId": "p18-aa516f4123", "page": 18, "quote": "= min\nζ∈C\nmax\nI∈I, ε∈{−1,1} Lζ,I,ε(sJ)." } ], "occurrenceCount": 4 }, { "symbol": "λ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-84ffa95f13", "page": 5, "quote": "The tuple is ordered by increasing k. Let Q be the set of all such queries of all types, and let\nP(i) ⊆U ∪V be the set of distinct variables appearing in i. The side of each variable is part of its\nidentity, since U and V are disjoint. Repeated entries in the tuple contribute only one variable to\nP(i); thus |P(i)| ≤d −1.\nA label λ for i assigns a symbol from the appropriate alphabet to each variable of P(i) and\nsatisfies every source constraint occurrence whose two endpoints belong to P(i). Write Λi for the\nset of these labels. This set may be empty, for example when constraints internal to the scope are\ninconsistent. There are at most q|P(i)| ≤r labels. Fix orders on variables and alphabet symbols,\nand list the valid assignments in lexicographic order to define an injection" }, "occurrences": [ { "blockId": "p5-84ffa95f13", "page": 5, "quote": "The tuple is ordered by increasing k. Let Q be the set of all such queries of all types, and let\nP(i) ⊆U ∪V be the set of distinct variables appearing in i. The side of each variable is part of its\nidentity, since U and V are disjoint. Repeated entries in the tuple contribute only one variable to\nP(i); thus |P(i)| ≤d −1.\nA label λ for i assigns a symbol from the appropriate alphabet to each variable of P(i) and\nsatisfies every source constraint occurrence whose two endpoints belong to P(i). Write Λi for the\nset of these labels. This set may be empty, for example when constraints internal to the scope are\ninconsistent. There are at most q|P(i)| ≤r labels. Fix orders on variables and alphabet symbols,\nand list the valid assignments in lexicographic order to define an injection" }, { "blockId": "p5-c0dae8c4e1", "page": 5, "quote": "P = {(i, λ) : i ∈Q, λ ∈Λi}." }, { "blockId": "p5-aca0ba4dde", "page": 5, "quote": "(i,λ)∈I\nP(i)." }, { "blockId": "p6-99c6986cec", "page": 6, "quote": "(i,λ)∈I\nzi,λ" }, { "blockId": "p6-96563cae0a", "page": 6, "quote": "z(i, s)i,λ = ssloti(λ)\nfor λ ∈Λi." }, { "blockId": "p7-65cbc81ba7", "page": 7, "quote": "(i,λ)∈I\nzi,λ" }, { "blockId": "p7-182f70c256", "page": 7, "quote": "∥z∥∗=\nmax\nI∈I:\n(i,λ)∈I =⇒i∈B" }, { "blockId": "p17-c91c50c010", "page": 17, "quote": "For I ∈I, let vI(i) ∈RJ×[r] have entry one at (j, slotij(λ)) if (ij, λ) ∈I, and zero elsewhere. It has\nat most one nonzero entry in each block j. We will represent the gradient using these incidence\nvectors. At exceptional weights where a derivative need not exist, the same representation will hold\nfor a prescribed value of the gradient rule." }, { "blockId": "p19-e116ff9258", "page": 19, "quote": "Lj(u, s) =\n\nλ ∈Λij(u) :\n¯gj,slotij(u)(λ)(u, s)\n ≥β" }, { "blockId": "p20-e5348c584f", "page": 20, "quote": "For an unsupported own question tuple, let the list be empty. At an actual draw write Lj = Lj(uj, sj),\nand define\nZ = max\nI∈I #{j ∈J : ∃λ ∈Lj with (ij, λ) ∈I}.\n(6.14)" }, { "blockId": "p20-068f4fb2a5", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≥βh −β" }, { "blockId": "p20-a0b7388e2c", "page": 20, "quote": "λ∈Lj\n|gj,slotij (λ)| ≤Z.\n(6.17)" }, { "blockId": "p21-9da6541b11", "page": 21, "quote": "Z = max\nI∈I #{j ∈J : some λ ∈Lj has (ij, λ) ∈I}.\n(7.3)" } ], "occurrenceCount": 13 }, { "symbol": "ν", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" }, "occurrences": [ { "blockId": "p9-7f11e26df2", "page": 9, "quote": "b0 =\n1\n2m,\nν = b0a2" }, { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" }, { "blockId": "p16-1533308f69", "page": 16, "quote": "β = ν" }, { "blockId": "p16-064cae7e41", "page": 16, "quote": "Since ν = 1/(27m3), both ℓand h are positive integers depending only on m. Assume from now on\nthat the number d of graph positions also satisfies" }, { "blockId": "p23-74a961993f", "page": 23, "quote": "a =\n1\n4m,\nb0 =\n1\n2m,\nν =\n1\n27m3 ," } ], "occurrenceCount": 5 }, { "symbol": "ρ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." }, "occurrences": [ { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." }, { "blockId": "p22-e0975c3db8", "page": 22, "quote": "For every actual remaining occurrence ce, the reconstructed tuples uxce\nj\n, uyce\nk\nare supported under\nthe original hidden-seed law: the full seed tuple comprising the fixed other seeds and this occurrence\nhas positive probability in that product law. They are exactly the own tuples at which the actual\nlists are evaluated.