{ "schemaVersion": 1, "generator": "readable-paper 0.1.4", "paperId": "e79e9ce01cf7", "version": "adc7f1241b42e322a6451854ab7e4b4c146bf78a", "sha256": "32a11e793f6e379c8d750f8f3560a6b0648bee56441efd25bceea9c2b7ffb849", "title": "A Direct Proof of Optimal Max-Cut Hardness", "status": "Unverified reading map", "nodes": [ { "id": "theorem-1", "kind": "theorem", "label": "Theorem 1.1", "source": { "blockId": "p1-9ee5bebb0f", "page": 1, "quote": "Theorem 1.1. For every fixed α with αGW < α ≤1, it is NP-hard to approximate Max-Cut\nwithin a factor of α. This holds for simple unweighted graphs." }, "statement": [ { "blockId": "p1-9ee5bebb0f", "page": 1, "quote": "Theorem 1.1. For every fixed α with αGW < α ≤1, it is NP-hard to approximate Max-Cut\nwithin a factor of α. This holds for simple unweighted graphs." } ], "proof": [ { "blockId": "p27-a71d41604b", "page": 27, "quote": "Proof of Theorem 1.1. Substituting ρ = −t into (1.1) gives" }, { "blockId": "p27-b50f9361a2", "page": 27, "quote": "αGW =\nmin\n−1 1 + B(t)" }, { "blockId": "p27-84df7a9631", "page": 27, "quote": "2\n." }, { "blockId": "p27-fdadd86d20", "page": 27, "quote": "Choose positive rational ε and ϵ′ small enough that ε < (t −B(t))/4 and" }, { "blockId": "p27-c6f29518ac", "page": 27, "quote": "α\n1 + t" }, { "blockId": "p27-193eb5417a", "page": 27, "quote": "2\n−ε −ϵ′\n\n> 1 + B(t)" }, { "blockId": "p27-f5698fca36", "page": 27, "quote": "2\n+ ε + ϵ′.\n(9.2)" }, { "blockId": "p27-7abf2de4cc", "page": 27, "quote": "Apply Theorem 1.2. Delete loops while retaining the original weight scale, so that the remaining\nweights sum to at most one. Apply Theorem B.1 with error ϵ′, producing a simple unweighted\ngraph whose maximum cut, on the common scale Q, differs from the original maximum cut\nweight by at most ϵ′. An α-approximation algorithm on that graph returns, in the YES case, a\ncut of scaled value at least the left side of (9.2). In the NO case, no cut has scaled value greater\nthan its right side. A fixed rational threshold strictly between them therefore distinguishes the\nNP-hard promise cases. This proves the theorem." } ], "guide": [ { "text": "The paper claims NP-hardness of approximation above the Goemans–Williamson threshold, for simple unweighted graphs. This map records the claim; it does not certify it.", "sources": [ { "blockId": "p1-9ee5bebb0f", "page": 1, "quote": "Theorem 1.1. For every fixed α with αGW < α ≤1, it is NP-hard to approximate Max-Cut\nwithin a factor of α. This holds for simple unweighted graphs." } ] }, { "text": "The final proof substitutes ρ = −t into the threshold formula, rewriting the ratio in terms of B(t). The following paragraph argues that the minimum is attained for 0 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" } ] } ], "guideStatus": "Agent-authored reading guide; not independently verified." }, { "id": "theorem-2", "kind": "theorem", "label": "Theorem 1.2", "source": { "blockId": "p2-2780d2e59a", "page": 2, "quote": "Theorem 1.2. Fix a rational t ∈(0, 1) and a constant 0 < ε < (t−B(t))/4. For graphs equipped\nwith a rational probability distribution on edges, it is NP-hard to distinguish" }, "statement": [ { "blockId": "p2-2780d2e59a", "page": 2, "quote": "Theorem 1.2. Fix a rational t ∈(0, 1) and a constant 0 < ε < (t−B(t))/4. For graphs equipped\nwith a rational probability distribution on edges, it is NP-hard to distinguish" }, { "blockId": "p2-8ccbd48dcc", "page": 2, "quote": "YES:\nVal ≥1 + t" }, { "blockId": "p2-b8ed68c6cc", "page": 2, "quote": "2\n−ε," }, { "blockId": "p2-9875f02b11", "page": 2, "quote": "NO:\nVal ≤1 + B(t)" }, { "blockId": "p2-3f56d83e08", "page": 2, "quote": "2\n+ ε," }, { "blockId": "p2-dffbfd6960", "page": 2, "quote": "where Val is the maximum cut weight." } ], "proof": [ { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, { "blockId": "p26-719578cdd9", "page": 26, "quote": "0 < b < 2−2s p∗θ" }, { "blockId": "p26-92eaaa5414", "page": 26, "quote": "2 ." }, { "blockId": "p26-842530a02d", "page": 26, "quote": "By Theorem 2.2, this fixes an alphabet dimension n for which the game has arbitrarily small\ncompleteness error. Choose an integer T satisfying" }, { "blockId": "p26-7a6c9e8358", "page": 26, "quote": "T ≥2n(2k−2s)" }, { "blockId": "p26-b35054634f", "page": 26, "quote": "(p∗θ)2 ." }, { "blockId": "p26-8659c32a43", "page": 26, "quote": "Finally choose a positive rational a < 1 −b small enough that 2Ta ≤ε/4, and start with the\nNP-hard game gap Val(G) ≥1 −a versus Val(G) ≤b." }, { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, { "blockId": "p27-74a15e2a80", "page": 27, "quote": "vertices. All factors except |X| are constants for fixed t, ε. There are polynomially many\nendpoint and edge tuples, and all word-generation sample spaces have constant size. Enumerate\nevery test outcome and add its probability to the corresponding edge weight. The initial game\nweights, t, and r are rational, so these weights have polynomial bit complexity. Conditional edge\nprobabilities also have polynomial bit complexity. All alphabets and words can be enumerated\nin constant time relative to the input size. The resulting graph is therefore computable in\ndeterministic polynomial time. It has a rational edge distribution and satisfies the asserted,\nweaker ε-loss bounds." } ], "guide": [] }, { "id": "lemma-3", "kind": "lemma", "label": "Lemma 2.1", "source": { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." }, "statement": [ { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." } ], "proof": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p5-ac48b29911", "page": 5, "quote": "0 ≤Stabt(h) −Stabt(hu) ≤sup\nj≥0\ntj(1 −u2j) −→0\n(u ↑1)." }, { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, { "blockId": "p5-3a3de3600b", "page": 5, "quote": "Apply the untruncated theorem to hu and add the stability error." } ], "guide": [ { "text": "The proof starts from the externally cited Majority Is Stablest result, [16, Theorem 4.4]. That external statement is not loaded into this map.", "sources": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," } ] }, { "text": "The substitution g=(h+1)/2 changes the mean from 0 to 1/2 and halves each nonconstant Fourier coefficient. The stated stability identity Stabt(h)=4 Stabt(g)−1 accounts for the changed normalization.", "sources": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," } ] }, { "text": "The proof then introduces hu=Tu h to pass from a degree-cutoff hypothesis to the untruncated assertion. Read the tail estimates after this paragraph before accepting that transition.", "sources": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," } ] } ], "guideStatus": "Agent-authored reading guide; not independently verified." }, { "id": "proposition-4", "kind": "proposition", "label": "Proposition 2.2", "source": { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" }, "statement": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" }, { "blockId": "p6-9fbf48bbae", "page": 6, "quote": "πe : Fn+1\n2\n−→Fn\n2" }, { "blockId": "p6-e527580c8a", "page": 6, "quote": "on each edge e = (x, y). An edge is satisfied when πe(λX(x)) = λY (y). The promise cases are" }, { "blockId": "p6-cadeb61e43", "page": 6, "quote": "YES :\nVal(G) ≥1 −a,\nNO :\nVal(G) ≤b." }, { "blockId": "p6-be51df5284", "page": 6, "quote": "The integer n depends on b alone. For fixed a, b, the reduction is deterministic and has polynomial\nrunning time and output size." } ], "proof": [], "guide": [] }, { "id": "definition-5", "kind": "definition", "label": "Definition 3.1", "source": { "blockId": "p6-f809474603", "page": 6, "quote": "Definition 3.1 (Tree code). Fix integers s ≥2 and m ≥1, and put" }, "statement": [ { "blockId": "p6-f809474603", "page": 6, "quote": "Definition 3.1 (Tree code). Fix integers s ≥2 and m ≥1, and put" }, { "blockId": "p6-9bad825090", "page": 6, "quote": "k = (2s)m,\nd = sm." }, { "blockId": "p6-552df38005", "page": 6, "quote": "Let T be the complete rooted 2s-ary tree of depth m. The children of each internal node v are\nindexed by (g, b) ∈[s] × {0, 1}. Its leaves are indexed by [k]. To each internal node assign a\ntable Fv : Fs\n2 →{−1, 1}, and to each leaf assign a row Ri ∈H. The word W(R, F) ∈{−1, 1}A" }, { "blockId": "p7-d0f3eb1713", "page": 7, "quote": "" }, { "blockId": "p7-6027f8900c", "page": 7, "quote": "is defined pointwise as follows. At a ∈A, leaf i has sign (−1)Ri(a). An internal node with child\nsigns (Zg,b) outputs\n" }, { "blockId": "p7-61f19e4119", "page": 7, "quote": "\ns\nY" }, { "blockId": "p7-477ca2f74c", "page": 7, "quote": "Fv(z),\n(−1)zg = Zg,0Zg,1.\n(3.1)" }, { "blockId": "p7-c9ba14f2dd", "page": 7, "quote": "g=1\nZg,0" }, { "blockId": "p7-e01a1112a3", "page": 7, "quote": "The root output is W(R, F)(a)." } ], "proof": [], "guide": [] }, { "id": "lemma-6", "kind": "lemma", "label": "Lemma 3.2", "source": { "blockId": "p7-aa2f88a795", "page": 7, "quote": "Lemma 3.2 (Complementary traversal and correlation). For every fixed collection of tables\nF, the root output as a multilinear polynomial in the k leaf signs has only degree-d monomials.\nEach such monomial chooses one child from every pair at every node that it reaches. Every\nresulting traversal meets every possible slice D at exactly one node. Consequently, negating all\nthe tables on a slice negates W.\nLet 0 < t < 1 and 0 < r < 1. Draw F uniformly, and construct F ′ by leaving the tables\noutside D unchanged and coupling each pair of corresponding table entries on D as uniform signs\nof correlation t, independently across entries. Independently draw R ∈Hk uniformly, and form\nR1, R2 by retaining each row of R with probability r and otherwise replacing it by an independent\nuniform row, independently in the two copies. Then, for every a ∈A and every fixed slice," }, "statement": [ { "blockId": "p7-aa2f88a795", "page": 7, "quote": "Lemma 3.2 (Complementary traversal and correlation). For every fixed collection of tables\nF, the root output as a multilinear polynomial in the k leaf signs has only degree-d monomials.\nEach such monomial chooses one child from every pair at every node that it reaches. Every\nresulting traversal meets every possible slice D at exactly one node. Consequently, negating all\nthe tables on a slice negates W.\nLet 0 < t < 1 and 0 < r < 1. Draw F uniformly, and construct F ′ by leaving the tables\noutside D unchanged and coupling each pair of corresponding table entries on D as uniform signs\nof correlation t, independently across entries. Independently draw R ∈Hk uniformly, and form\nR1, R2 by retaining each row of R with probability r and otherwise replacing it by an independent\nuniform row, independently in the two copies. Then, for every a ∈A and every fixed slice," }, { "blockId": "p7-6aee721e9d", "page": 7, "quote": "E\nW(R1, F)(a)W(R2, F ′)(a) | D\n = tr2d.\n(3.2)" }, { "blockId": "p7-0bbbca177d", "page": 7, "quote": "In particular, if r = 1 −µ/d with 0 < µ < 1/2, then r2d ≥1 −2µ." } ], "proof": [ { "blockId": "p8-5a286b54c0", "page": 8, "quote": "Proof. For a fixed gate table, expand in its argument z: Fv(z) = P\nI⊆[s] bFv(I)(−1)\nP" }, { "blockId": "p8-64d8f6ec74", "page": 8, "quote": "g∈I zg.\nSubstituting into Equation (3.1), the term indexed by I is" }, { "blockId": "p8-57d2659481", "page": 8, "quote": "bFv(I)\nY" }, { "blockId": "p8-54055171f0", "page": 8, "quote": "g∈I\nZg,1." }, { "blockId": "p8-0d5161f6e9", "page": 8, "quote": "g /∈I\nZg,0\nY" }, { "blockId": "p8-e0de6afd1d", "page": 8, "quote": "Iterating this expansion gives" }, { "blockId": "p8-f585bdc4f9", "page": 8, "quote": "WF (x) =\nX" }, { "blockId": "p8-4efa298442", "page": 8, "quote": "i∈P\nxi,\n|P| = d,\n(3.3)" }, { "blockId": "p8-6af3299821", "page": 8, "quote": "P∈P\naP (F)\nY" }, { "blockId": "p8-bcf0ef5bfe", "page": 8, "quote": "where P is the family of leaf sets produced by choosing one child per pair at each reached internal\nnode. The leaf set determines all these choices: a selected child has a nonempty intersection\nwith that leaf set, whereas an unselected child has none. Thus different traversals give different\nmonomials, and aP (F) is the product of one Walsh coefficient of Fv for each node reached by\nthe traversal.\nAt the root both a monomial traversal and the slice exploration are present. At any common\nnode above the slice depth, exactly one child is selected by both explorations. Induction gives\nexactly one common node at the slice depth. Hence each aP contains exactly one factor from a\ntable on D. Negating every such table negates each coefficient in Equation (3.3) and therefore\nnegates the word. Moreover, each Walsh coefficient of a slice table satisfies" }, { "blockId": "p8-1d56f60d7e", "page": 8, "quote": "E[ bF ′\nv(I) | Fv] = t bFv(I)." }, { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, { "blockId": "p8-74ab4e8894", "page": 8, "quote": "The correlation identity describes a cut that evaluates the word at one label. To analyze\narbitrary cuts, we also need functions that can depend on the entire word but do not see its\ngenerating rows or tables. We call f a function of the code if" }, { "blockId": "p8-7efa825524", "page": 8, "quote": "f(R, F) = ϕ(W(R, F))\n(3.4)" }, { "blockId": "p8-4481b3a5f3", "page": 8, "quote": "for a fixed function ϕ on words. This class is closed under averaging over fixed choices of ϕ,\nincluding choices that first pull a word back through a fixed affine map. For r ∈[0, 1], let\nTr denote independent row resampling with retention probability r, and write ¯f = Trf. The\noperator preserves bounds on the range of f. If ϕ is odd, Lemma 3.2 shows that both f and ¯f\nare odd under simultaneous negation of the slice tables. In particular, they have mean zero in\nthose table entries when all rows and outside tables are fixed.\nAt these fixed data, the remaining random input is the cube of |D|2s independent entries,\nindexed by (v, z) ∈D × Fs\n2. Fourier degree and influence on this cube refer to subsets of\nentries. This differs from the Walsh expansion of a fixed gate in its s-bit argument z used in the\ncorrelation proof." } ], "guide": [] }, { "id": "lemma-7", "kind": "lemma", "label": "Lemma 4.1", "source": { "blockId": "p9-a96718c718", "page": 9, "quote": "Lemma 4.1 (Completeness). If Val(G) ≥1 −a, the graph has a cut of value at least" }, "statement": [ { "blockId": "p9-a96718c718", "page": 9, "quote": "Lemma 4.1 (Completeness). If Val(G) ≥1 −a, the graph has a cut of value at least" }, { "blockId": "p9-04c0cecab0", "page": 9, "quote": "(1 −2Ta)1 + tr2d" }, { "blockId": "p9-4ccbfbe000", "page": 9, "quote": "2\n≥1 + t" }, { "blockId": "p9-42fcfc6700", "page": 9, "quote": "2\n−tµ −2Ta.