\nSuppose Ejk occurs and choose the ranks ρ, τ of labels witnessing it. The endpoint xce belongs\nto P(ij), and yce to P(ik). Compatibility requires the induced labeling to satisfy every source\nconstraint with both endpoints in the union of the selected scopes. In particular it satisfies the\nsampled occurrence ce. Its endpoint labels are exactly those used by Aρ,τ, so" } ], "occurrenceCount": 2 }, { "symbol": "τ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, "occurrences": [ { "blockId": "p1-96bdb33e48", "page": 1, "quote": "A vertex cover of a graph is a set of vertices meeting every edge. Its complement is an independent\nset, a set containing no two adjacent vertices. This elementary duality lets us express the hardness\ngap in terms of the densities of independent sets. For a finite simple undirected graph G, write τ(G)\nfor the minimum size of a vertex cover and n = |V (G)|. An α-approximation returns an actual\ncover of size at most ατ(G). The endpoints of a maximal matching give a factor-2 approximation.\nWe prove that every fixed improvement in this factor is NP-hard." }, { "blockId": "p1-0040e41d56", "page": 1, "quote": "the formula is satisfiable\n=⇒\nτ(G) <\n1" }, { "blockId": "p1-b633dc36e4", "page": 1, "quote": "the formula is unsatisfiable\n=⇒\nτ(G) >\n\n1 −1" }, { "blockId": "p9-faaf97dc8c", "page": 9, "quote": "τ(G) <\n1" }, { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." }, { "blockId": "p22-943351b541", "page": 22, "quote": "\nπc(Aρ,τ(xc)) = Aρ,τ(yc)\n\n≤σ.\n(7.5)" }, { "blockId": "p22-e0975c3db8", "page": 22, "quote": "For every actual remaining occurrence ce, the reconstructed tuples uxce\nj\n, uyce\nk\nare supported under\nthe original hidden-seed law: the full seed tuple comprising the fixed other seeds and this occurrence\nhas positive probability in that product law. They are exactly the own tuples at which the actual\nlists are evaluated.\nSuppose Ejk occurs and choose the ranks ρ, τ of labels witnessing it. The endpoint xce belongs\nto P(ij), and yce to P(ik). Compatibility requires the induced labeling to satisfy every source\nconstraint with both endpoints in the union of the selected scopes. In particular it satisfies the\nsampled occurrence ce. Its endpoint labels are exactly those used by Aρ,τ, so" }, { "blockId": "p22-7166649a70", "page": 22, "quote": "πce(Aρ,τ(xce)) = Aρ,τ(yce)." }, { "blockId": "p22-9e3b3dd195", "page": 22, "quote": "Corollary 7.2 (Soundness). For the graph constructed in Section 3, assume (6.2). If val(Φ) ≤σ,\nevery independent set has size strictly less than n/m. Consequently τ(G) > (1 −1/m)n." }, { "blockId": "p23-ed115d16da", "page": 23, "quote": "Suppose an algorithm always returns a vertex cover of size at most ατ(G). On a completeness\ninstance its output has size strictly less than" } ], "occurrenceCount": 10 }, { "symbol": "ω", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p13-ae279f868f", "page": 13, "quote": "The nested ranges just proved make the three projections in the second line pairwise orthogonal,\nand their sum is the identity. For a choice ω of one displayed projection at every seed, let fS,ω be\ntheir product applied to fS. For fixed S, these components give an orthogonal decomposition of\nfS. The seed projections commute with all weight projections, so every fS,ω remains in ran(ΠS).\nConsequently the components for different S’s are also orthogonal, even though the endpoint used\nto refine a seed can change with S. Thus\nX" }, "occurrences": [ { "blockId": "p13-ae279f868f", "page": 13, "quote": "The nested ranges just proved make the three projections in the second line pairwise orthogonal,\nand their sum is the identity. For a choice ω of one displayed projection at every seed, let fS,ω be\ntheir product applied to fS. For fixed S, these components give an orthogonal decomposition of\nfS. The seed projections commute with all weight projections, so every fS,ω remains in ran(ΠS).\nConsequently the components for different S’s are also orthogonal, even though the endpoint used\nto refine a seed can change with S. Thus\nX" }, { "blockId": "p13-3cfa9c14d3", "page": 13, "quote": "S,ω\n∥fS,ω∥2\n2 = ∥f∥2\n2,\n(5.6)" }, { "blockId": "p13-319e53e23f", "page": 13, "quote": "where henceforth these sums range over nonempty S and its allowed choices ω.\nLet TS,ω ⊆E be the seeds where the chosen projection is not P 0\ne . Let NS,ω ⊆[d] \\ {j(S)} be\nthe positions k for which the projection at e = {j(S), k} is Id −Oe,j(S). These are the incident\ncomponents not retained by the designated position’s own question. For each fixed e, applying\nId −P 0\ne keeps exactly the components with e ∈TS,ω. Orthogonality and the one-seed oscillation\nbound therefore give" }, { "blockId": "p13-46470989f4", "page": 13, "quote": "S,ω\n|TS,ω|∥fS,ω∥2\n2 =\nX" }, { "blockId": "p14-11c91b21e1", "page": 14, "quote": "For fixed S and k ̸= j(S), its seed decomposition shows that (Id −O{k,j(S)},j(S))fS is the sum of\nexactly the components with k ∈NS,ω. Consequently\nX" }, { "blockId": "p14-2de9842507", "page": 14, "quote": "S,ω\n|NS,ω|∥fS,ω∥2\n2 =\nX" }, { "blockId": "p15-5747044d54", "page": 15, "quote": "This identity uses only independent seed coordinates. Conditioning on the own questions restricts\neach incident hidden seed to its own fiber, as in the definition of Hj.\nConsider fS,ω and its designated position j = j(S). This component is preserved by the term\nKJ,j whenever" }, { "blockId": "p15-aba5150e1a", "page": 15, "quote": "S ∩J = {j},\nNS,ω ∩J = ∅,\nTS,ω ∩\nJ \\ {j}\n2" }, { "blockId": "p15-a71b6ad9bd", "page": 15, "quote": "S,ω\n1{conditions (5.12) hold}∥fS,ω∥2\n2.\n(5.13)" }, { "blockId": "p16-b015ba1dc9", "page": 16, "quote": "The first two conditions in (5.12) fail only if one of S \\ {j(S)} or NS,ω is included. The third can fail\nonly at a pair in TS,ω avoiding j(S). A union bound therefore gives conditional survival probability\nat least\n1 −p1\n\n|S| −1 + |NS,ω|\n\n−p2|TS,ω|.\n(5.14)" }, { "blockId": "p16-4b32130281", "page": 16, "quote": "Charging all of TS,ω only decreases this lower bound. The bound remains valid when the displayed\nexpression is negative.\nBy (5.5), (5.9), and the orthogonal refinement of each fS,\nX" }, { "blockId": "p16-7a2e765530", "page": 16, "quote": "\n|S| −1 + |NS,ω|\n\n∥fS,ω∥2\n2 ≤8d." }, { "blockId": "p16-889a85ba2b", "page": 16, "quote": "S,ω" } ], "occurrenceCount": 13 } ], "issues": [ { "id": "issue-1", "kind": "label-mismatch", "message": "Theorem 1.2 is used in a proof heading; the statement is Corollary 1.2.", "source": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "related": [ { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." } ] }, { "id": "issue-2", "kind": "label-mismatch", "message": "Text says Theorem 4.1; candidate heading is Lemma 4.1. Check the mismatch.", "source": { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, "related": [ { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" } ] }, { "id": "issue-3", "kind": "label-mismatch", "message": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch.", "source": { "blockId": "p3-04bc2f4316", "page": 3, "quote": "Retaining variance in own-question means.\nDiscard from a large independent set its at most\none vertex with vector norm at most t. The remaining vectors are separated from the negatives of\none another. The half-difference of their distance functions gives one fixed odd Lipschitz function\nF. Replace the grid weights by independent continuous uniform weights on cubes; rounding and\nthe Lipschitz bound control the error. On fresh random sums, F then has variance at least a fixed\nmultiple of d, as proved in Theorem 4.1.\nChoose a small batch J of h positions, freeze the weights outside it and the constraints on\npairs not contained in it, and average F conditional only on one remaining position’s own endpoint\nquestions and weights. Such averaging might erase the variance. Theorem 5.1 shows that a suitable\nrestriction retains total variance at least a fixed multiple of h in these own-question means. The\nproof decomposes over the independent weights and pair seeds. Its crucial estimate resamples every\nseed incident to one position while preserving the questions received at all other positions. Only the\nfirst position’s query can then change. This bounds the energy lost when the conditional means\nforget other positions’ questions, independently of the label alphabets." }, "related": [ { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" } ] }, { "id": "issue-4", "kind": "label-mismatch", "message": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch.", "source": { "blockId": "p3-e75cdd5634", "page": 3, "quote": "From variance to compatible lists.