\n(4.2)" }, { "blockId": "p9-c3890e6117", "page": 9, "quote": "The first lower bound is useful when 2Ta ≤1, which is the regime used below." } ], "proof": [ { "blockId": "p9-c569ea464e", "page": 9, "quote": "Proof. Choose game labels ℓX, ℓY satisfying an edge with probability at least 1 −a. Label each\nword vertex by evaluation at the tuple of left labels:" }, { "blockId": "p9-ae0ee21fa9", "page": 9, "quote": "σx(w) = w\nℓX(x1), . . . , ℓX(xT )\n." }, { "blockId": "p10-0e07627ed1", "page": 10, "quote": "Each coordinate of either sampled edge tuple has the original edge law.\nA union bound\nshows that all 2T sampled constraints are satisfied except with probability at most 2Ta.\nWhen all these constraints are satisfied, the two queried tuples project to the same label\nl = (ℓY (y1), . . . , ℓY (yT )) ∈A. The cut signs then have product −W(R1, F)(l)W(R2, F ′)(l). The\nword-generation data are independent of the game edges, so Theorem 3.2 gives conditional cut\nprobability (1 + tr2d)/2. On the remaining event the cut probability is nonnegative. This proves\nthe first bound when 2Ta ≤1; if 2Ta > 1 that bound is negative and is automatic. Finally,\nr2d ≥1 −2µ, and the loss due to the prefactor is at most 2Ta, giving the second bound." }, { "blockId": "p10-8edb124402", "page": 10, "quote": "4.2\nA large cut produces excess stability" }, { "blockId": "p10-d2c31bf4c3", "page": 10, "quote": "An arbitrary cut need not change sign when its queried word is negated. We separate its two\nparities because the negative second query makes odd dependence increase the cut value and\neven dependence decrease it. For each cut table define its even and odd parts by" }, { "blockId": "p10-47320fe400", "page": 10, "quote": "σeven(w) = σ(w) + σ(−w)" }, { "blockId": "p10-67e4e14f7d", "page": 10, "quote": "2\n,\nσo(w) = σ(w) −σ(−w)" }, { "blockId": "p10-f3004d6c04", "page": 10, "quote": "2\n." }, { "blockId": "p10-5aaa4b15bd", "page": 10, "quote": "For an edge tuple and a right-endpoint tuple, set" }, { "blockId": "p10-faaa210733", "page": 10, "quote": "Ge(R, F) = σo\nx(e)(W(R, F) ◦πe),\n(4.3)" }, { "blockId": "p10-872453b709", "page": 10, "quote": "Gy(R, F) = Ee|yGe(R, F),\n¯Gy = TrGy.\n(4.4)" }, { "blockId": "p10-c7b1b1841b", "page": 10, "quote": "Write Ay and ¯Ay = TrAy for the analogous averages using σeven. These functions are bounded\nin absolute value by one. Moreover," }, { "blockId": "p10-914b23de3b", "page": 10, "quote": "Gy(R, F) = ϕy(W(R, F)),\nϕy(w) = Ee|yσo\nx(e)(w ◦πe)," }, { "blockId": "p10-12c4ff12f9", "page": 10, "quote": "where ϕy is odd and depends on no slice or row data. Thus ¯Gy has mean zero in the slice entries\nfor every fixed y, D, R, Fout, by the observation following Theorem 3.2. The later extraction\nargument, Theorem 7.1, applies to precisely such bounded odd functions of the generated word." } ], "guide": [] }, { "id": "lemma-8", "kind": "lemma", "label": "Lemma 4.2", "source": { "blockId": "p10-b68ce20ed0", "page": 10, "quote": "Lemma 4.2 (Stability identity). For every cut of the constructed graph," }, "statement": [ { "blockId": "p10-b68ce20ed0", "page": 10, "quote": "Lemma 4.2 (Stability identity). For every cut of the constructed graph," }, { "blockId": "p10-063154c490", "page": 10, "quote": "cut(σ) = 1" }, { "blockId": "p10-8550f1789b", "page": 10, "quote": "2 + 1" }, { "blockId": "p10-84cd1fc5fb", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy) −1" }, { "blockId": "p10-542d9b17d9", "page": 10, "quote": "2E StabFD\nt\n( ¯Ay),\n(4.5)" }, { "blockId": "p10-4dd6fad46a", "page": 10, "quote": "where both expectations are over y, D, R, Fout. In particular," }, { "blockId": "p10-b7dbc8a9a7", "page": 10, "quote": "cut(σ) ≤1" }, { "blockId": "p10-a3dc9360a3", "page": 10, "quote": "2 + 1" }, { "blockId": "p10-1301175d92", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy).\n(4.6)" } ], "proof": [ { "blockId": "p10-c4b16ddf34", "page": 10, "quote": "Proof. Condition on y, D, R, Fout. The sampled edge tuples and the two row resamplings are\nindependent between copies. After averaging over them, the expected first sign as a function\nof FD is ¯Ay + ¯Gy. The minus sign in the second queried word makes the expected second sign\n¯Ay −¯Gy at F ′\nD. The two slice inputs are ordinary t-correlated uniform cube inputs. The rows\nin these functions are the common latent array R; the two independent row resamplings have\nalready been absorbed into Tr. Negating every slice entry leaves ¯Ay unchanged and negates ¯Gy.\nTheir Fourier supports therefore have opposite total parities, so the cross terms vanish. The\nexpected sign product is Stabt( ¯Ay) −Stabt( ¯Gy). An edge is cut with probability one half of one\nminus its sign product, giving (4.5). Nonnegativity of stability for t > 0 gives (4.6)." }, { "blockId": "p11-164121ec4c", "page": 11, "quote": "If a cut has value at least (1 + B(t))/2 + δ/2, then its odd average has expected slice stability\nat least B(t) + δ. The next three sections show that this excess stability yields a label carrying\nsubstantial Fourier mass, with constants independent of the game dimension and the number of\nrepetitions; the formal statement is Theorem 7.1. Section 8 then turns that mass into a labeling\nof a single copy of the original game." } ], "guide": [] }, { "id": "lemma-9", "kind": "lemma", "label": "Lemma 5.1", "source": { "blockId": "p11-9c0d842b01", "page": 11, "quote": "Lemma 5.1 (Row symmetries). Let f be a function of the code. Every nonzero row Fourier\ncoefficient of f satisfies, at each internal node v," }, "statement": [ { "blockId": "p11-9c0d842b01", "page": 11, "quote": "Lemma 5.1 (Row symmetries). Let f be a function of the code. Every nonzero row Fourier\ncoefficient of f satisfies, at each internal node v," }, { "blockId": "p11-826b929997", "page": 11, "quote": "qv,g0 + qv,g1 = qv\n(g ∈[s]).\n(5.2)" }, { "blockId": "p11-3833abcf8b", "page": 11, "quote": "For a table collection F, let F v,−negate Fv alone, and let F v,z replace Fv(u) by Fv(u + z),\nleaving all other tables unchanged. Then" }, { "blockId": "p11-fe2c2859c8", "page": 11, "quote": "bfC(F v,−) = (−1)qv(1) bfC(F),\n(5.3)" }, { "blockId": "p11-ef6839a7ad", "page": 11, "quote": "bfC(F v,z) = (−1)\nP" }, { "blockId": "p11-49471e5bc7", "page": 11, "quote": "g zgqv,g1(1) bfC(F).\n(5.4)" }, { "blockId": "p11-0fd347ca6e", "page": 11, "quote": "All these statements remain true after row smoothing. They are also preserved by row Fourier\nprojections and by projection to any range of total Fourier degrees in the entries of a specified\ncollection of whole gate tables." } ], "proof": [ { "blockId": "p12-197aaf8a57", "page": 12, "quote": "Proof. For each g, shifting the generators of both children in pair g by h has exactly the same\neffect on the subtree at v as shifting P(v) by h. Consequently the two row translations have the\nsame effect on the whole word. Taking row Fourier coefficients yields" }, { "blockId": "p12-b13f87e64f", "page": 12, "quote": "χqv,g0+qv,g1(h) bfC(F) = χqv(h) bfC(F)\n(h ∈H)." }, { "blockId": "p12-6c5c7c7ebd", "page": 12, "quote": "Distinct characters are distinct functions, proving Equation (5.2).\nNegating Fv has the same effect on the word as adding 1 to all rows in P(v). This proves\nEquation (5.3). To translate the input of Fv by z, add 1 to the generator of its second child\nin each pair with zg = 1. This changes precisely those gate input bits and leaves its prefactor\nunchanged. Fourier transformation gives Equation (5.4).\nRow smoothing and row Fourier projections act diagonally on C, so preserve all three\nidentities. On the cube of table-entry signs, a domain translation permutes coordinates, and\nnegating a gate table multiplies each monomial by its parity sign. Both operations preserve\ntotal degree in any specified collection of whole gate tables and therefore commute with the\nstated degree projections." } ], "guide": [] }, { "id": "lemma-10", "kind": "lemma", "label": "Lemma 5.2", "source": { "blockId": "p12-77d0658e91", "page": 12, "quote": "Lemma 5.2 (Average cost of the unsplit projection). Let f be a function of the code with |f| ≤1,\nlet r = 1 −µ/d with 0 < µ < 1/2, and let ¯f = Trf. Choose the random slice independently of f.\nThen\nEJ0,D\nX" }, "statement": [ { "blockId": "p12-77d0658e91", "page": 12, "quote": "Lemma 5.2 (Average cost of the unsplit projection). Let f be a function of the code with |f| ≤1,\nlet r = 1 −µ/d with 0 < µ < 1/2, and let ¯f = Trf. Choose the random slice independently of f.\nThen\nEJ0,D\nX" }, { "blockId": "p12-091c8a397c", "page": 12, "quote": "v∈D\n∥(I −Sv) ¯f∥2\n2,R,F ≤\ns\n2emµ.\n(5.5)" }, { "blockId": "p12-986ee343c1", "page": 12, "quote": "The same bound holds after averaging over additional background data determining f, provided\nthe slice is chosen independently of those data." } ], "proof": [ { "blockId": "p12-20009794ba", "page": 12, "quote": "Proof. Fix an index C satisfying Equation (5.2), and let Nj be the number of nodes at depth\nj with nonzero index qv. If qv ̸= 0, each child pair contains at least one nonzero index. A\nfailure of the unsplit condition at such a node supplies at least one additional nonzero child. If\nqv = 0, a failure supplies at least two nonzero children, whereas its minimum contribution is\nzero. Therefore the number Bj of failures at depth j satisfies" }, { "blockId": "p12-eb5659cc8e", "page": 12, "quote": "Bj ≤Nj+1 −sNj." }, { "blockId": "p13-c9992c7f30", "page": 13, "quote": "Using the slice inclusion probability and telescoping gives" }, { "blockId": "p13-1d94de894b", "page": 13, "quote": "Nj+1 −sNj" }, { "blockId": "p13-bdbb0e0bfa", "page": 13, "quote": "m−1\nX" }, { "blockId": "p13-56287dce16", "page": 13, "quote": "EJ0,D\nX" }, { "blockId": "p13-9b61544320", "page": 13, "quote": "m" }, { "blockId": "p13-59440fee2c", "page": 13, "quote": "sj" }, { "blockId": "p13-4a419cc61f", "page": 13, "quote": "j=0" }, { "blockId": "p13-0445728a8f", "page": 13, "quote": "v∈D\n1{C is split at v} ≤1" }, { "blockId": "p13-3de080b04d", "page": 13, "quote": "Nm" }, { "blockId": "p13-f4501905b7", "page": 13, "quote": "\n≤s|C|" }, { "blockId": "p13-4b4ab824ae", "page": 13, "quote": "= s" }, { "blockId": "p13-48cc739616", "page": 13, "quote": "m" }, { "blockId": "p13-3d4fa5dfc4", "page": 13, "quote": "sm −N0" }, { "blockId": "p13-8ce880716b", "page": 13, "quote": "md ." }, { "blockId": "p13-7a47c97d6f", "page": 13, "quote": "Here Nm = |C|. By Equation (5.1) and Parseval, the left side of Equation (5.5) is at most" }, { "blockId": "p13-f4dfe0ed90", "page": 13, "quote": "X" }, { "blockId": "p13-5bdf4ab73d", "page": 13, "quote": "\nr2|C| |C|" }, { "blockId": "p13-da78f816db", "page": 13, "quote": "\nEF | bfC(F)|2." }, { "blockId": "p13-982239b9e8", "page": 13, "quote": "s\nm" }, { "blockId": "p13-dcaf016a3b", "page": 13, "quote": "d" }, { "blockId": "p13-43c431e9b3", "page": 13, "quote": "C" }, { "blockId": "p13-7c81722455", "page": 13, "quote": "For x ≥0, xe−2µx ≤(2eµ)−1, and log r ≤−µ/d. Hence every parenthesized factor is at most\n(2eµ)−1, while P\nC EF | bfC(F)|2 = ∥f∥2\n2 ≤1. This proves the bound. The proof is pointwise in\nthe background data, so its average satisfies the same inequality." } ], "guide": [] }, { "id": "lemma-11", "kind": "lemma", "label": "Lemma 5.3", "source": { "blockId": "p13-1062f9bea6", "page": 13, "quote": "Lemma 5.3 (Free-shift coordinates). The change of coordinates" }, "statement": [ { "blockId": "p13-1062f9bea6", "page": 13, "quote": "Lemma 5.3 (Free-shift coordinates). The change of coordinates" }, { "blockId": "p13-10cb301361", "page": 13, "quote": "Hk −→H2s × Hk−2s,\nR 7−→((Lg,b), Γ)" }, { "blockId": "p13-bbf043b794", "page": 13, "quote": "is a linear bijection. In particular, under uniform rows the free shifts are independent uniforms\nconditional on Γ. The Fourier index of Lg,b corresponding to a full row index C is Qg,b = qv,gb.\nConsequently, at every fixed Γ, the operator Sv acts as the Fourier projection in the 2s shifts\nonto precisely the unsplit shift indices. For a function of the code or its row smoothing, the shift\ncoefficients vanish unless the sums Qg,0 + Qg,1 are the same for every g. The table actions in\nEquations (5.3) and (5.4) hold for each shift coefficient at this fixed context, with qv = Q1,0 +Q1,1\nand qv,g1 = Qg,1." } ], "proof": [ { "blockId": "p13-a07f9ca181", "page": 13, "quote": "Proof. The inverse map sets the representative row equal to Lg,b and recovers every other row in\nits generator by adding Lg,b to its recorded difference. This proves bijectivity. In a row character,\nthe coefficient of Lg,b is exactly P\ni∈P(vg,b) qi = qv,gb. Thus the condition defining Sv depends\nonly on the shift index Q, not on any Fourier index of Γ. Grouping the full Fourier expansion\nby Q proves its asserted action on each fixed fiber. The same grouping proves the support and\ntable-action identities: all full row coefficients with a given Q satisfy the same identities and\ncarry the same signs." }, { "blockId": "p14-44722fe4bf", "page": 14, "quote": "Thus, for a nonzero common index q, the retained shift indices are specified by the subset\nI ⊆[s] of pairs in which q occurs in the second child. Their characters are\nY" }, { "blockId": "p14-d51f5145b4", "page": 14, "quote": "g∈I\nχq(Lg1)." }, { "blockId": "p14-e2312596f1", "page": 14, "quote": "g /∈I\nχq(Lg0)\nY" }, { "blockId": "p14-443399ff79", "page": 14, "quote": "If q = 0, there is only the all-zero unsplit index, with character 1. Negating Fv multiplies the\ncorresponding coefficient by (−1)q(1). On the part odd in Fv, q is therefore a label and hence\nnonzero, so the description by q and I is unique. For such a label, translating the argument of\nFv by z ∈Fs\n2 multiplies its coefficient by (−1)\nP" }, { "blockId": "p14-7624f50e24", "page": 14, "quote": "g∈I zg. Thus the same subset I records both the\nrow character and the table translation character. This is the local Fourier description used in\nSection 7, valid at each fixed context." }, { "blockId": "p14-8c5a6d5d48", "page": 14, "quote": "6\nA vector-valued hashing lemma" }, { "blockId": "p14-7efcadf5c3", "page": 14, "quote": "The following lemma turns dispersion among characters into dispersion among table entries. Its\nconstants are independent of the dimension of the character space and of the dimension of the\nvectors being hashed. In the application, q will index candidate labels and z will index entries of\na gate table. The resulting hashed vector will collect the Fourier coefficients contributing to one\nentry’s influence, so its squared norm will be that influence. This is why we need a vector-valued\nstatement rather than a scalar estimate." } ], "guide": [] }, { "id": "lemma-12", "kind": "lemma", "label": "Lemma 6.1", "source": { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." }, "statement": [ { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." }, { "blockId": "p14-bd99ad447a", "page": 14, "quote": "Set\nwq =\nX" }, { "blockId": "p14-edbd4f63d7", "page": 14, "quote": "q∈H∗\nwq." }, { "blockId": "p14-89104d356c", "page": 14, "quote": "z∈Fs\n2\n∥c(q, z)∥2,\nM =\nX" }, { "blockId": "p14-0e6f947b09", "page": 14, "quote": "Choose L, J1, . . . , Js independently and uniformly in H, and define" }, { "blockId": "p14-8e87797cb0", "page": 14, "quote": "J(q) = (q(J1), . . . , q(Js)),\nU(z) =\nX" }, { "blockId": "p14-a0193972bf", "page": 14, "quote": "q∈H∗\nχq(L)c(q, z + J(q))." }, { "blockId": "p14-6293ddf1d4", "page": 14, "quote": "Then\nE\nX" }, { "blockId": "p14-b98783d5c4", "page": 14, "quote": "z∈Fs\n2\n∥U(z)∥2 = M.