\nFor fixed batch queries, differentiate F composed with the\nsum of the position vectors. The full gradient lies in the convex hull of signed incidence vectors of\ncompatible label selections (Theorem 6.1). It carries a compatible representation, but can depend\non all batch inputs. Conversely, the derivatives of the own-question means have the required\ninformation restriction, but need not together lie in that same hull.\nThe cube variance bound in Theorem 6.2 gives enough squared energy in the own-question\ngradients. Each block has ℓ1 norm at most one, so thresholding its entries produces a short list\nwithout an alphabet-size factor. Conditional Jensen then transfers the retained list mass back to\nthe single full gradient before its compatible-incidence bound is used. If Z is the largest number of\nlists hit by one compatible selection, this gives EZ ≥2 (Theorem 6.3). The pair-decoding argument\nabove, summed over pairs of positions, gives EZ < 3/2 under source soundness (Theorem 7.1), a\ncontradiction." }, "related": [ { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" } ] }, { "id": "issue-5", "kind": "label-mismatch", "message": "Text says Theorem 6.2; candidate heading is Lemma 6.2. Check the mismatch.", "source": { "blockId": "p3-e75cdd5634", "page": 3, "quote": "From variance to compatible lists.\nFor fixed batch queries, differentiate F composed with the\nsum of the position vectors. The full gradient lies in the convex hull of signed incidence vectors of\ncompatible label selections (Theorem 6.1). It carries a compatible representation, but can depend\non all batch inputs. Conversely, the derivatives of the own-question means have the required\ninformation restriction, but need not together lie in that same hull.\nThe cube variance bound in Theorem 6.2 gives enough squared energy in the own-question\ngradients. Each block has ℓ1 norm at most one, so thresholding its entries produces a short list\nwithout an alphabet-size factor. Conditional Jensen then transfers the retained list mass back to\nthe single full gradient before its compatible-incidence bound is used. If Z is the largest number of\nlists hit by one compatible selection, this gives EZ ≥2 (Theorem 6.3). The pair-decoding argument\nabove, summed over pairs of positions, gives EZ < 3/2 under source soundness (Theorem 7.1), a\ncontradiction." }, "related": [ { "blockId": "p18-0ff91ffdb7", "page": 18, "quote": "Lemma 6.2 (Conditional cube variance). For each j ∈J and supported own tuple u, the function\nHj(u, ·) is 1-Lipschitz with respect to ∥· ∥∞. For almost every own s ∈[−1, 1]r," } ] }, { "id": "issue-6", "kind": "label-mismatch", "message": "Text says Theorem 6.3; candidate heading is Proposition 6.3. Check the mismatch.", "source": { "blockId": "p3-e75cdd5634", "page": 3, "quote": "From variance to compatible lists.\nFor fixed batch queries, differentiate F composed with the\nsum of the position vectors. The full gradient lies in the convex hull of signed incidence vectors of\ncompatible label selections (Theorem 6.1). It carries a compatible representation, but can depend\non all batch inputs. Conversely, the derivatives of the own-question means have the required\ninformation restriction, but need not together lie in that same hull.\nThe cube variance bound in Theorem 6.2 gives enough squared energy in the own-question\ngradients. Each block has ℓ1 norm at most one, so thresholding its entries produces a short list\nwithout an alphabet-size factor. Conditional Jensen then transfers the retained list mass back to\nthe single full gradient before its compatible-incidence bound is used. If Z is the largest number of\nlists hit by one compatible selection, this gives EZ ≥2 (Theorem 6.3). The pair-decoding argument\nabove, summed over pairs of positions, gives EZ < 3/2 under source soundness (Theorem 7.1), a\ncontradiction." }, "related": [ { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," } ] }, { "id": "issue-7", "kind": "label-mismatch", "message": "Text says Theorem 7.1; candidate heading is Lemma 7.1. Check the mismatch.", "source": { "blockId": "p3-e75cdd5634", "page": 3, "quote": "From variance to compatible lists.\nFor fixed batch queries, differentiate F composed with the\nsum of the position vectors. The full gradient lies in the convex hull of signed incidence vectors of\ncompatible label selections (Theorem 6.1). It carries a compatible representation, but can depend\non all batch inputs. Conversely, the derivatives of the own-question means have the required\ninformation restriction, but need not together lie in that same hull.\nThe cube variance bound in Theorem 6.2 gives enough squared energy in the own-question\ngradients. Each block has ℓ1 norm at most one, so thresholding its entries produces a short list\nwithout an alphabet-size factor. Conditional Jensen then transfers the retained list mass back to\nthe single full gradient before its compatible-incidence bound is used. If Z is the largest number of\nlists hit by one compatible selection, this gives EZ ≥2 (Theorem 6.3). The pair-decoding argument\nabove, summed over pairs of positions, gives EZ < 3/2 under source soundness (Theorem 7.1), a\ncontradiction." }, "related": [ { "blockId": "p21-70de857a8d", "page": 21, "quote": "Lemma 7.1 (Decoding a pair). For every fixed J, a and fixed list maps satisfying Equation (7.2)," } ] }, { "id": "issue-8", "kind": "label-mismatch", "message": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch.", "source": { "blockId": "p7-df63b72b49", "page": 7, "quote": "Proof. Let L be the source bit length. The fixed integers m, d determine p and the rational number t\nby Equation (2.2). The fixed alphabets determine q and r = qd−1. All these quantities are therefore\ncomputable constants independent of L. Since the constraint list is explicit, M = O(L), and (3.6)\ngives n = O(L(d\n2)). A vertex description contains a fixed number of source indices and grid entries.\nThus all vertices can be enumerated and assigned identifiers of O(1 + log n) bits.\nApply Theorem 3.2 to each unordered pair of distinct vertices. All label and selection enu-\nmerations have a fixed number of choices. Variable comparisons, sorting, constraint scans, and\nprojection-map evaluations take polynomial bit time in L. The arithmetic of the edge test uses\nonly rational grid entries. In fact, every coordinate of Sv + Sw has denominator p, after summing\nall repeated-block contributions. The local maximum is therefore ζ/p for a nonnegative integer ζ.\nBy (3.2) and the triangle inequality, ζ/p ≤2d, so its numerator and denominator have bit lengths\nbounded solely in terms of the fixed parameters. Since t =\n√" }, "related": [ { "blockId": "p7-bd0f6459db", "page": 7, "quote": "Lemma 3.2 (Local evaluation of the norm). Let B ⊆Q be a set of query blocks supporting z ∈RP.\nThen" } ] }, { "id": "issue-9", "kind": "label-mismatch", "message": "Text says Theorem 3.4; candidate heading is Lemma 3.4. Check the mismatch.", "source": { "blockId": "p9-c90bc9285c", "page": 9, "quote": "For a uniform vertex, conditional on the seed tuple, the selected slot in each weight vector is fixed.\nIts entries are independent uniform elements of T , one from each independent sj. Coincident\nendpoint questions or slot numbers do not change this independence. By Theorem 3.4, more than a\n(1/2 −1/m) fraction of vertices are kept.\nFor two distinct kept vertices v, w, evaluating Sv + Sw on IA gives exactly the sum of their two\nexpressions in (3.9). This includes all contributions even when the same query appears in both\nvertices. The sum exceeds 2t, so (3.1) rules out the edge in (3.5). The kept vertices are therefore\nindependent. Their complement is a vertex cover of size less than (1/2 + 1/m)n." }, "related": [ { "blockId": "p8-e6ad8608a9", "page": 8, "quote": "Lemma 3.4 (Midpoint-grid tail). If T1, . . . , Td are independent uniform elements of T , then" } ] }, { "id": "issue-10", "kind": "label-mismatch", "message": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch.", "source": { "blockId": "p16-3ddfbe0fc9", "page": 16, "quote": "The joint parameter choice in Section 8 will ensure these conditions. Applying Theorem 5.1 to the\nfunction in Theorem 4.1 supplies a fixed set J ⊂[d] of size h and fixed data a consisting of the\nweights outside J and every seed outside EJ =\nJ\n2\n\n. Write f∗(cJ, sJ) for f with these values plugged\nin. In the rest of this section the probability law is the product of independent uniform hidden seeds\n(ce)e∈EJ and independent uniform weight blocks (sj)j∈J on [−1, 1]r. The choice of the fixed data\ndoes not condition these remaining draws on any event of success.