\n(6.1)" }, { "blockId": "p14-7d2e24e8ce", "page": 14, "quote": "For every ζ > 0, there are γ > 0 and s0 ∈N, depending only on ζ, such that for every s ≥s0," }, { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" } ], "proof": [ { "blockId": "p14-445c75ee85", "page": 14, "quote": "Proof. All expectations over H and Fs\n2 below use uniform probability measure; the displayed\nsums defining wq and M are unnormalized. Conditional on the Jg, orthogonality of the characters\nof L gives\nEL∥U(z)∥2 =\nX" }, { "blockId": "p14-13220ba711", "page": 14, "quote": "q\n∥c(q, z + J(q))∥2." }, { "blockId": "p15-3a6aba3253", "page": 15, "quote": "Summing over z proves (6.1). If M = 0, every vector c(q, z) is zero and (6.2) is immediate.\nHenceforth assume M > 0.\nSet h = 2s and ρ = maxq wq/M. We will prove the quantitative bound below for every\nC, D > 0, and then choose its parameters. The main estimate bounds the maximum squared\nnorm by the square root of the sum of fourth powers. After a Fourier change of variables,\nmost terms in that fourth moment force the same character q in all four factors; their total is\ncontrolled by the dispersion of the character masses. Two truncations will control the remaining\nterms and the cost of this change to the array:" }, { "blockId": "p15-e39283385d", "page": 15, "quote": "" }, { "blockId": "p15-569ca26ddb", "page": 15, "quote": "" }, { "blockId": "p15-bc6a793a36", "page": 15, "quote": "s" }, { "blockId": "p15-3c08e33737", "page": 15, "quote": "2\n+ 1" }, { "blockId": "p15-7ebae8fadb", "page": 15, "quote": "2ρ + 2C2" }, { "blockId": "p15-843f6b6a9c", "page": 15, "quote": "" }, { "blockId": "p15-670fa7848e", "page": 15, "quote": ".\n(6.3)" }, { "blockId": "p15-afd7da3e46", "page": 15, "quote": "E max\nz ∥U(z)∥2 ≤2M" }, { "blockId": "p15-10e448dc3e", "page": 15, "quote": "D2 + 3D4" }, { "blockId": "p15-0e40b33fca", "page": 15, "quote": "h\n+\n 1\n√" }, { "blockId": "p15-8402ef9569", "page": 15, "quote": "D" }, { "blockId": "p15-3d26a9f225", "page": 15, "quote": "h" }, { "blockId": "p15-d06b44edd7", "page": 15, "quote": "C\n+ 1" }, { "blockId": "p15-960ace62e8", "page": 15, "quote": "Fourier representation.\nFor p ∈Fs\n2 and u ∈H, define" }, { "blockId": "p15-bc026517c6", "page": 15, "quote": "s\nX" }, { "blockId": "p15-ac39e00d34", "page": 15, "quote": "αq(p) =\nX" }, { "blockId": "p15-b80f04e6a6", "page": 15, "quote": "g=1\npgJg." }, { "blockId": "p15-3b1a12fcb4", "page": 15, "quote": "q∈H∗\nχq(u)αq(p),\nSp = L +" }, { "blockId": "p15-7cd34b9c28", "page": 15, "quote": "z∈Fs\n2\n(−1)p·zc(q, z),\nAp(u) =\nX" }, { "blockId": "p15-e5cedc4c6c", "page": 15, "quote": "Fourier inversion in z gives\nU(z) = Ep(−1)p·zAp(Sp).\n(6.4)" }, { "blockId": "p15-f537a34b14", "page": 15, "quote": "Writing mp = Eu∥Ap(u)∥2, Parseval’s identity in the two groups gives" }, { "blockId": "p15-94505c12b8", "page": 15, "quote": "mp =\nX" }, { "blockId": "p15-21f25791dc", "page": 15, "quote": "q\n∥αq(p)∥2,\nEpmp = M,\nEp∥αq(p)∥2 = wq.\n(6.5)" }, { "blockId": "p15-167b2af07b", "page": 15, "quote": "For distinct p, p′, the variables Sp, Sp′ are independent and uniform in H: the coefficient\nvectors (1, p), (1, p′) are linearly independent over F2. Similarly, any three distinct p1, p2, p3 give\nthree independent uniform variables Sp1, Sp2, Sp3. Indeed, any nonempty subcollection of their\ncoefficient vectors whose sum is zero must have even cardinality, and hence would consist of two\nequal vectors." }, { "blockId": "p15-f90af860d4", "page": 15, "quote": "Two truncations.\nWe first discard indices p with unusually large mean squared norm mp, and\nthen discard unusually large values of Ap(u) on the remaining indices. The first cutoff makes\nthe change to each individual character coefficient uniformly small under the second cutoff. The\nsecond gives the pointwise bound needed when the fourth-moment indices coincide. We will\ncontrol the discarded part using a first-moment bound and a variance bound from the pairwise\nindependence of the samples Sp. Define" }, { "blockId": "p15-c9810b8cdb", "page": 15, "quote": "(\nAp(u),\nmp ≤CM and ∥Ap(u)∥≤D\n√" }, { "blockId": "p15-58dc2906d1", "page": 15, "quote": "A′\np(u) =" }, { "blockId": "p15-23126a94dd", "page": 15, "quote": "M,\n0,\notherwise," }, { "blockId": "p15-44d2d058a4", "page": 15, "quote": "and put" }, { "blockId": "p15-192503b724", "page": 15, "quote": "α′\nq(p) = Euχq(u)A′\np(u),\nw′\nq = Ep∥α′\nq(p)∥2,\nU′(z) = Ep(−1)p·zA′\np(Sp)." }, { "blockId": "p15-afbe23d3c0", "page": 15, "quote": "Pointwise truncation does not increase the total squared norm, so Parseval gives P\nq w′\nq ≤M. If\nmp ≤CM, then" }, { "blockId": "p15-202173c823", "page": 15, "quote": "∥αq(p) −α′\nq(p)∥≤Eu∥Ap(u)∥1{∥Ap(u)∥>D\n√" }, { "blockId": "p15-3ce88ce786", "page": 15, "quote": "M\nD\n." }, { "blockId": "p15-36ef70d880", "page": 15, "quote": "M} ≤\nmp\nD\n√" }, { "blockId": "p15-e4e231f994", "page": 15, "quote": "M\n≤C\n√" }, { "blockId": "p16-4c5b556ff5", "page": 16, "quote": "If mp > CM, then α′\nq(p) = 0. Consequently," }, { "blockId": "p16-d06e3b92e2", "page": 16, "quote": "X" }, { "blockId": "p16-21fe70f778", "page": 16, "quote": "D2\n.\n(6.6)" }, { "blockId": "p16-d08d31fb5a", "page": 16, "quote": "q\nw′\nq ≤M,\nw′\nq ≤2wq + 2C2M" }, { "blockId": "p16-07b05b513e", "page": 16, "quote": "Let\nXp = ∥(Ap −A′\np)(Sp)∥,\nX = EpXp." }, { "blockId": "p16-29e6db68c4", "page": 16, "quote": "Equation (6.4) and the triangle inequality imply maxz∥U(z) −U′(z)∥≤X. The contribution to\nEX from the indices with mp > CM is at most" }, { "blockId": "p16-227a3a958d", "page": 16, "quote": "M/C." }, { "blockId": "p16-105190767d", "page": 16, "quote": "Ep1{mp>CM}\n√mp ≤Epmp\n√" }, { "blockId": "p16-1b11e5d234", "page": 16, "quote": "CM\n=\nq" }, { "blockId": "p16-07cd8e01a3", "page": 16, "quote": "M) =\n√" }, { "blockId": "p16-3f65aa7f28", "page": 16, "quote": "The contribution from the remaining indices is at most Epmp/(D\n√" }, { "blockId": "p16-1d632d1379", "page": 16, "quote": "M/D. Moreover,\nthe Xp are pairwise independent, by the pairwise independence of the Sp. Since Ap −A′\np is\neither Ap or zero at each point," }, { "blockId": "p16-762981b84e", "page": 16, "quote": "Var(X) = h−2 X" }, { "blockId": "p16-be6d973a69", "page": 16, "quote": "h ." }, { "blockId": "p16-256d7952e5", "page": 16, "quote": "p\nVar(Xp) ≤h−2 X" }, { "blockId": "p16-3ac1900157", "page": 16, "quote": "p\nEu∥Ap(u) −A′\np(u)∥2 ≤M" }, { "blockId": "p16-0d54a98b08", "page": 16, "quote": "We have proved" }, { "blockId": "p16-614486c749", "page": 16, "quote": "#" }, { "blockId": "p16-7f5dc619d5", "page": 16, "quote": "\" 1\n√" }, { "blockId": "p16-0f2778f3fc", "page": 16, "quote": "2\n+ 1" }, { "blockId": "p16-f67cba1c4e", "page": 16, "quote": ".\n(6.7)" }, { "blockId": "p16-9f579ae464", "page": 16, "quote": "E max\nz ∥U(z) −U′(z)∥2 ≤M" }, { "blockId": "p16-6642ab58f7", "page": 16, "quote": "D" }, { "blockId": "p16-5b2eb04688", "page": 16, "quote": "h" }, { "blockId": "p16-675625b1f7", "page": 16, "quote": "C\n+ 1" }, { "blockId": "p16-06d5e24e4d", "page": 16, "quote": "The fourth moment.\nFor brevity write Bp = A′\np(Sp). Expanding the fourth power of the\nnorm and summing the characters in z gives the exact identity" }, { "blockId": "p16-34f6aa8d11", "page": 16, "quote": "E\nX" }, { "blockId": "p16-611205ecdb", "page": 16, "quote": "z\n∥U′(z)∥4 = h−3\nX" }, { "blockId": "p16-3b57fa6b54", "page": 16, "quote": "p1,p2,p3\nE\n⟨Bp1, Bp2⟩⟨Bp3, Bp1+p2+p3⟩\n" }, { "blockId": "p16-961cce19bb", "page": 16, "quote": "= Ep1,p2,p3E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n,\np4 = p1 + p2 + p3.\n(6.8)" }, { "blockId": "p16-73cffa56cb", "page": 16, "quote": "Here p1, p2, p3 are independent uniform indices in the second line. The factor h−3 in the first\nline comes from four factors h−1 in Fourier inversion and the sum of h characters over z.\nIf any two of p1, p2, p3, p4 coincide, then some two of p1, p2, p3 coincide. This event has\nprobability at most 3/h. Since ∥Bp∥≤D\n√" }, { "blockId": "p16-2f7dca4364", "page": 16, "quote": "M, its contribution to (6.8) is at most 3D4M2/h in\nabsolute value.\nNow fix four distinct indices with p4 = p1 + p2 + p3. The first three Spi are independent\nuniform elements of H, and Sp4 = Sp1 + Sp2 + Sp3. Expanding the four A′\npi in characters and\naveraging over those three independent elements yields" }, { "blockId": "p16-35bfa1fe78", "page": 16, "quote": "E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n =\nX" }, { "blockId": "p16-6e934d6f5f", "page": 16, "quote": "q\n⟨α′\nq(p1), α′\nq(p2)⟩⟨α′\nq(p3), α′\nq(p4)⟩.\n(6.9)" }, { "blockId": "p16-bba3569c1b", "page": 16, "quote": "In detail, characters indexed by q1, q2, q3, q4 produce the factor" }, { "blockId": "p16-a05b5672d9", "page": 16, "quote": "3\nY" }, { "blockId": "p16-c6c053c5de", "page": 16, "quote": "i=1\nχqi+q4(Spi)," }, { "blockId": "p16-34b5c828e7", "page": 16, "quote": "whose expectation is zero unless q1 = q2 = q3 = q4." }, { "blockId": "p17-6b00892076", "page": 17, "quote": "The right side of (6.9) need not be nonnegative. For an upper bound, replace each summand\nby Q4\ni=1∥α′\nq(pi)∥, which is nonnegative, and then drop the restriction that the four indices be\ndistinct. For every fixed q, Cauchy–Schwarz gives" }, { "blockId": "p17-8a42c3ee0d", "page": 17, "quote": "4\nY" }, { "blockId": "p17-7ffeb93a9a", "page": 17, "quote": "Ep1,p2,p3" }, { "blockId": "p17-c22ad44175", "page": 17, "quote": "i=1\n∥α′\nq(pi)∥≤\n\nE∥α′\nq(p1)∥2∥α′\nq(p2)∥21/2 \nE∥α′\nq(p3)∥2∥α′\nq(p4)∥21/2" }, { "blockId": "p17-8d36a1a88b", "page": 17, "quote": "= (w′\nq)2." }, { "blockId": "p17-bcf33326e9", "page": 17, "quote": "The last equality uses the unconditioned independent uniform law of each pair (p1, p2) and (p3, p4);\nit does not assert independence after conditioning on distinctness. Thus the nondegenerate\nfourth moment is bounded by P\nq(w′\nq)2: dispersion among characters now controls concentration\namong table entries. Combining these estimates with (6.6) proves" }, { "blockId": "p17-a89add6fb0", "page": 17, "quote": "E\nX" }, { "blockId": "p17-dac695ef40", "page": 17, "quote": "h" }, { "blockId": "p17-fd58efeff6", "page": 17, "quote": "z\n∥U′(z)∥4 ≤\nX" }, { "blockId": "p17-4ba293a407", "page": 17, "quote": "q\n(w′\nq)2 + 3D4M2" }, { "blockId": "p17-64777fb10d", "page": 17, "quote": "!" }, { "blockId": "p17-e60be528d4", "page": 17, "quote": "≤M2" }, { "blockId": "p17-3fb9b1859f", "page": 17, "quote": "." }, { "blockId": "p17-4743fa2f17", "page": 17, "quote": "2ρ + 2C2" }, { "blockId": "p17-0bb51ec97b", "page": 17, "quote": "D2 + 3D4" }, { "blockId": "p17-36d0030bf3", "page": 17, "quote": "h" }, { "blockId": "p17-d9bce41dc4", "page": 17, "quote": "Therefore, using Cauchy–Schwarz once more," }, { "blockId": "p17-9739b715e0", "page": 17, "quote": "s" }, { "blockId": "p17-d3200f345b", "page": 17, "quote": "E\nX" }, { "blockId": "p17-6b32b3b65b", "page": 17, "quote": "2ρ + 2C2" }, { "blockId": "p17-e09a1b19d1", "page": 17, "quote": "E max\nz ∥U′(z)∥2 ≤" }, { "blockId": "p17-3b2ce259c4", "page": 17, "quote": "D2 + 3D4" }, { "blockId": "p17-a5d7732e54", "page": 17, "quote": "h .\n(6.10)" }, { "blockId": "p17-c9db8d03d2", "page": 17, "quote": "z\n∥U′(z)∥4\n!1/2\n≤M" }, { "blockId": "p17-4bdc769804", "page": 17, "quote": "The inequality (a + b)2 ≤2a2 + 2b2, together with (6.7) and (6.10), proves (6.3).\nFinally, set ε = min{ζ, 1} and choose" }, { "blockId": "p17-ea99596a91", "page": 17, "quote": ")" }, { "blockId": "p17-b9ed0f5afa", "page": 17, "quote": "(\n192D4" }, { "blockId": "p17-abc74c2d37", "page": 17, "quote": "." }, { "blockId": "p17-9b2eb207d3", "page": 17, "quote": "C = 64" }, { "blockId": "p17-812837a320", "page": 17, "quote": "ε ,\nD = 8C" }, { "blockId": "p17-999f55e22e", "page": 17, "quote": "ε ,\nγ = ε2" }, { "blockId": "p17-48fb9381b9", "page": 17, "quote": "ε2\n, 16" }, { "blockId": "p17-0314d0205a", "page": 17, "quote": "ε" }, { "blockId": "p17-909bb27fde", "page": 17, "quote": "128,\n2s0 ≥max" }, { "blockId": "p17-9e2b210abb", "page": 17, "quote": "If s ≥s0 and ρ ≤γ, the square-root term in (6.3) is at most ε/4. Also D ≥\n√" }, { "blockId": "p17-c646c8dd58", "page": 17, "quote": "C, so the other two\nterms in its brackets are at most ε/16 each. The resulting bound is 3εM/4 ≤ζM, establishing\n(6.2) with constants depending only on ζ." } ], "guide": [] }, { "id": "proposition-13", "kind": "proposition", "label": "Proposition 7.1", "source": { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, "statement": [ { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, { "blockId": "p18-01097b485f", "page": 18, "quote": "Here we sample ω, the random slice D, and a uniform node v ∈D; conditional on these, we\nsample Fout and Γ independently and uniformly." } ], "proof": [ { "blockId": "p18-0ab17157e8", "page": 18, "quote": "Proof. Excess stability first produces an influential slice entry. We retain that influence after the\nunsplit projection, discard the part even in its gate table, and express the remaining influence\nas an affine hash of vectors indexed by labels. The hashing lemma then forces one label to carry\nsubstantial conditional mass.