\nFor j ∈J, write uj = (qjk)k∈J\\{j}, in increasing order of k, for the tuple of questions received\nby j from its incident hidden seeds. Together with a, it specifies the whole query ij; denote the\ncompleted query by ij(uj). The conditional averages supplied by the restriction lemma satisfy\nX" }, "related": [ { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" } ] }, { "id": "issue-11", "kind": "label-mismatch", "message": "Text says Theorem 4.1; candidate heading is Lemma 4.1. Check the mismatch.", "source": { "blockId": "p16-3ddfbe0fc9", "page": 16, "quote": "The joint parameter choice in Section 8 will ensure these conditions. Applying Theorem 5.1 to the\nfunction in Theorem 4.1 supplies a fixed set J ⊂[d] of size h and fixed data a consisting of the\nweights outside J and every seed outside EJ =\nJ\n2\n\n. Write f∗(cJ, sJ) for f with these values plugged\nin. In the rest of this section the probability law is the product of independent uniform hidden seeds\n(ce)e∈EJ and independent uniform weight blocks (sj)j∈J on [−1, 1]r. The choice of the fixed data\ndoes not condition these remaining draws on any event of success.\nFor j ∈J, write uj = (qjk)k∈J\\{j}, in increasing order of k, for the tuple of questions received\nby j from its incident hidden seeds. Together with a, it specifies the whole query ij; denote the\ncompleted query by ij(uj). The conditional averages supplied by the restriction lemma satisfy\nX" }, "related": [ { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" } ] }, { "id": "issue-12", "kind": "label-mismatch", "message": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch.", "source": { "blockId": "p18-058a961704", "page": 18, "quote": "Here the notation for gj means that the seed tuple is used only to form the queries supplied to the\nrule in Theorem 6.1. Set ¯gj(u, s) = 0 for unsupported own tuples. For supported u, its query ij(u)\nis fixed throughout the integral. Convexity and (6.6) give, at every own input," }, "related": [ { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" } ] }, { "id": "issue-13", "kind": "label-mismatch", "message": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch.", "source": { "blockId": "p19-a8b14455ed", "page": 19, "quote": "Proof. Replacing the own block s by s′, with a seed configuration and other weights fixed, changes\nthe integrand in (6.4) by at most ∥s −s′∥∞. The same bound holds after averaging. In particular,\nevery scalar-coordinate difference quotient of the integrand is bounded in absolute value by one.\nFor each fixed hidden seed tuple, the full gradient identification in Theorem 6.1 holds outside\na null set in all batch weights. Fubini implies that for almost every own s it holds for almost all\nother weight blocks. The conditional seed law κj(u) has finite support and does not depend on s, so\nthe exceptional sets may be united over its support and over the finitely many weight coordinates.\nAt the remaining own values, the coordinate difference quotients converge almost surely to gj,t′.\nDominated convergence gives (6.10); the boundary of the cube has measure zero.\nEvery coordinate-line restriction of Hj(u, ·) is Lipschitz and hence absolutely continuous. By\nFubini applied to (6.10), for almost every fixing of the other own coordinates its derivative agrees\nalmost everywhere with the corresponding entry of ¯gj. The fundamental theorem of calculus\ntherefore expresses differences between any two values on such a line as the integral of that entry.\nLet w0 be the restriction of Hj(u, ·) to one of these lines. For x, y ∈[−1, 1], Cauchy–Schwarz\ngives" }, "related": [ { "blockId": "p17-13be592474", "page": 17, "quote": "Lemma 6.1 (A gradient rule in the compatibility hull). There is a deterministic measurable rule\ng = (gj)j∈J ∈RJ×[r], defined at every batch query and real-weight tuple, that equals ∇ψi outside a\nfinite union of proper affine hyperplanes for each fixed query tuple and fixed frozen data. At every\ntuple,\ng ∈conv{±vI(i) : I ∈I}.\n(6.5)" } ] }, { "id": "issue-14", "kind": "label-mismatch", "message": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch.", "source": { "blockId": "p21-ae56abe745", "page": 21, "quote": "and assume throughout this section that val(Φ) ≤σ. The source promise bounds every labeling\nof U ∪V , so the decoders below may use arbitrary fixed instance-dependent data. Their essential\nrestriction is that the label returned at a variable does not also depend on the sampled constraint\noccurrence.