\nAll constants chosen below are independent of A and of the law of ω. The degree cutoff\nK and influence threshold τ are chosen first. We leave s, m unspecified during the estimates:\nStep 4 will choose s to control the even and hashed contributions, and then m to control the\ncost of the unsplit projection." }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p18-65f0ea897b", "page": 18, "quote": "some entry z ∈Fs\n2 of some v ∈D has Inf≤K\nv,z ( ¯fω) ≥τ. Indeed, if the probability of this event is\np, the expected stability is at most B(t) + δ/2 + p." }, { "blockId": "p19-e5b0903627", "page": 19, "quote": "Let Sv be the orthogonal row projection onto indices unsplit at v. Put" }, { "blockId": "p19-ec2b872581", "page": 19, "quote": "Hv = P FD\n≤KSv ¯fω." }, { "blockId": "p19-52d05b19a0", "page": 19, "quote": "By Theorem 5.2, the required bound on the projection loss is" }, { "blockId": "p19-fd1e8fb7b5", "page": 19, "quote": "E\nX" }, { "blockId": "p19-2048fcebc1", "page": 19, "quote": "8 .\n(7.5)" }, { "blockId": "p19-f2ddd11aec", "page": 19, "quote": "v∈D\n∥(1 −Sv) ¯fω∥2\n2,R,F ≤\ns\n2emµ ≤ητ" }, { "blockId": "p19-d0b22816e9", "page": 19, "quote": "For any fixed s, the last inequality can be ensured by increasing m. The same expectation is the\nexpectation over ω, D, R, Fout of the sum of the corresponding conditional squared norms in FD.\nMarkov’s inequality shows that this sum is at most τ/4 except on an event of probability η/2.\nOn the intersection with the influential-entry event, the triangle inequality for the projection\ndefining that influence gives" }, { "blockId": "p19-fa3dee0c84", "page": 19, "quote": "q" }, { "blockId": "p19-9606c52284", "page": 19, "quote": "Infv,z(Hv) ≥\nq" }, { "blockId": "p19-d3d4a81378", "page": 19, "quote": "Inf≤K\nv,z ( ¯fω) −∥(1 −Sv) ¯fω∥2,FD ≥\n√τ" }, { "blockId": "p19-56f1af5a37", "page": 19, "quote": "2 ." }, { "blockId": "p19-7c4264bd30", "page": 19, "quote": "Consequently,\nE\nX" }, { "blockId": "p19-2bd4dfb0c5", "page": 19, "quote": "8 =: Λ.\n(7.6)" }, { "blockId": "p19-377627618d", "page": 19, "quote": "v∈D\nmax\nz\nInfv,z(Hv) ≥ητ" }, { "blockId": "p19-22db67cb9b", "page": 19, "quote": "X" }, { "blockId": "p19-c3c840fe26", "page": 19, "quote": "We will also use\nE\nX" }, { "blockId": "p19-1fcabf9810", "page": 19, "quote": "z\nInfv,z(Hv) ≤K.\n(7.7)" }, { "blockId": "p19-d9f997fc30", "page": 19, "quote": "v∈D" }, { "blockId": "p19-d7a06cebf5", "page": 19, "quote": "To verify it, integrate over all rows. Each Sv is an orthogonal projection in the rows and\ncommutes with the slice-bit Fourier projections. It can therefore only decrease the integrated\ninfluence at its entries. After dropping each Sv separately, the left side is bounded by the total\ninfluence of P FD\n≤K ¯fω, averaged over the remaining variables. Equation (2.3) and | ¯fω| ≤1 give\n(7.7). No pointwise boundedness of Hv is asserted or needed." }, { "blockId": "p19-7c4b091a0b", "page": 19, "quote": "Step 2: discard even dependence on a gate.\nFix v ∈D, and split Hv into its even and\nodd parts under negation of Fv alone. These parts have disjoint Fourier supports in the slice\nbits, also after retaining the monomials containing any specified entry. Thus" }, { "blockId": "p19-9105296e0b", "page": 19, "quote": "Infv,z(Hv) = Infv,z(Heven\nv\n) + Infv,z(Hodd\nv\n)." }, { "blockId": "p19-20ff4ebca5", "page": 19, "quote": "By the sign action in Theorem 5.1, the even part has qv(1) = 0 on row support. Since the\nindex is unsplit, all its child indices are either 0 or qv, so they too take value zero on 1. The\ndomain-translation action in that lemma shows that the even part is invariant under translations\nof the domain of Fv, with the rows and outside tables fixed. These translations act transitively\non its entries. All entry influences at v are therefore equal. By (7.7)," }, { "blockId": "p19-d0cb574ac1", "page": 19, "quote": "E\nX" }, { "blockId": "p19-5abdbe5cbd", "page": 19, "quote": "2s .\n(7.8)" }, { "blockId": "p19-af991044f0", "page": 19, "quote": "v∈D\nmax\nz\nInfv,z(Heven\nv\n) ≤K" }, { "blockId": "p20-56ef8758e5", "page": 20, "quote": "Step 3: express odd influences as affine hashes.\nFix ω, D, v, Γ, Fout. The remaining row\ncoordinates Lgb are independent uniform elements of H. On the odd part, the common pair\nsum q is a label, by (2.1), and hence is nonzero. An unsplit shift index with common sum q\nputs q in the second child on a subset I ⊆[s] and in the first child elsewhere. Let aq,I(FD) be\nits coefficient in Hodd\nv\n.\nSet" }, { "blockId": "p20-99d9930127", "page": 20, "quote": "s\nX" }, { "blockId": "p20-0ea458ab03", "page": 20, "quote": "LΣ =" }, { "blockId": "p20-66948052d6", "page": 20, "quote": "g=1\nLg0,\nJg = Lg0 + Lg1,\nJ(q) = (q(J1), . . . , q(Js))." }, { "blockId": "p20-068291a42a", "page": 20, "quote": "The s + 1 forms LΣ, J1, . . . , Js are independent and uniform: the displayed linear map from H2s" }, { "blockId": "p20-31a5650d69", "page": 20, "quote": "onto Hs+1 is surjective. The remaining s −1 coordinates do not occur in the unsplit expansion.\nThe character of the shift index (q, I) is" }, { "blockId": "p20-33b4c62dc4", "page": 20, "quote": "g∈I q(Jg)." }, { "blockId": "p20-fff1270b5c", "page": 20, "quote": "χq(LΣ)(−1)\nP" }, { "blockId": "p20-21eb663700", "page": 20, "quote": "On the other hand, the domain-translation identity gives" }, { "blockId": "p20-6c2a0b8054", "page": 20, "quote": "g∈I zgaq,I(FD)." }, { "blockId": "p20-402de8fc02", "page": 20, "quote": "aq,I(Fv(· + z), FD\\{v}) = (−1)\nP" }, { "blockId": "p20-025f9c3b7f", "page": 20, "quote": "With Ψq = P\nI⊆[s] aq,I we obtain" }, { "blockId": "p20-fc407884f3", "page": 20, "quote": "Hodd\nv\n=\nX" }, { "blockId": "p20-3e08f51158", "page": 20, "quote": "q∈A\nχq(LΣ)Ψq\nFv(· + J(q)), FD\\{v}\n.\n(7.9)" }, { "blockId": "p20-651ecd9795", "page": 20, "quote": "The functions aq,I for distinct I are orthogonal in the uniform slice bits: a domain translation\nwhose two characters differ preserves their inner product and changes its sign. Consequently," }, { "blockId": "p20-2a3d64fbf7", "page": 20, "quote": "∥Ψq∥2\n2,FD =\nX" }, { "blockId": "p20-dbdc944b36", "page": 20, "quote": "I\n∥aq,I∥2\n2,FD ≤mq( ¯fω; Γ, Fout).\n(7.10)" }, { "blockId": "p20-b92cb4dec5", "page": 20, "quote": "The inequality follows because the unsplit row projection is determined by the free shift index,\nand the degree and parity projections are orthogonal projections in the slice bits at each fixed\ncontext. Every Ψq has degree at most K.\nEquation (7.9) has converted the free-shift characters into translations of a gate table. We\nnext collect its Boolean Fourier coefficients into vectors whose squared norms are entry influences.\nThese Fourier coefficients are indexed by sets of table entries; the earlier index I ⊆[s] instead\nspecified a character of the table’s domain Fs\n2.\nDenote by bΨq(B, D) the slice Fourier coefficient with entry set B ⊆Fs\n2 at gate v and entry\nset D on the other slice gates. For z ∈Fs\n2 define a real vector c(q, z), with coordinates indexed\nby pairs (A, D) where 0 ∈A ⊆Fs\n2, by" }, { "blockId": "p20-d48a46c1ab", "page": 20, "quote": "c(q, z)A,D = bΨq(z + A, D)." }, { "blockId": "p20-5019de2b8d", "page": 20, "quote": "Using entry sets relative to z makes this vector space the same for every z and every label q.\nSet c(q, z) = 0 for functionals q which are not labels. For the influence at z, a slice Fourier\nset B containing z has the unique form B = z + A with 0 ∈A. Translation by J(q) sends its\ncoefficient to bΨq(z + A + J(q), D). Hence, exactly," }, { "blockId": "p20-d4d90b5fb9", "page": 20, "quote": "X" }, { "blockId": "p20-ef57cb6081", "page": 20, "quote": "2\n.\n(7.11)" }, { "blockId": "p20-1f7f1c905a", "page": 20, "quote": "Infv,z(Hodd\nv\n) =" }, { "blockId": "p20-cd64589e0f", "page": 20, "quote": "q\nχq(LΣ)c(q, z + J(q))" }, { "blockId": "p21-05000909ec", "page": 21, "quote": "Writing wq = P\nz∥c(q, z)∥2 and Mv = P\nq wq, we have" }, { "blockId": "p21-991df0479c", "page": 21, "quote": "wq =\nX" }, { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" }, { "blockId": "p21-2a433dcc94", "page": 21, "quote": "The factor |B| counts the entries whose influences contain that Fourier monomial; it is at most\nthe monomial’s total degree, which is at most K. Thus Mv ≤K at every context. The total-mass\nidentity in Theorem 6.1, together with (7.11), also gives" }, { "blockId": "p21-b797e6b637", "page": 21, "quote": "X" }, { "blockId": "p21-20567f8e19", "page": 21, "quote": "E\nX" }, { "blockId": "p21-e512194a97", "page": 21, "quote": "z\nInfv,z(Hodd\nv\n) ≤K.\n(7.13)" }, { "blockId": "p21-b19f677e7e", "page": 21, "quote": "v∈D\nMv = E\nX" }, { "blockId": "p21-952accfd4a", "page": 21, "quote": "v∈D" }, { "blockId": "p21-bffe34457f", "page": 21, "quote": "For this identity the free coordinates are integrated separately for each node; their different\ncoordinate systems do not affect the uniform row law." }, { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, { "blockId": "p21-de23f92e07", "page": 21, "quote": "M0 =\nΛ\n4 · 2m ,\nθ = γM0" }, { "blockId": "p21-b8e4cf0492", "page": 21, "quote": "K ,\np∗=\nΛ\n4K 2m .\n(7.14)" }, { "blockId": "p21-a65ea046f4", "page": 21, "quote": "For each node, condition on ω, D, v, Γ, Fout and integrate the free shifts. Partition these\ncontexts into three classes:" }, { "blockId": "p21-1e104952eb", "page": 21, "quote": "(i) wq ≤γMv for every q;" }, { "blockId": "p21-1bab181aeb", "page": 21, "quote": "(ii) the first condition fails and Mv < M0;" }, { "blockId": "p21-125cee3bff", "page": 21, "quote": "(iii) Mv ≥M0 and wq > γMv for some q." }, { "blockId": "p21-b276b21d53", "page": 21, "quote": "On class (i), the hashing lemma bounds the expected maximum odd influence by ζMv. Summing\nand using (7.13) costs at most Λ/4. On class (ii), the maximum is bounded in expectation\nby the total influence, namely Mv < M0. Since |D| ≤2m, these contexts cost at most Λ/4.\nOn class (iii), the same total-mass bound is at most K per node. Equation (7.8) bounds all\neven contributions by Λ/4. Let Niii count the nodes whose induced contexts lie in class (iii).\nComparing all four contributions with (7.6) gives" }, { "blockId": "p21-cbfd209c78", "page": 21, "quote": "Λ ≤K2−s + ζK + 2mM0 + KENiii ≤3Λ" }, { "blockId": "p21-8b82079776", "page": 21, "quote": "4 + KENiii." }, { "blockId": "p21-ae8861c37e", "page": 21, "quote": "Thus ENiii ≥Λ/(4K). At each such node (7.12) supplies a label with mq > γM0/K = θ.\nChoosing a uniform node from the slice loses at most a factor 2m and gives (7.4). This proves\nthe proposition." } ], "guide": [] }, { "id": "lemma-14", "kind": "lemma", "label": "Lemma 8.1", "source": { "blockId": "p22-955a28fdcb", "page": 22, "quote": "Lemma 8.1 (Removing smoothing). For every fixed y, D, v, Γ, Fout, and label q ∈A," }, "statement": [ { "blockId": "p22-955a28fdcb", "page": 22, "quote": "Lemma 8.1 (Removing smoothing). For every fixed y, D, v, Γ, Fout, and label q ∈A," }, { "blockId": "p22-470d569f6e", "page": 22, "quote": "mq( ¯Gy; Γ, Fout) ≤EE, e|ymq(Ge; Γ + EΓ, Fout).\n(8.1)" }, { "blockId": "p22-5acddaee14", "page": 22, "quote": "Here the noise E is independent of the conditionally sampled edge tuple e." } ], "proof": [ { "blockId": "p22-7f90375d31", "page": 22, "quote": "Proof. For the fixed context, regard a function of (L, FD) as an element of the real Hilbert space\nwith the uniform probability measure. Let Πq be the orthogonal projection onto the shift Fourier\nindices Q satisfying Qg0 + Qg1 = q for every g. Then the corresponding mass is the squared\nnorm of this projection. The identity" }, { "blockId": "p22-f931f31145", "page": 22, "quote": "¯Gy(L, Γ, F) = EE, e|yGe(L + EL, Γ + EΓ, F)" }, { "blockId": "p22-01fb0ceced", "page": 22, "quote": "and convexity of the squared Hilbert-space norm give the desired inequality. Indeed, translation\nby EL commutes with Πq and preserves its norm: on each shift Fourier coefficient it only\nmultiplies by a sign." }, { "blockId": "p22-20ee83acee", "page": 22, "quote": "Choose once and for all an ordering of each label space, and define" }, { "blockId": "p22-7eded40296", "page": 22, "quote": "q∗= q∗(y, D, v, Γ, Fout) ∈arg max\nq∈A\nmq( ¯Gy; Γ, Fout)\n(8.2)" }, { "blockId": "p22-62d9b730ba", "page": 22, "quote": "using that ordering to break ties. This definition uses the original conditional edge average and\nthe original row-smoothing operator. It will remain unchanged when we modify the distribution\nof the row data below.\nSample y, the slice, its uniform node, the context, and the outside tables as in Proposition 7.1,\nand then sample Rnew, B, and e | y. Combining that Proposition with Lemma 8.1 gives" }, { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" }, { "blockId": "p22-3867cb7d85", "page": 22, "quote": "The mass in this expectation lies in [0, 1], since Ge is bounded in absolute value by one." } ], "guide": [] }, { "id": "lemma-15", "kind": "lemma", "label": "Lemma 8.2", "source": { "blockId": "p23-e015ef835b", "page": 23, "quote": "Lemma 8.2 (Hiding the planted coordinate). Let U be the unplanted law and P the planted law\nafter forgetting the mark. Both laws are taken on the full data" }, "statement": [ { "blockId": "p23-e015ef835b", "page": 23, "quote": "Lemma 8.2 (Hiding the planted coordinate). Let U be the unplanted law and P the planted law\nafter forgetting the mark. Both laws are taken on the full data" }, { "blockId": "p23-5cf61811f3", "page": 23, "quote": "e, J0, D, v, Fout, Γ, Rnew, B\n." }, { "blockId": "p23-5ff6cd6496", "page": 23, "quote": "Then" }, { "blockId": "p23-6da92718fb", "page": 23, "quote": "s" }, { "blockId": "p23-43c0a12025", "page": 23, "quote": "1 −p0" }, { "blockId": "p23-04c4b640a3", "page": 23, "quote": "TV(P, U) ≤1" }, { "blockId": "p23-8ea2a6fbf1", "page": 23, "quote": "2" }, { "blockId": "p23-266720f7f6", "page": 23, "quote": "Tp0\n≤\n1\n2√Tp0\n.\n(8.5)" }, { "blockId": "p23-fda0329fed", "page": 23, "quote": "Consequently, if" }, { "blockId": "p23-d3c148ea4c", "page": 23, "quote": "T ≥\n1\np0(p∗θ)2 ,\n(8.6)" }, { "blockId": "p23-b1d8c89607", "page": 23, "quote": "then under the planted law," }, { "blockId": "p23-46c96a798b", "page": 23, "quote": "EP mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ" }, { "blockId": "p23-555402b059", "page": 23, "quote": "2 .\n(8.7)" } ], "proof": [ { "blockId": "p23-38adbae09d", "page": 23, "quote": "Proof. Let Z = PT\na=1 1Za. Averaging the conditional densities over the uniform mark gives" }, { "blockId": "p23-a7c4fa153b", "page": 23, "quote": "dP\ndU =\nZ\nTp0\n." }, { "blockId": "p23-4d47b40518", "page": 23, "quote": "Conditional on the tree data, Z has the binomial distribution with parameters T, p0; these\nparameters do not depend on the tree data. Thus Cauchy–Schwarz gives" }, { "blockId": "p23-7a59ba92ab", "page": 23, "quote": "s" }, { "blockId": "p23-85090c45b5", "page": 23, "quote": "1 −p0" }, { "blockId": "p23-21e0eefd1c", "page": 23, "quote": "TV(P, U) = 1" }, { "blockId": "p23-195866d846", "page": 23, "quote": "2EU| Z" }, { "blockId": "p23-4395b5b196", "page": 23, "quote": "2" }, { "blockId": "p23-284e4db2e3", "page": 23, "quote": "Tp0\n−1| ≤1" }, { "blockId": "p23-b742d300a1", "page": 23, "quote": "Tp0\n." }, { "blockId": "p23-11ce6c955c", "page": 23, "quote": "The score in Equation (8.3) is a [0, 1]-valued function of the displayed data without the mark.\nIn particular, q∗is the fixed function defined in Equation (8.2); it does not use the mark, the\nsampled incident edges beyond their Y -endpoints, or the sampled noise. Its statistical dependence\non zero slots causes no difficulty. Expectations of this score under the two laws differ by at most\ntheir total variation distance. Equations (8.3) and (8.6) therefore imply Equation (8.7)." } ], "guide": [] }, { "id": "lemma-16", "kind": "lemma", "label": "Lemma 8.3", "source": { "blockId": "p24-d75bb0e3c1", "page": 24, "quote": "Lemma 8.3 (Restriction of shift mass). Fix a context and the outside tables. Let ef(eL, FD) be a\nfunction on eH2s and the slice tables. Suppose its shift Fourier support has the common-pair-sum\nproperty\neQg0 + eQg1 = eQh0 + eQh1\nfor all g, h ∈[s]." }, "statement": [ { "blockId": "p24-d75bb0e3c1", "page": 24, "quote": "Lemma 8.3 (Restriction of shift mass). Fix a context and the outside tables. Let ef(eL, FD) be a\nfunction on eH2s and the slice tables. Suppose its shift Fourier support has the common-pair-sum\nproperty\neQg0 + eQg1 = eQh0 + eQh1\nfor all g, h ∈[s]." }, { "blockId": "p24-e555a043e6", "page": 24, "quote": "Set f(L, FD) = ef(ιL, FD), where ι acts on each shift. For every label q ∈A," }, { "blockId": "p24-a0c196f7d5", "page": 24, "quote": "mq(f) ≤22s X" }, { "blockId": "p24-242a8fe97e", "page": 24, "quote": "emeq( ef).\n(8.8)" }, { "blockId": "p24-17f2024d92", "page": 24, "quote": "eq∈e\nA\nρ(eq)=q" }, { "blockId": "p24-ff91954647", "page": 24, "quote": "The masses on both sides use the common-pair-sum definition of Equation (7.1), with the\nrespective affine row spaces." } ], "proof": [ { "blockId": "p24-ddcc7f1cf8", "page": 24, "quote": "Proof. For every fixed choice of slice tables, Fourier inversion and character orthogonality give" }, { "blockId": "p24-3273ea4d82", "page": 24, "quote": "bf(Q) =\nX" }, { "blockId": "p24-6a14c19954", "page": 24, "quote": "bef( eQ)." }, { "blockId": "p24-ec8f6287b9", "page": 24, "quote": "e\nQ\nι∗e\nQgb=Qgb ∀g,b" }, { "blockId": "p24-a2f9a37a9b", "page": 24, "quote": "The sum has 22s indices. Hence its squared absolute value is at most 22s times the sum of the\nsquared absolute values of its terms. Now sum over all Q with pair sums q. Every nonzero\ncoefficient on the right has a common pair sum eq whose restriction is q. Moreover," }, { "blockId": "p24-dae4f91596", "page": 24, "quote": "eq(1) = eq(ι(1)) = q(1) = 1," }, { "blockId": "p24-e0bbb65da1", "page": 24, "quote": "so eq is an evaluation functional, hence a label in e\nA, and ρ(eq) = q. Each lifted index is counted\nonly once. Averaging over the slice tables proves Equation (8.8)." } ], "guide": [] }, { "id": "proposition-17", "kind": "proposition", "label": "Proposition 8.4", "source": { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" }, "statement": [ { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" }, { "blockId": "p24-44276ea803", "page": 24, "quote": "Val(G) ≥2−2s p∗θ" }, { "blockId": "p24-37148e8ed5", "page": 24, "quote": "2 .\n(8.9)" } ], "proof": [ { "blockId": "p24-f0f700ed40", "page": 24, "quote": "Proof. We construct randomized labelers that use only their own endpoint and shared randomness.\nThe entire game and the fixed cut tables are available as background data. No efficiency claim\nabout the labelers is needed to lower-bound game value." }, { "blockId": "p25-03898445eb", "page": 25, "quote": "Shared data and the Y -labeler.\nGiven a random input edge e = (x, y) from the original\nedge law, choose a uniform public mark a∗∈[T]. At every other slot, publicly sample an\nindependent edge, including its endpoints and projection. Insert the input edge at the marked\nslot. Its projection is used only in the analysis; it is not revealed to either labeler. Conditional on\nany mark, the resulting full edge tuple has the product edge law; hence the tuple is independent\nof the mark. Each labeler knows its own endpoint tuple, and both know all edges outside the\nmarked slot.\nPublicly sample the depth, slice, node, outside tables, context, latent rows, and masks\naccording to the planted experiment above. The row sampling uses the fixed alphabet dimensions,\nthe public tree coordinates, and the mark, and is independent of the input edge. Together with\nthe edge tuple, these data have exactly the planted law used in Equation (8.7). The Y -labeler\nforms its endpoint tuple y, computes the label q∗from Equation (8.2), and outputs its marked\ncomponent q∗\na∗. All quantities used in that computation are available to this labeler: in particular\nthe conditional average defining ¯Gy can be computed from the game and cut as fixed background\ndata. It does not use the actual marked edge or its projection." }, { "blockId": "p25-c56d73301c", "page": 25, "quote": "The X-labeler.\nTo define the X-labeler, replace the marked factor of A by the X-alphabet:" }, { "blockId": "p25-b0c7d04e05", "page": 25, "quote": "e\nA =\nY" }, { "blockId": "p25-5305981634", "page": 25, "quote": "a̸=a∗\nFn\n2 × Fn+1\n2\n," }, { "blockId": "p25-dd8d66dbbc", "page": 25, "quote": "with factors in their original slot order. Let B = (Fn+1\n2\n)T . The full projection factors as" }, { "blockId": "p25-b92708bed6", "page": 25, "quote": "B\nκ\n−−→e\nA\nρ\n−−→A." }, { "blockId": "p25-c16f205107", "page": 25, "quote": "Here κ is the public edge projection at every unmarked slot and the identity at the marked slot;\nρ is the identity at every unmarked slot and the unknown map πe at the marked slot. Thus κ is\nknown to the X-labeler, while ρ is used only in the analysis.\nThe planted zeros imply that every row in Γ and Rnew has zero linear coefficient block in\nthe marked slot. The same holds for E, EΓ, and Γ′ = Γ + EΓ, since the noise masks and context\nchange only multiply by scalars and add rows. Consequently eΓ′ = ρ∗Γ′ is known to the X-labeler:\nit is obtained by leaving every unmarked coefficient and constant coefficient unchanged and\nsetting all marked coefficients to zero. This description does not require πe, including its affine\ntranslation part.\nThe X-labeler forms the bounded raw function" }, { "blockId": "p25-64801e2186", "page": 25, "quote": "eG( eR, F) = σo\nx" }, { "blockId": "p25-bc667bad25", "page": 25, "quote": "W e\nA( eR, F) ◦κ\n,\neR ∈eHk." }, { "blockId": "p25-b66d45fd9b", "page": 25, "quote": "Fix its context to eΓ′ and its outside tables to the public Fout, leaving all 2s shifts free in eH. Let\nemeq be its masses. They are nonnegative and their sum over labels is at most one, by Parseval and\n| eG| ≤1. The labeler samples a label eq with probability at least emeq for each label, assigning any\nleftover probability to a fixed label. It uses an additional independent public uniform random\nnumber to sample from this distribution and outputs the marked component of eq. This sampling\ndoes not use q∗." }, { "blockId": "p25-fd64645ce0", "page": 25, "quote": "Comparison on the sampled edge.\nWe verify the restriction hypothesis needed to compare\nthese probabilities with the score. For any h ∈eH, shifting both child generators in one pair by" }, { "blockId": "p26-3843e9cd2b", "page": 26, "quote": "h multiplies the output of gate v by (−1)h(el) at every label el. Applying this shift in two distinct\npairs leaves that output, and hence the whole word, unchanged. These operations only translate\nthe free shifts and preserve the context. Character orthogonality therefore shows that every\nshift Fourier index of eG has a common pair sum. This argument holds for each fixed choice of\nall gate tables.\nWord generation commutes with pullback of rows, pointwise in the label. Since πe = ρ ◦κ,\nrestricting each free shift of eG to ρ∗H yields exactly Ge with context Γ′ and the same tables.\nThe map ρ has two-element fibers, so Lemma 8.3 applies and gives\nX" }, { "blockId": "p26-5741bc9606", "page": 26, "quote": "ρ(eq)=q∗\nemeq ≥2−2smq∗(Ge; Γ′, Fout)." }, { "blockId": "p26-1de248c808", "page": 26, "quote": "Conditional on all the planted data, the left side lower-bounds the probability that the sampled\nfull label projects to q∗. That event in particular implies satisfaction of the marked input edge.\nAveraging and using Equation (8.7) gives success probability at least 2−2sp∗θ/2.\nFinally, fix all shared randomness, including the uniform number used to sample the X-label.\nThe same fixed data define the two strategies at every endpoint, so they give a deterministic\nlabeling of the original game. Their average value over the shared randomness has the lower\nbound just proved, so some fixed choice attains it. This proves Equation (8.9)." }, { "blockId": "p26-852017d85d", "page": 26, "quote": "The decoding loss depends on s, p∗, θ, and hence on the target cut gap, but not on the\nalphabet dimension or the repetition length. After choosing a game soundness smaller than the\nright side of Equation (8.9), one may therefore fix the alphabet dimension and then choose T by\nEquation (8.6)." } ], "guide": [] }, { "id": "lemma-18", "kind": "lemma", "label": "Lemma A.1", "source": { "blockId": "p31-89e490beb0", "page": 31, "quote": "Lemma A.1 (The invariant-list assertion on admissible advice spaces). Let X have dimension\n3k0, let H ≤X have dimension k0, and let h : H →F2 be linear. Suppose a table F assigns\na linear function to every ℓ-space, and that for every (ℓ+ k0)-space K ⊇H there is a linear\nfunction fK : K →F2 such that fK|H = h and F[L] = fK|L whenever L + H = K. Fix a\nq-space Q with Q ∩H = {0}, and set" }, "statement": [ { "blockId": "p31-89e490beb0", "page": 31, "quote": "Lemma A.1 (The invariant-list assertion on admissible advice spaces). Let X have dimension\n3k0, let H ≤X have dimension k0, and let h : H →F2 be linear. Suppose a table F assigns\na linear function to every ℓ-space, and that for every (ℓ+ k0)-space K ⊇H there is a linear\nfunction fK : K →F2 such that fK|H = h and F[L] = fK|L whenever L + H = K. Fix a\nq-space Q with Q ∩H = {0}, and set" }, { "blockId": "p31-ed3855f4fc", "page": 31, "quote": "ek0 = 2ℓ+r−2k0+1." }, { "blockId": "p31-12962946c5", "page": 31, "quote": "Assume q ≤ℓ, 3k0 −r > ℓ, (A.6), and ek0 ≤τ0/4. For Q ⊆W ⊆X of codimension i ≤r, say\nthat (g, W) occurs if g : W →F2 is linear and" }, { "blockId": "p31-688afbcee3", "page": 31, "quote": "Pr\nQ⊆L⊆W, dim L=ℓ[F[L] = g|L] > τi." }, { "blockId": "p31-bfbce5ead0", "page": 31, "quote": "Call it maximal if it has no occurring extension to a strictly larger space. Every maximal\noccurring pair satisfies H ⊆W and g|H = h." } ], "proof": [ { "blockId": "p31-7ddb6dc30c", "page": 31, "quote": "Proof. Put H′ = H ∩W. In W/Q, the image (H′ + Q)/Q has dimension dim H′ because\nQ ∩H = {0}. Counting nonzero vectors in a uniform (ℓ−q)-space gives" }, { "blockId": "p31-6fcb94c4dc", "page": 31, "quote": "dW :=\nPr\nQ⊆L⊆W[L ∩H ̸= {0}]" }, { "blockId": "p31-e7f6244d22", "page": 31, "quote": "≤(2dim H′ −1)(2ℓ−q −1)" }, { "blockId": "p31-b166cf97de", "page": 31, "quote": "2dim W−q −1\n≤2ℓ+dim H′−dim W+1 ≤ek0." }, { "blockId": "p31-380e6ce143", "page": 31, "quote": "The same bound holds with W replaced by any containing space of codimension at most r." }, { "blockId": "p32-6cf9d2c867", "page": 32, "quote": "Partition the valid spaces L with Q ⊆L ⊆W by Y = L + H′. Within a fixed class, the\ntable is the restriction of one linear function on Y respecting h|H′; this follows by restricting\nfL+H. If g|H′ ̸= h|H′, the difference of these two functions is nonzero on H′. A complement to\nH′ in Y containing Q is a graph of a linear map on an (ℓ−q)-dimensional quotient. Each of its\nℓ−q independent lift choices must satisfy one nontrivial linear equation for the two functions to\nagree. Thus agreement within this class is at most 2q−ℓ(and is zero if the functions already\ndisagree on Q). Total agreement is at most" }, { "blockId": "p32-32ec19b100", "page": 32, "quote": "dW + (1 −dW )2q−ℓ≤ek0 + 2q−ℓ≤τ0/2 < τi," }, { "blockId": "p32-83f71e51c1", "page": 32, "quote": "a contradiction. Hence g|H′ = h|H′.\nThere is consequently a unique extension eg to f\nW = W + H respecting h. Under either of\nthe two uniform distributions on valid ℓ-spaces containing Q in W and in f\nW, the image of L in" }, { "blockId": "p32-7df251af67", "page": 32, "quote": "W/(W ∩H) ≃f\nW/H" }, { "blockId": "p32-cbe513b5ca", "page": 32, "quote": "is uniform among the ℓ-spaces containing the image of Q. Indeed, every such image has the\nsame number of complements containing Q, namely 2(ℓ−q) dim(H∩W) in W and 2(ℓ−q) dim H in f\nW.\nMoreover, agreement depends only on this image: fL+H and eg agree on H, so their difference\nvanishes on one complement precisely when it vanishes on every complement. The conditional\nagreement on valid spaces is therefore identical in the two ambient spaces. Since agreement in\nW is at least τi, agreement in f\nW is at least" }, { "blockId": "p32-f7058861a5", "page": 32, "quote": "1 −dW\n≥(1 −ek0)(τi −ek0) ≥τi −2ek0 ≥τi/2." }, { "blockId": "p32-ec9e33264f", "page": 32, "quote": "(1 −d e\nW )τi −dW" }, { "blockId": "p32-b74edd162a", "page": 32, "quote": "If H ̸⊆W, the codimension j of f\nW is strictly smaller than i, and τj ≤τi−1 < τi/2. This gives\nan occurring extension, contradicting maximality." }, { "blockId": "p32-8cac9e4355", "page": 32, "quote": "Applying this lemma with X = VU, H = HU, and h = hU shows that every maximal pair\ndecoded from FU respects all equations in U, provided Q ∩HU = {0}. We next estimate the\nprobability of that condition under the distribution used by the source’s soundness proof." }, { "blockId": "p32-e4f1444ea5", "page": 32, "quote": "A.3\nSoundness, completeness, and the parameter order" }, { "blockId": "p32-c85fcfca35", "page": 32, "quote": "The advice distribution.