\nFix a set J ⊆[d] of size h and frozen data a as in Theorem 5.1. The remaining seeds ce, e ∈\nJ\n2\n\n,\nare independent uniform elements of [M]; the weight blocks sj, j ∈J, are independent uniforms on\n[−1, 1]r, independent of the seeds. Let uj be position j’s tuple of questions from its incident hidden\nseeds. Together with a, this tuple determines its query ij.\nConsider any fixed total measurable maps" }, "related": [ { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" } ] }, { "id": "issue-15", "kind": "label-mismatch", "message": "Text says Theorem 6.3; candidate heading is Proposition 6.3. Check the mismatch.", "source": { "blockId": "p21-7b7e7e18ec", "page": 21, "quote": "Each map receives only the indicated own questions and weights; on an unsupported own-question\ntuple it returns the empty list. Order each list by its slots. The lists in Theorem 6.3 have exactly\nthese properties.\nFor distinct j, k ∈J, let Ejk be the event that some I ∈I hits both lists: it contains (ij, λj)\nand (ik, λk) for labels λj ∈Lj, λk ∈Lk. Define" }, "related": [ { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," } ] }, { "id": "issue-16", "kind": "label-mismatch", "message": "Text says Theorem 6.3; candidate heading is Proposition 6.3. Check the mismatch.", "source": { "blockId": "p21-e0837c3d44", "page": 21, "quote": "Proof. Fix j < k and write e = {j, k}. Fix all batch weights and all hidden seeds except ce. By\nindependence, the remaining occurrence ce is still uniform on [M]. Position j’s only varying question\nis xce ∈U, and position k’s only varying question is yce ∈V . Every other argument of each list\nmap is now a constant, while the maps themselves remain fixed. For the maps constructed in\nTheorem 6.3, their conditional averages are not recomputed: the actual values just fixed are not\nsupplied as new arguments to those averages.\nFor x ∈U, form the tuple ux\nj by putting x in the distinguished e-coordinate of uj and retaining\nall its other, now fixed, coordinates. Define uy\nk for y ∈V in the same way. Choose default symbols\naU ∈ΣU and aV ∈ΣV . For each pair of ranks ρ, τ ∈[ℓ], define a labeling Aρ,τ of the entire source\ninstance as follows. At a variable x ∈U, evaluate Lj(ux\nj , sj). If rank ρ is present, give x its value\nin that label; otherwise give it aU. At a variable y ∈V , use its value in the label at rank τ of\nLk(uy\nk, sk), or aV if the rank is absent." }, "related": [ { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," } ] }, { "id": "issue-17", "kind": "label-mismatch", "message": "Text says Theorem 4.1; candidate heading is Lemma 4.1. Check the mismatch.", "source": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "related": [ { "blockId": "p10-196f045160", "page": 10, "quote": "Lemma 4.1 (Separation, variance, and oscillation). Suppose an independent set contains at least\nb0n vertices with ∥Sv∥∗> t. There is a fixed nonempty finite set A∗⊂RP such that" } ] }, { "id": "issue-18", "kind": "label-mismatch", "message": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch.", "source": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "related": [ { "blockId": "p12-8d649b928d", "page": 12, "quote": "Lemma 5.1 (Own-question variance under a restriction). Let ν > 0 and let h be an integer with\n2 ≤h ≤d. Suppose the bounded Borel measurable function f on Ωsatisfies" } ] }, { "id": "issue-19", "kind": "label-mismatch", "message": "Text says Theorem 6.3; candidate heading is Proposition 6.3. Check the mismatch.", "source": { "blockId": "p22-4e91acb171", "page": 22, "quote": "Theorem 4.1 therefore supplies the single fixed odd Lipschitz function F and its variance lower\nbound. Applying Theorem 5.1 and then the list construction of Theorem 6.3 and Equation (6.15)\ngives fixed J, a and local lists with" }, "related": [ { "blockId": "p20-a53d542367", "page": 20, "quote": "Proposition 6.3 (Local lists and compatible hits). The functions Lj are deterministic total\nmeasurable maps returning at most ℓvalid labels. Once Φ, F, J, a and the conditional kernels above\nare fixed, each list uses only its own questions uj and weights sj. In particular it has no additional\naccess to the actual hidden seed occurrences or to any other actual batch input. Under the fixed\nrestricted product law," } ] }, { "id": "issue-20", "kind": "label-mismatch", "message": "Text says Theorem 7.1; candidate heading is Lemma 7.1. Check the mismatch.", "source": { "blockId": "p22-a7124c215a", "page": 22, "quote": "This contradicts Theorem 7.1, which gives EZ < 3/2 under the same remaining product law.