\nThe outer game in [6, Section 5.2] first samples (U, V ), then takes\nQ uniformly from the q-spaces of VV . It sends (U, Q) and (V, Q) to its two players and asks\nfor assignments on var(U) and V , respectively. It accepts when those assignments agree on V\nand the first satisfies every equation of U. If f0 equations contributed all three variables, then\ndim VV = k0 + 2f0 and dim H(V ) = f0. Consequently, under this actual distribution, uniformly\nin U, V ," }, { "blockId": "p32-413730e086", "page": 32, "quote": "Pr[Q ∩HU ̸= {0} | U, V ] ≤(2f0 −1)(2q −1)" }, { "blockId": "p32-85bfea9df7", "page": 32, "quote": "2k0+2f0 −1\n≤2q−k0.\n(A.7)" }, { "blockId": "p32-f24fc582c3", "page": 32, "quote": "We further increase k0 until" }, { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" }, { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p33-f5108712f2", "page": 33, "quote": "Pr[Q ⊆L | V ] =\nNq(ℓ)\nNq(dim VV )." }, { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, { "blockId": "p33-fe1104a565", "page": 33, "quote": "TV(LU,Q, L′\nU,Q) ≤2ℓ+5p" }, { "blockId": "p33-8960e84e9f", "page": 33, "quote": "β k1/4\n0\n," }, { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, { "blockId": "p33-c98c99efb8", "page": 33, "quote": "(" }, { "blockId": "p33-6794ebf1cb", "page": 33, "quote": "Tγ[L/Q] =" }, { "blockId": "p33-308d8f1893", "page": 33, "quote": "FV [L] −pγ|L,\nFV [L]|Q = γ,\n0,\nFV [L]|Q ̸= γ." }, { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." }, { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p34-7df6b3632a", "page": 34, "quote": "codimVV (W ∩VV ) = i,\nPr\nQ⊆L⊆W∩VV [FV [L] = f|L] > τi/2." }, { "blockId": "p34-be24debe49", "page": 34, "quote": "Here L has dimension ℓ. Set S0 = W ∩VV and γ = f|Q. Every equality FV [L] = f|L in this\ndisplay is retained in Tγ. Since ζi ≤τ0/2 ≤τi/2, the quotient pair" }, { "blockId": "p34-78524b1e91", "page": 34, "quote": "f|S0 −pγ|S0, S0/Q\n" }, { "blockId": "p34-56088078a9", "page": 34, "quote": "occurs for Tγ and hence has a maximal occurring extension in that auxiliary table. Lifting any\nsuch extension by the same pγ gives a function agreeing with f on all of S0. This conclusion\nconcerns function extension and is unaffected by the extra agreements that zero completion may\nintroduce.\nThe second player selects this tag and one such maximal extension with probability at\nleast 1/M, where M := 2qM0 is independent of k0. Its eventual answer then agrees with f\non W ∩VV . The first player’s uniform extension agrees with this answer on all of VV with\nprobability 2−i ≥2−r: the quotient VV /(W ∩VV ) has dimension i. Thus, conditionally on\nE1 ∩E2 ∩E3 ∩E4, the strategy wins with probability at least" }, { "blockId": "p34-2d55acb379", "page": 34, "quote": "c∗:= 2−r 0.3τ0" }, { "blockId": "p34-04997566be", "page": 34, "quote": "M\n> 0." }, { "blockId": "p34-380ad96976", "page": 34, "quote": "The overall success probability is at least δα(ℓ)c∗/8, a positive constant independent of k0. The\nupper bound in [6, Lemma 5.4] tends to zero as βk0 →∞. Choosing k0 sufficiently large gives\nthe same contradiction as in the source and proves that raw valid acceptance is at most δ." }, { "blockId": "p34-80a9431745", "page": 34, "quote": "Fixing soundness before completeness.\nFor the game in Theorem 2.2, take δ = b/4 and\nretain the valid raw weights. Their total mass qedge is at least 1/2 by the preceding rejection\nestimates. Normalization therefore gives" }, { "blockId": "p34-20357bc1a9", "page": 34, "quote": "Val ≤δ/qedge ≤b/2 < b." }, { "blockId": "p34-d3a9576af9", "page": 34, "quote": "Every restriction above concerns only b and parameters fixed from it. After choosing ℓ, take\nk0 = j4 and β = j−3 with j sufficiently large. Then βk0 →∞, β√k0 →0, and √β k1/4\n0\n→0.\nThese limits satisfy the outer-game soundness and covering requirements, including 2ℓβ ≤1/8,\nas well as all the displayed margins. In particular, the alphabet dimension n = ℓ−1 has been\nfixed using only b." }, { "blockId": "p35-4807687585", "page": 35, "quote": "Completeness with the alphabet fixed.\nLet α : Z →F2 satisfy all but an ε fraction\nof the input equations, and also write α : V0 →F2 for its linear extension. Give the folded\nleft question C the answer α|RC and give (V, L′) the answer α|L′. Whenever every equation of\nsampled U is satisfied, hU = α|HU , so (A.2) is exactly α|RC+HU . The edge is therefore satisfied.\nThis argument concerns the sampled edge; it does not require every member of its folding class\nto involve satisfied equations.\nBefore either conditioning step, a union bound gives probability at most k0ε of sampling an\nunsatisfied equation. Since the total retained probability qall is at least 1/2, the conditioned\ngame’s value is at least 1 −k0ε/qall ≥1 −2k0ε. After ℓ, k0 have been fixed for soundness, choose\nε ≤a/(2k0) in the initial 3Lin gap. This gives the asserted completeness without altering n." }, { "blockId": "p35-5bbe01a21e", "page": 35, "quote": "Effectivity and size.\nAll spaces in an individual question have bounded dimension. There\nare polynomially many k0-tuples of input equations, and each tuple has only constantly many\nsubspaces, bases, and local sampling outcomes. The folding equivalence relation is explicitly\ntestable by linear algebra. To find RC effectively, start from any (U, L) ∈C and enumerate\nthe ℓ-spaces inside L + HU, whose dimension is the fixed constant ℓ+ k0; test each candidate\nagainst (A.1) for every class member. Lemma 4.2 guarantees a successful candidate. Gaussian\nelimination then computes bases and the affine edge maps. The finite sampling distributions,\ntheir conditioning, and the aggregation of parallel edges have rational weights of polynomial\nbit length. These observations give the deterministic polynomial reduction claimed in the\nproposition. This completes the proof of Theorem 2.2." } ], "guide": [] }, { "id": "lemma-19", "kind": "lemma", "label": "Lemma B.1", "source": { "blockId": "p35-be0e319110", "page": 35, "quote": "Lemma B.1. Let G be a graph on N vertices with nonnegative rational weights wuv on unordered\ndistinct vertex pairs, and P\nu 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" }, "statement": [ { "blockId": "p35-be0e319110", "page": 35, "quote": "Lemma B.1. Let G be a graph on N vertices with nonnegative rational weights wuv on unordered\ndistinct vertex pairs, and P\nu 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" }, { "blockId": "p35-b21848e281", "page": 35, "quote": "Q\n−MaxCut(G)\n ≤ϵ′." }, { "blockId": "p35-9c43ca15b1", "page": 35, "quote": "The same conclusion holds when the input graph has loops, after deleting them without renor-\nmalizing its other weights." } ], "proof": [ { "blockId": "p35-a5b2600e3a", "page": 35, "quote": "Proof. Choose a prime p > max{2, 2N2/ϵ′}. A prime of polynomial size can be found determin-\nistically by testing successive integers, using Bertrand’s postulate to bound the search. Replace\nevery vertex by a cluster indexed by F6\np. For each u < v let Iuv ⊆Fp be any fixed set of ⌊pwuv⌋\nresidues, for example the first such residues in the usual ordering. Join x in cluster u to y in\ncluster v exactly when\nx · y ∈Iuv." }, { "blockId": "p36-5f81808622", "page": 36, "quote": "There are no edges within clusters. This defines a simple graph on Np6 vertices, constructible\nby enumerating all possible pairs. Take Q = p12.\nWe first estimate its edge count on arbitrary rectangles between two clusters. Fix a residue\nset I and subsets S, T ⊆F6\np. Writing ψ(a) = exp(2πia/p), Fourier inversion gives" }, { "blockId": "p36-07973c5c0d", "page": 36, "quote": "X" }, { "blockId": "p36-958ee9d27e", "page": 36, "quote": "1I(a) =\nX" }, { "blockId": "p36-8cf87e4236", "page": 36, "quote": "p" }, { "blockId": "p36-66fc5872b6", "page": 36, "quote": "a∈I\nψ(−ha),\n|βh| ≤1." }, { "blockId": "p36-bd161adfea", "page": 36, "quote": "h∈Fp\nβhψ(ha),\nβh = 1" }, { "blockId": "p36-9ce14bea1c", "page": 36, "quote": "For h ̸= 0, the matrix Mh(x, y) = ψ(hx · y) satisfies" }, { "blockId": "p36-bdee2d4429", "page": 36, "quote": "MhM∗\nh = p6I," }, { "blockId": "p36-b6b27453f9", "page": 36, "quote": "|S||T|" }, { "blockId": "p36-a033e1bdfc", "page": 36, "quote": "p |S||T|\n ≤\nX" }, { "blockId": "p36-87947acdb3", "page": 36, "quote": "by character orthogonality, so its operator norm is p3. Consequently,\n#{(x, y) ∈S × T : x · y ∈I} −|I|" }, { "blockId": "p36-4ee1626576", "page": 36, "quote": "h̸=0\n|βh|p3q" }, { "blockId": "p36-b05edcdd9f", "page": 36, "quote": "≤p10." }, { "blockId": "p36-cd18b0c605", "page": 36, "quote": "p12\n−wuv\n|S|" }, { "blockId": "p36-54fe45d855", "page": 36, "quote": "p6\n|T|" }, { "blockId": "p36-da57b18f6b", "page": 36, "quote": "p6" }, { "blockId": "p36-4579cd2c8e", "page": 36, "quote": "p2 .\n(B.1)" }, { "blockId": "p36-15379ba8bb", "page": 36, "quote": "p + 1" }, { "blockId": "p36-1f2c375563", "page": 36, "quote": "For I = Iuv, division by p12 and\n|I|/p −wuv\n ≤1/p show that\n\neuv(S, T)" }, { "blockId": "p36-d285345cbd", "page": 36, "quote": "≤1" }, { "blockId": "p36-82d91c1daf", "page": 36, "quote": "Now fix an arbitrary cut of G′, and let au ∈[0, 1] be the fraction of cluster u on its positive\nside. Its crossing edges between clusters u, v form two rectangles. Equation (B.1) therefore\ncompares their scaled count with" }, { "blockId": "p36-cbf457085c", "page": 36, "quote": "wuv\nau(1 −av) + (1 −au)av\n" }, { "blockId": "p36-301ecd5146", "page": 36, "quote": "to error at most 2/p + 2/p2. Summing over the at most N(N −1)/2 pairs costs at most\nN(N −1)(1/p + 1/p2). The displayed weighted expression, summed over pairs, is the expected\ncut weight obtained by independently assigning vertex u a positive sign with probability au. It\nis at most MaxCut(G). This proves the desired upper bound for every cut of G′.\nConversely, give all vertices in each cluster the sign of a maximum cut of G. The same\nrectangle bound compares its scaled weight with MaxCut(G) to at most the same error, giving\nthe reverse inequality. Our choice of p ensures N(N −1)(1/p + 1/p2) < ϵ′, which proves the\nassertion. Loops contribute zero to every cut and can be deleted at the outset." } ], "guide": [] }, { "id": "external-20", "kind": "external", "label": "Reference [1]", "source": { "blockId": "p36-1fb3382d1e", "page": 36, "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": "p36-1fb3382d1e", "page": 36, "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-21", "kind": "external", "label": "Reference [2]", "source": { "blockId": "p36-988b98a811", "page": 36, "quote": "[2] Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization\nof NP. Journal of the ACM, 45(1):70–122, 1998. Preliminary version in FOCS 1992." }, "statement": [ { "blockId": "p36-988b98a811", "page": 36, "quote": "[2] Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization\nof NP. 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-22", "kind": "external", "label": "Reference [3]", "source": { "blockId": "p36-9ecb96a4e1", "page": 36, "quote": "[3] Boaz Barak, Pravesh K. Kothari, and David Steurer. 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Journal of the ACM, 48(4):798–859,\n2001." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-30", "kind": "external", "label": "Reference [11]", "source": { "blockId": "p37-af50ade833", "page": 37, "quote": "[11] 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, New York, 1972." }, "statement": [ { "blockId": "p37-af50ade833", "page": 37, "quote": "[11] 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, New York, 1972." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-31", "kind": "external", "label": "Reference [12]", "source": { "blockId": "p37-1d0448d49b", "page": 37, "quote": "[12] Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the\nThirty-Fourth Annual ACM Symposium on Theory of Computing, pages 767–775. ACM,\n2002." }, "statement": [ { "blockId": "p37-1d0448d49b", "page": 37, "quote": "[12] Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the\nThirty-Fourth Annual ACM Symposium on Theory of Computing, pages 767–775. ACM,\n2002." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-32", "kind": "external", "label": "Reference [13]", "source": { "blockId": "p37-0e5abec217", "page": 37, "quote": "[13] Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapprox-\nimability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing,\n37(1):319–357, 2007." }, "statement": [ { "blockId": "p37-0e5abec217", "page": 37, "quote": "[13] Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapprox-\nimability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing,\n37(1):319–357, 2007." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-33", "kind": "external", "label": "Reference [14]", "source": { "blockId": "p37-1db23b8bfb", "page": 37, "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." }, "statement": [ { "blockId": "p37-1db23b8bfb", "page": 37, "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." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-34", "kind": "external", "label": "Reference [15]", "source": { "blockId": "p37-9f6572f634", "page": 37, "quote": "[15] Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games and\nGrassmann graphs. Theory of Computing, 21(10):1–55, 2025. Preliminary version in STOC\n2017." }, "statement": [ { "blockId": "p37-9f6572f634", "page": 37, "quote": "[15] Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games and\nGrassmann graphs. Theory of Computing, 21(10):1–55, 2025. Preliminary version in STOC\n2017." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-35", "kind": "external", "label": "Reference [16]", "source": { "blockId": "p37-771f692f0f", "page": 37, "quote": "[16] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions\nwith low influences: Invariance and optimality. Annals of Mathematics, 171(1):295–341,\n2010." }, "statement": [ { "blockId": "p37-771f692f0f", "page": 37, "quote": "[16] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions\nwith low influences: Invariance and optimality. Annals of Mathematics, 171(1):295–341,\n2010." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-36", "kind": "external", "label": "Reference [17]", "source": { "blockId": "p37-96b6c1fd67", "page": 37, "quote": "[17] Ryan O’Donnell and Yi Wu. An optimal SDP algorithm for Max-Cut, and equally optimal\nlong code tests. In Proceedings of the Fortieth Annual ACM Symposium on Theory of\nComputing, pages 335–344. ACM, 2008." }, "statement": [ { "blockId": "p37-96b6c1fd67", "page": 37, "quote": "[17] Ryan O’Donnell and Yi Wu. An optimal SDP algorithm for Max-Cut, and equally optimal\nlong code tests. In Proceedings of the Fortieth Annual ACM Symposium on Theory of\nComputing, pages 335–344. ACM, 2008." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-37", "kind": "external", "label": "Reference [18]", "source": { "blockId": "p38-f69daa4e2b", "page": 38, "quote": "[18] Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP?\nIn Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pages\n245–254. ACM, 2008." }, "statement": [ { "blockId": "p38-f69daa4e2b", "page": 38, "quote": "[18] Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP?\nIn Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pages\n245–254. ACM, 2008." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "external-38", "kind": "external", "label": "Reference [19]", "source": { "blockId": "p38-38319912c0", "page": 38, "quote": "[19] Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, and David P. Williamson. Gadgets,\napproximation, and linear programming. SIAM Journal on Computing, 29(6):2074–2097,\n2000." }, "statement": [ { "blockId": "p38-38319912c0", "page": 38, "quote": "[19] Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, and David P. Williamson. Gadgets,\napproximation, and linear programming. SIAM Journal on Computing, 29(6):2074–2097,\n2000." } ], "proof": [], "guide": [], "externalStatus": "Bibliographic entry only; external source not retrieved." }, { "id": "equation-39", "kind": "equation", "label": "Equation (1.1)", "source": { "blockId": "p1-1700164c77", "page": 1, "quote": "π(1 −ρ) = 0.878567 . . . .\n(1.1)" }, "statement": [ { "blockId": "p1-1700164c77", "page": 1, "quote": "π(1 −ρ) = 0.878567 . . . .\n(1.1)" } ], "proof": [], "guide": [] }, { "id": "equation-40", "kind": "equation", "label": "Equation (1.2)", "source": { "blockId": "p1-875921d5b7", "page": 1, "quote": "π arcsin t,\n0 < t < 1.\n(1.2)" }, "statement": [ { "blockId": "p1-875921d5b7", "page": 1, "quote": "π arcsin t,\n0 < t < 1.\n(1.2)" } ], "proof": [], "guide": [] }, { "id": "equation-41", "kind": "equation", "label": "Equation (2.1)", "source": { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" }, "statement": [ { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" } ], "proof": [], "guide": [] }, { "id": "equation-42", "kind": "equation", "label": "Equation (2.2)", "source": { "blockId": "p5-537b4ee14b", "page": 5, "quote": "d\nTrf(C) = r|C| bf(C).\n(2.2)" }, "statement": [ { "blockId": "p5-537b4ee14b", "page": 5, "quote": "d\nTrf(C) = r|C| bf(C).\n(2.2)" } ], "proof": [], "guide": [] }, { "id": "equation-43", "kind": "equation", "label": "Equation (2.3)", "source": { "blockId": "p5-54e1c6b4a5", "page": 5, "quote": "a∈Ω\nInfa(P≤Kh) ≤K∥h∥2\n2.\n(2.3)" }, "statement": [ { "blockId": "p5-54e1c6b4a5", "page": 5, "quote": "a∈Ω\nInfa(P≤Kh) ≤K∥h∥2\n2.\n(2.3)" } ], "proof": [], "guide": [] }, { "id": "equation-44", "kind": "equation", "label": "Equation (2.4)", "source": { "blockId": "p5-5609145f4b", "page": 5, "quote": "S⊆Ω\nt|S|bh(S)2.\n(2.4)" }, "statement": [ { "blockId": "p5-5609145f4b", "page": 5, "quote": "S⊆Ω\nt|S|bh(S)2.\n(2.4)" } ], "proof": [], "guide": [] }, { "id": "equation-45", "kind": "equation", "label": "Equation (3.1)", "source": { "blockId": "p7-477ca2f74c", "page": 7, "quote": "Fv(z),\n(−1)zg = Zg,0Zg,1.\n(3.1)" }, "statement": [ { "blockId": "p7-477ca2f74c", "page": 7, "quote": "Fv(z),\n(−1)zg = Zg,0Zg,1.\n(3.1)" } ], "proof": [], "guide": [] }, { "id": "equation-46", "kind": "equation", "label": "Equation (3.2)", "source": { "blockId": "p7-6aee721e9d", "page": 7, "quote": "E\nW(R1, F)(a)W(R2, F ′)(a) | D\n = tr2d.\n(3.2)" }, "statement": [ { "blockId": "p7-6aee721e9d", "page": 7, "quote": "E\nW(R1, F)(a)W(R2, F ′)(a) | D\n = tr2d.\n(3.2)" } ], "proof": [], "guide": [] }, { "id": "equation-47", "kind": "equation", "label": "Equation (3.3)", "source": { "blockId": "p8-4efa298442", "page": 8, "quote": "i∈P\nxi,\n|P| = d,\n(3.3)" }, "statement": [ { "blockId": "p8-4efa298442", "page": 8, "quote": "i∈P\nxi,\n|P| = d,\n(3.3)" } ], "proof": [], "guide": [] }, { "id": "equation-48", "kind": "equation", "label": "Equation (3.4)", "source": { "blockId": "p8-7efa825524", "page": 8, "quote": "f(R, F) = ϕ(W(R, F))\n(3.4)" }, "statement": [ { "blockId": "p8-7efa825524", "page": 8, "quote": "f(R, F) = ϕ(W(R, F))\n(3.4)" } ], "proof": [], "guide": [] }, { "id": "equation-49", "kind": "equation", "label": "Equation (4.1)", "source": { "blockId": "p9-b5d19f87a9", "page": 9, "quote": ",\nx′, −W(R2, F ′) ◦πe′.\n(4.1)" }, "statement": [ { "blockId": "p9-b5d19f87a9", "page": 9, "quote": ",\nx′, −W(R2, F ′) ◦πe′.\n(4.1)" } ], "proof": [], "guide": [] }, { "id": "equation-50", "kind": "equation", "label": "Equation (4.2)", "source": { "blockId": "p9-42fcfc6700", "page": 9, "quote": "2\n−tµ −2Ta.\n(4.2)" }, "statement": [ { "blockId": "p9-42fcfc6700", "page": 9, "quote": "2\n−tµ −2Ta.\n(4.2)" } ], "proof": [], "guide": [] }, { "id": "equation-51", "kind": "equation", "label": "Equation (4.3)", "source": { "blockId": "p10-faaa210733", "page": 10, "quote": "Ge(R, F) = σo\nx(e)(W(R, F) ◦πe),\n(4.3)" }, "statement": [ { "blockId": "p10-faaa210733", "page": 10, "quote": "Ge(R, F) = σo\nx(e)(W(R, F) ◦πe),\n(4.3)" } ], "proof": [], "guide": [] }, { "id": "equation-52", "kind": "equation", "label": "Equation (4.4)", "source": { "blockId": "p10-872453b709", "page": 10, "quote": "Gy(R, F) = Ee|yGe(R, F),\n¯Gy = TrGy.\n(4.4)" }, "statement": [ { "blockId": "p10-872453b709", "page": 10, "quote": "Gy(R, F) = Ee|yGe(R, F),\n¯Gy = TrGy.\n(4.4)" } ], "proof": [], "guide": [] }, { "id": "equation-53", "kind": "equation", "label": "Equation (4.5)", "source": { "blockId": "p10-542d9b17d9", "page": 10, "quote": "2E StabFD\nt\n( ¯Ay),\n(4.5)" }, "statement": [ { "blockId": "p10-542d9b17d9", "page": 10, "quote": "2E StabFD\nt\n( ¯Ay),\n(4.5)" } ], "proof": [], "guide": [] }, { "id": "equation-54", "kind": "equation", "label": "Equation (4.6)", "source": { "blockId": "p10-1301175d92", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy).\n(4.6)" }, "statement": [ { "blockId": "p10-1301175d92", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy).\n(4.6)" } ], "proof": [], "guide": [] }, { "id": "equation-55", "kind": "equation", "label": "Equation (5.1)", "source": { "blockId": "p11-f3c7473fca", "page": 11, "quote": "bfC(F)χC(R),\nb¯fC(F) = r|C| bfC(F).\n(5.1)" }, "statement": [ { "blockId": "p11-f3c7473fca", "page": 11, "quote": "bfC(F)χC(R),\nb¯fC(F) = r|C| bfC(F).\n(5.1)" } ], "proof": [], "guide": [] }, { "id": "equation-56", "kind": "equation", "label": "Equation (5.2)", "source": { "blockId": "p11-826b929997", "page": 11, "quote": "qv,g0 + qv,g1 = qv\n(g ∈[s]).\n(5.2)" }, "statement": [ { "blockId": "p11-826b929997", "page": 11, "quote": "qv,g0 + qv,g1 = qv\n(g ∈[s]).\n(5.2)" } ], "proof": [], "guide": [] }, { "id": "equation-57", "kind": "equation", "label": "Equation (5.3)", "source": { "blockId": "p11-fe2c2859c8", "page": 11, "quote": "bfC(F v,−) = (−1)qv(1) bfC(F),\n(5.3)" }, "statement": [ { "blockId": "p11-fe2c2859c8", "page": 11, "quote": "bfC(F v,−) = (−1)qv(1) bfC(F),\n(5.3)" } ], "proof": [], "guide": [] }, { "id": "equation-58", "kind": "equation", "label": "Equation (5.4)", "source": { "blockId": "p11-49471e5bc7", "page": 11, "quote": "g zgqv,g1(1) bfC(F).\n(5.4)" }, "statement": [ { "blockId": "p11-49471e5bc7", "page": 11, "quote": "g zgqv,g1(1) bfC(F).\n(5.4)" } ], "proof": [], "guide": [] }, { "id": "equation-59", "kind": "equation", "label": "Equation (5.5)", "source": { "blockId": "p12-091c8a397c", "page": 12, "quote": "v∈D\n∥(I −Sv) ¯f∥2\n2,R,F ≤\ns\n2emµ.\n(5.5)" }, "statement": [ { "blockId": "p12-091c8a397c", "page": 12, "quote": "v∈D\n∥(I −Sv) ¯f∥2\n2,R,F ≤\ns\n2emµ.\n(5.5)" } ], "proof": [], "guide": [] }, { "id": "equation-60", "kind": "equation", "label": "Equation (6.1)", "source": { "blockId": "p14-b98783d5c4", "page": 14, "quote": "z∈Fs\n2\n∥U(z)∥2 = M.\n(6.1)" }, "statement": [ { "blockId": "p14-b98783d5c4", "page": 14, "quote": "z∈Fs\n2\n∥U(z)∥2 = M.\n(6.1)" } ], "proof": [], "guide": [] }, { "id": "equation-61", "kind": "equation", "label": "Equation (6.2)", "source": { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" }, "statement": [ { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" } ], "proof": [], "guide": [] }, { "id": "equation-62", "kind": "equation", "label": "Equation (6.3)", "source": { "blockId": "p15-670fa7848e", "page": 15, "quote": ".\n(6.3)" }, "statement": [ { "blockId": "p15-670fa7848e", "page": 15, "quote": ".\n(6.3)" } ], "proof": [], "guide": [] }, { "id": "equation-63", "kind": "equation", "label": "Equation (6.4)", "source": { "blockId": "p15-e5cedc4c6c", "page": 15, "quote": "Fourier inversion in z gives\nU(z) = Ep(−1)p·zAp(Sp).\n(6.4)" }, "statement": [ { "blockId": "p15-e5cedc4c6c", "page": 15, "quote": "Fourier inversion in z gives\nU(z) = Ep(−1)p·zAp(Sp).\n(6.4)" } ], "proof": [], "guide": [] }, { "id": "equation-64", "kind": "equation", "label": "Equation (6.5)", "source": { "blockId": "p15-21f25791dc", "page": 15, "quote": "q\n∥αq(p)∥2,\nEpmp = M,\nEp∥αq(p)∥2 = wq.\n(6.5)" }, "statement": [ { "blockId": "p15-21f25791dc", "page": 15, "quote": "q\n∥αq(p)∥2,\nEpmp = M,\nEp∥αq(p)∥2 = wq.\n(6.5)" } ], "proof": [], "guide": [] }, { "id": "equation-65", "kind": "equation", "label": "Equation (6.6)", "source": { "blockId": "p16-21fe70f778", "page": 16, "quote": "D2\n.\n(6.6)" }, "statement": [ { "blockId": "p16-21fe70f778", "page": 16, "quote": "D2\n.\n(6.6)" } ], "proof": [], "guide": [] }, { "id": "equation-66", "kind": "equation", "label": "Equation (6.7)", "source": { "blockId": "p16-f67cba1c4e", "page": 16, "quote": ".\n(6.7)" }, "statement": [ { "blockId": "p16-f67cba1c4e", "page": 16, "quote": ".\n(6.7)" } ], "proof": [], "guide": [] }, { "id": "equation-67", "kind": "equation", "label": "Equation (6.8)", "source": { "blockId": "p16-961cce19bb", "page": 16, "quote": "= Ep1,p2,p3E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n,\np4 = p1 + p2 + p3.\n(6.8)" }, "statement": [ { "blockId": "p16-961cce19bb", "page": 16, "quote": "= Ep1,p2,p3E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n,\np4 = p1 + p2 + p3.\n(6.8)" } ], "proof": [], "guide": [] }, { "id": "equation-68", "kind": "equation", "label": "Equation (6.9)", "source": { "blockId": "p16-6e934d6f5f", "page": 16, "quote": "q\n⟨α′\nq(p1), α′\nq(p2)⟩⟨α′\nq(p3), α′\nq(p4)⟩.\n(6.9)" }, "statement": [ { "blockId": "p16-6e934d6f5f", "page": 16, "quote": "q\n⟨α′\nq(p1), α′\nq(p2)⟩⟨α′\nq(p3), α′\nq(p4)⟩.\n(6.9)" } ], "proof": [], "guide": [] }, { "id": "equation-69", "kind": "equation", "label": "Equation (6.10)", "source": { "blockId": "p17-a5d7732e54", "page": 17, "quote": "h .\n(6.10)" }, "statement": [ { "blockId": "p17-a5d7732e54", "page": 17, "quote": "h .\n(6.10)" } ], "proof": [], "guide": [] }, { "id": "equation-70", "kind": "equation", "label": "Equation (7.1)", "source": { "blockId": "p18-19a1cb39ea", "page": 18, "quote": "| bf shifts(Q)|2.\n(7.1)" }, "statement": [ { "blockId": "p18-19a1cb39ea", "page": 18, "quote": "| bf shifts(Q)|2.\n(7.1)" } ], "proof": [], "guide": [] }, { "id": "equation-71", "kind": "equation", "label": "Equation (7.2)", "source": { "blockId": "p18-1a521f670f", "page": 18, "quote": "q∈A\nmq(f; Γ, Fout) ≤1.\n(7.2)" }, "statement": [ { "blockId": "p18-1a521f670f", "page": 18, "quote": "q∈A\nmq(f; Γ, Fout) ≤1.\n(7.2)" } ], "proof": [], "guide": [] }, { "id": "equation-72", "kind": "equation", "label": "Equation (7.3)", "source": { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, "statement": [ { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" } ], "proof": [], "guide": [] }, { "id": "equation-73", "kind": "equation", "label": "Equation (7.4)", "source": { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, "statement": [ { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" } ], "proof": [], "guide": [] }, { "id": "equation-74", "kind": "equation", "label": "Equation (7.5)", "source": { "blockId": "p19-2048fcebc1", "page": 19, "quote": "8 .\n(7.5)" }, "statement": [ { "blockId": "p19-2048fcebc1", "page": 19, "quote": "8 .\n(7.5)" } ], "proof": [], "guide": [] }, { "id": "equation-75", "kind": "equation", "label": "Equation (7.6)", "source": { "blockId": "p19-2bd4dfb0c5", "page": 19, "quote": "8 =: Λ.\n(7.6)" }, "statement": [ { "blockId": "p19-2bd4dfb0c5", "page": 19, "quote": "8 =: Λ.\n(7.6)" } ], "proof": [], "guide": [] }, { "id": "equation-76", "kind": "equation", "label": "Equation (7.7)", "source": { "blockId": "p19-1fcabf9810", "page": 19, "quote": "z\nInfv,z(Hv) ≤K.\n(7.7)" }, "statement": [ { "blockId": "p19-1fcabf9810", "page": 19, "quote": "z\nInfv,z(Hv) ≤K.\n(7.7)" } ], "proof": [], "guide": [] }, { "id": "equation-77", "kind": "equation", "label": "Equation (7.8)", "source": { "blockId": "p19-5abdbe5cbd", "page": 19, "quote": "2s .\n(7.8)" }, "statement": [ { "blockId": "p19-5abdbe5cbd", "page": 19, "quote": "2s .\n(7.8)" } ], "proof": [], "guide": [] }, { "id": "equation-78", "kind": "equation", "label": "Equation (7.9)", "source": { "blockId": "p20-3e08f51158", "page": 20, "quote": "q∈A\nχq(LΣ)Ψq\nFv(· + J(q)), FD\\{v}\n.\n(7.9)" }, "statement": [ { "blockId": "p20-3e08f51158", "page": 20, "quote": "q∈A\nχq(LΣ)Ψq\nFv(· + J(q)), FD\\{v}\n.\n(7.9)" } ], "proof": [], "guide": [] }, { "id": "equation-79", "kind": "equation", "label": "Equation (7.10)", "source": { "blockId": "p20-dbdc944b36", "page": 20, "quote": "I\n∥aq,I∥2\n2,FD ≤mq( ¯fω; Γ, Fout).\n(7.10)" }, "statement": [ { "blockId": "p20-dbdc944b36", "page": 20, "quote": "I\n∥aq,I∥2\n2,FD ≤mq( ¯fω; Γ, Fout).\n(7.10)" } ], "proof": [], "guide": [] }, { "id": "equation-80", "kind": "equation", "label": "Equation (7.11)", "source": { "blockId": "p20-ef57cb6081", "page": 20, "quote": "2\n.