\nThe lower bound just used averages over the hidden seeds and continuous batch weights for\nthose fixed J, a. The upper bound fixes further weights and seeds only to decode a single occurrence" }, "related": [ { "blockId": "p21-70de857a8d", "page": 21, "quote": "Lemma 7.1 (Decoding a pair). For every fixed J, a and fixed list maps satisfying Equation (7.2)," } ] }, { "id": "issue-21", "kind": "label-mismatch", "message": "Text says Theorem 1.2; candidate heading is Corollary 1.2. Check the mismatch.", "source": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "related": [ { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." } ] }, { "id": "issue-22", "kind": "label-mismatch", "message": "Text says Theorem 3.3; candidate heading is Proposition 3.3. Check the mismatch.", "source": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "related": [ { "blockId": "p7-44df83b758", "page": 7, "quote": "Proposition 3.3 (Deterministic polynomial construction). For a fixed integer m ≥4, a fixed even\nsquare d satisfying Equation (2.1), and fixed source alphabets, the graph G can be output explicitly\nin deterministic time polynomial in the source bit length. Every edge test is exact." } ] }, { "id": "issue-23", "kind": "label-mismatch", "message": "Text says Theorem 3.5; candidate heading is Proposition 3.5. Check the mismatch.", "source": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "related": [ { "blockId": "p9-0650d48b6b", "page": 9, "quote": "Proposition 3.5 (Completeness). If val(Φ) = 1, then G has an independent set of size greater than\n(1/2 −1/m)n. Consequently," } ] }, { "id": "issue-24", "kind": "label-mismatch", "message": "Text says Theorem 7.2; candidate heading is Corollary 7.2. Check the mismatch.", "source": { "blockId": "p23-f8f1712bec", "page": 23, "quote": "Proof of Theorems 1.1 and 1.2. Fix m ≥4 and the constants in Equation (8.1). Apply Theorem 2.1\nat this σ to obtain fixed source alphabets and a polynomial-time reduction from 3SAT. Now define\nq, r from those alphabets and construct the graph. Theorem 3.3 makes its vertex and edge lists\nexplicit in deterministic polynomial time. All the hypotheses of Theorems 3.5 and 7.2 have been\nverified above, so these results give the two cover bounds in Theorem 1.1.\nFor a fixed real α ∈[1, 2), choose a fixed integer m ≥4 such that" }, "related": [ { "blockId": "p22-9e3b3dd195", "page": 22, "quote": "Corollary 7.2 (Soundness). For the graph constructed in Section 3, assume (6.2). If val(Φ) ≤σ,\nevery independent set has size strictly less than n/m. Consequently τ(G) > (1 −1/m)n." } ] }, { "id": "issue-25", "kind": "label-mismatch", "message": "Text says Theorem 1.2; candidate heading is Corollary 1.2. Check the mismatch.", "source": { "blockId": "p24-38dac4ff9d", "page": 24, "quote": "uses only integer arithmetic: multiply the cardinality by m and compare with (m −1)n. The real\nnumber α is used only to choose the fixed integer m; it is not an input to the reduction. This proves\nTheorem 1.2." }, "related": [ { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." } ] }, { "id": "issue-26", "kind": "label-mismatch", "message": "Text says Theorem 1.2; candidate heading is Corollary 1.2. Check the mismatch.", "source": { "blockId": "p24-8bda18762a", "page": 24, "quote": "For completeness, the matching upper bound on the approximation factor is elementary. Find a\nmaximal matching and return the set of all its endpoints. Maximality makes this set a vertex cover:\nan edge with neither endpoint selected could be added to the matching. Every vertex cover contains\nan endpoint of each matching edge, and these edges are disjoint. The returned cover therefore has\nsize at most twice optimum. Together with Theorem 1.2, this gives the constant approximation\nthreshold 2, unless P = NP." }, "related": [ { "blockId": "p1-2674c769e7", "page": 1, "quote": "Corollary 1.2. For every fixed real α with 1 ≤α < 2, approximating Vertex Cover within factor α\nis NP-hard. Thus, unless P = NP, no deterministic polynomial-time algorithm achieves factor 2 −ε\nfor any fixed ε ∈(0, 1]." } ] } ], "coverage": { "pages": 25, "blocks": 529, "method": "Numbered headings, explicit cross-references, nearby proof boundaries, and candidate notation contexts. PDF text can be incomplete.", "externalSourcesRetrieved": false, "enrichment": "Agent-authored source-linked notes on selected results. Remaining extraction is automatic.", "reviewedReferenceCorrections": 0 } }