\n(7.11)" }, "statement": [ { "blockId": "p20-ef57cb6081", "page": 20, "quote": "2\n.\n(7.11)" } ], "proof": [], "guide": [] }, { "id": "equation-81", "kind": "equation", "label": "Equation (7.12)", "source": { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" }, "statement": [ { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" } ], "proof": [], "guide": [] }, { "id": "equation-82", "kind": "equation", "label": "Equation (7.13)", "source": { "blockId": "p21-e512194a97", "page": 21, "quote": "z\nInfv,z(Hodd\nv\n) ≤K.\n(7.13)" }, "statement": [ { "blockId": "p21-e512194a97", "page": 21, "quote": "z\nInfv,z(Hodd\nv\n) ≤K.\n(7.13)" } ], "proof": [], "guide": [] }, { "id": "equation-83", "kind": "equation", "label": "Equation (7.14)", "source": { "blockId": "p21-b8e4cf0492", "page": 21, "quote": "K ,\np∗=\nΛ\n4K 2m .\n(7.14)" }, "statement": [ { "blockId": "p21-b8e4cf0492", "page": 21, "quote": "K ,\np∗=\nΛ\n4K 2m .\n(7.14)" } ], "proof": [], "guide": [] }, { "id": "equation-84", "kind": "equation", "label": "Equation (8.1)", "source": { "blockId": "p22-470d569f6e", "page": 22, "quote": "mq( ¯Gy; Γ, Fout) ≤EE, e|ymq(Ge; Γ + EΓ, Fout).\n(8.1)" }, "statement": [ { "blockId": "p22-470d569f6e", "page": 22, "quote": "mq( ¯Gy; Γ, Fout) ≤EE, e|ymq(Ge; Γ + EΓ, Fout).\n(8.1)" } ], "proof": [], "guide": [] }, { "id": "equation-85", "kind": "equation", "label": "Equation (8.2)", "source": { "blockId": "p22-7eded40296", "page": 22, "quote": "q∗= q∗(y, D, v, Γ, Fout) ∈arg max\nq∈A\nmq( ¯Gy; Γ, Fout)\n(8.2)" }, "statement": [ { "blockId": "p22-7eded40296", "page": 22, "quote": "q∗= q∗(y, D, v, Γ, Fout) ∈arg max\nq∈A\nmq( ¯Gy; Γ, Fout)\n(8.2)" } ], "proof": [], "guide": [] }, { "id": "equation-86", "kind": "equation", "label": "Equation (8.3)", "source": { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" }, "statement": [ { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" } ], "proof": [], "guide": [] }, { "id": "equation-87", "kind": "equation", "label": "Equation (8.4)", "source": { "blockId": "p23-8d3bd8ebde", "page": 23, "quote": "p0 := Pr(Za) = 2−n(2k−2s).\n(8.4)" }, "statement": [ { "blockId": "p23-8d3bd8ebde", "page": 23, "quote": "p0 := Pr(Za) = 2−n(2k−2s).\n(8.4)" } ], "proof": [], "guide": [] }, { "id": "equation-88", "kind": "equation", "label": "Equation (8.5)", "source": { "blockId": "p23-266720f7f6", "page": 23, "quote": "Tp0\n≤\n1\n2√Tp0\n.\n(8.5)" }, "statement": [ { "blockId": "p23-266720f7f6", "page": 23, "quote": "Tp0\n≤\n1\n2√Tp0\n.\n(8.5)" } ], "proof": [], "guide": [] }, { "id": "equation-89", "kind": "equation", "label": "Equation (8.6)", "source": { "blockId": "p23-d3c148ea4c", "page": 23, "quote": "T ≥\n1\np0(p∗θ)2 ,\n(8.6)" }, "statement": [ { "blockId": "p23-d3c148ea4c", "page": 23, "quote": "T ≥\n1\np0(p∗θ)2 ,\n(8.6)" } ], "proof": [], "guide": [] }, { "id": "equation-90", "kind": "equation", "label": "Equation (8.7)", "source": { "blockId": "p23-555402b059", "page": 23, "quote": "2 .\n(8.7)" }, "statement": [ { "blockId": "p23-555402b059", "page": 23, "quote": "2 .\n(8.7)" } ], "proof": [], "guide": [] }, { "id": "equation-91", "kind": "equation", "label": "Equation (8.8)", "source": { "blockId": "p24-242a8fe97e", "page": 24, "quote": "emeq( ef).\n(8.8)" }, "statement": [ { "blockId": "p24-242a8fe97e", "page": 24, "quote": "emeq( ef).\n(8.8)" } ], "proof": [], "guide": [] }, { "id": "equation-92", "kind": "equation", "label": "Equation (8.9)", "source": { "blockId": "p24-37148e8ed5", "page": 24, "quote": "2 .\n(8.9)" }, "statement": [ { "blockId": "p24-37148e8ed5", "page": 24, "quote": "2 .\n(8.9)" } ], "proof": [], "guide": [] }, { "id": "equation-93", "kind": "equation", "label": "Equation (9.1)", "source": { "blockId": "p27-c007b94b4a", "page": 27, "quote": "1 + t\n,\n(9.1)" }, "statement": [ { "blockId": "p27-c007b94b4a", "page": 27, "quote": "1 + t\n,\n(9.1)" } ], "proof": [], "guide": [] }, { "id": "equation-94", "kind": "equation", "label": "Equation (9.2)", "source": { "blockId": "p27-f5698fca36", "page": 27, "quote": "2\n+ ε + ϵ′.\n(9.2)" }, "statement": [ { "blockId": "p27-f5698fca36", "page": 27, "quote": "2\n+ ε + ϵ′.\n(9.2)" } ], "proof": [], "guide": [] }, { "id": "equation-95", "kind": "equation", "label": "Equation (A.1)", "source": { "blockId": "p28-a23e84293d", "page": 28, "quote": "RC + HU = L + HU\nfor every (U, L) ∈C.\n(A.1)" }, "statement": [ { "blockId": "p28-a23e84293d", "page": 28, "quote": "RC + HU = L + HU\nfor every (U, L) ∈C.\n(A.1)" } ], "proof": [], "guide": [] }, { "id": "equation-96", "kind": "equation", "label": "Equation (A.2)", "source": { "blockId": "p29-e4deb03dd4", "page": 29, "quote": "Both sides have dimension ℓ+ k0; consequently RC ∩HU = {0} as well. A folded answer is\na linear function σ : RC →F2. Its unfolding on (U, L) is the restriction to L of the unique\nextension\neσU(r + h) = σ(r) + hU(h),\nr ∈RC,\nh ∈HU.\n(A.2)" }, "statement": [ { "blockId": "p29-e4deb03dd4", "page": 29, "quote": "Both sides have dimension ℓ+ k0; consequently RC ∩HU = {0} as well. A folded answer is\na linear function σ : RC →F2. Its unfolding on (U, L) is the restriction to L of the unique\nextension\neσU(r + h) = σ(r) + hU(h),\nr ∈RC,\nh ∈HU.\n(A.2)" } ], "proof": [], "guide": [] }, { "id": "equation-97", "kind": "equation", "label": "Equation (A.3)", "source": { "blockId": "p29-2422708018", "page": 29, "quote": "We condition the sampling on legitimacy and L ∩HU = {0}. The discarded probability is at\nmost\nκ = O(k2\n0/|Z|) + 2ℓ−k0 + 2ℓ−2k0.\n(A.3)" }, "statement": [ { "blockId": "p29-2422708018", "page": 29, "quote": "We condition the sampling on legitimacy and L ∩HU = {0}. The discarded probability is at\nmost\nκ = O(k2\n0/|Z|) + 2ℓ−k0 + 2ℓ−2k0.\n(A.3)" } ], "proof": [], "guide": [] }, { "id": "equation-98", "kind": "equation", "label": "Equation (A.4)", "source": { "blockId": "p30-5c2cf5223d", "page": 30, "quote": "HU ∩VV = H(V ).\n(A.4)" }, "statement": [ { "blockId": "p30-5c2cf5223d", "page": 30, "quote": "HU ∩VV = H(V ).\n(A.4)" } ], "proof": [], "guide": [] }, { "id": "equation-99", "kind": "equation", "label": "Equation (A.5)", "source": { "blockId": "p30-9a2aac86b9", "page": 30, "quote": "FU[L] = FV [L]\n(L ⊆VV , dim L = ℓ).\n(A.5)" }, "statement": [ { "blockId": "p30-9a2aac86b9", "page": 30, "quote": "FU[L] = FV [L]\n(L ⊆VV , dim L = ℓ).\n(A.5)" } ], "proof": [], "guide": [] }, { "id": "equation-100", "kind": "equation", "label": "Equation (A.6)", "source": { "blockId": "p31-d15ae732c7", "page": 31, "quote": "2q−ℓ≤τ0/4.\n(A.6)" }, "statement": [ { "blockId": "p31-d15ae732c7", "page": 31, "quote": "2q−ℓ≤τ0/4.\n(A.6)" } ], "proof": [], "guide": [] }, { "id": "equation-101", "kind": "equation", "label": "Equation (A.7)", "source": { "blockId": "p32-85bfea9df7", "page": 32, "quote": "2k0+2f0 −1\n≤2q−k0.\n(A.7)" }, "statement": [ { "blockId": "p32-85bfea9df7", "page": 32, "quote": "2k0+2f0 −1\n≤2q−k0.\n(A.7)" } ], "proof": [], "guide": [] }, { "id": "equation-102", "kind": "equation", "label": "Equation (A.8)", "source": { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" }, "statement": [ { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" } ], "proof": [], "guide": [] }, { "id": "equation-103", "kind": "equation", "label": "Equation (B.1)", "source": { "blockId": "p36-4579cd2c8e", "page": 36, "quote": "p2 .\n(B.1)" }, "statement": [ { "blockId": "p36-4579cd2c8e", "page": 36, "quote": "p2 .\n(B.1)" } ], "proof": [], "guide": [] }, { "id": "external-104", "kind": "external", "label": "[16] Theorem 4.4", "source": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "statement": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p37-771f692f0f", "page": 37, "quote": "[16] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions\nwith low influences: Invariance and optimality. Annals of Mathematics, 171(1):295–341,\n2010." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-105", "kind": "external", "label": "[16] Theorem 4.1", "source": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "statement": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p37-771f692f0f", "page": 37, "quote": "[16] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions\nwith low influences: Invariance and optimality. Annals of Mathematics, 171(1):295–341,\n2010." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-106", "kind": "external", "label": "[16] Corollary 4.3", "source": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "statement": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p37-771f692f0f", "page": 37, "quote": "[16] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions\nwith low influences: Invariance and optimality. Annals of Mathematics, 171(1):295–341,\n2010." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-107", "kind": "external", "label": "[3] Theorem 9", "source": { "blockId": "p28-6ccfb497c5", "page": 28, "quote": "and Steurer [3, Theorems 9–11]; the introduction of the published version of [6] records this\ncompletion.\nWe first describe the folded game and its affine projections. The source’s vertex definition\nexcludes certain subspaces that its displayed sampler can produce. We specify the resulting\nconditioning, complete the tables used in its soundness proof, and give the invariant-list argument\nwith the needed exceptional-subspace estimates. For the second player we use bounded lists\non quotient spaces, keeping each restriction to the advice space separate. Completeness then\nfollows by choosing the initial 3Lin error last." }, "statement": [ { "blockId": "p28-6ccfb497c5", "page": 28, "quote": "and Steurer [3, Theorems 9–11]; the introduction of the published version of [6] records this\ncompletion.\nWe first describe the folded game and its affine projections. The source’s vertex definition\nexcludes certain subspaces that its displayed sampler can produce. We specify the resulting\nconditioning, complete the tables used in its soundness proof, and give the invariant-list argument\nwith the needed exceptional-subspace estimates. For the second player we use bounded lists\non quotient spaces, keeping each restriction to the advice space separate. Completeness then\nfollows by choosing the initial 3Lin error last." }, { "blockId": "p36-9ecb96a4e1", "page": 36, "quote": "[3] Boaz Barak, Pravesh K. Kothari, and David Steurer. Small-set expansion in shortcode\ngraph and the 2-to-2 Conjecture. In Avrim Blum, editor, 10th Innovations in Theoretical\nComputer Science Conference (ITCS 2019), volume 124 of Leibniz International Proceedings\nin Informatics, pages 9:1–9:12. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2019." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-108", "kind": "external", "label": "[6] Theorem 4.1", "source": { "blockId": "p28-0d80eef515", "page": 28, "quote": "Starting instance.\nTheorem 4.1 of [6], obtained from Håstad’s theorem [10], supplies an\nabsolute constant s∗< 1 such that for every sufficiently small ε > 0 it is NP-hard to distinguish\nregular binary 3Lin instances of value at least 1 −ε from those of value at most s∗. Regularity\nincludes bounded variable occurrence, three distinct variables per equation, and at most one\nvariable shared by two distinct equations.\nFor now let ℓ≥2, k0 ≥ℓ, and β ∈(0, 1) be fixed parameters, with β rational. The two\nalphabet dimensions will be ℓand ℓ−1, so the dimension in Theorem 2.2 is n = ℓ−1. The\nsoundness argument below chooses ℓ, k0, β using only b and the absolute constant s∗." }, "statement": [ { "blockId": "p28-0d80eef515", "page": 28, "quote": "Starting instance.\nTheorem 4.1 of [6], obtained from Håstad’s theorem [10], supplies an\nabsolute constant s∗< 1 such that for every sufficiently small ε > 0 it is NP-hard to distinguish\nregular binary 3Lin instances of value at least 1 −ε from those of value at most s∗. Regularity\nincludes bounded variable occurrence, three distinct variables per equation, and at most one\nvariable shared by two distinct equations.\nFor now let ℓ≥2, k0 ≥ℓ, and β ∈(0, 1) be fixed parameters, with β rational. The two\nalphabet dimensions will be ℓand ℓ−1, so the dimension in Theorem 2.2 is n = ℓ−1. The\nsoundness argument below chooses ℓ, k0, β using only b and the absolute constant s∗." }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result explicitly named with a citation; statement and proof not retrieved." }, { "id": "unresolved-109", "kind": "unresolved", "label": "External context: Lemma 4.2", "source": { "blockId": "p28-507fd52723", "page": 28, "quote": "Lemma 4.2 there proves that this is an equivalence relation and supplies, for every class C, a\nfixed ℓ-dimensional space RC such that" }, "statement": [ { "blockId": "p28-507fd52723", "page": 28, "quote": "Lemma 4.2 there proves that this is an equivalence relation and supplies, for every class C, a\nfixed ℓ-dimensional space RC such that" } ], "proof": [], "guide": [], "externalStatus": "The word “there” suggests an external reference; unresolved." }, { "id": "external-110", "kind": "external", "label": "[6] Claim 4.4", "source": { "blockId": "p29-c7b0fb6e28", "page": 29, "quote": "an affine bijection from R∗\nC to L∗. Composing it with the surjective restriction L∗→(L′)∗gives\na surjective affine map with a one-dimensional kernel in its linear part. Every fiber has exactly\ntwo elements. This proves the required affine promise after bases are fixed. Claim 4.4 of [6]\nshows that the projections agree when several unfolded edges are aggregated into the same\nfolded edge. Equivalently, these edges may be kept separately with their individual weights." }, "statement": [ { "blockId": "p29-c7b0fb6e28", "page": 29, "quote": "an affine bijection from R∗\nC to L∗. Composing it with the surjective restriction L∗→(L′)∗gives\na surjective affine map with a one-dimensional kernel in its linear part. Every fiber has exactly\ntwo elements. This proves the required affine promise after bases are fixed. Claim 4.4 of [6]\nshows that the projections agree when several unfolded edges are aggregated into the same\nfolded edge. Equivalently, these edges may be kept separately with their individual weights." }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result explicitly named with a citation; statement and proof not retrieved." }, { "id": "external-111", "kind": "external", "label": "[6] Lemma 3.7", "source": { "blockId": "p31-2563015c96", "page": 31, "quote": "Fix a soundness parameter δ ∈(0, 1).\nThe agreement lemma [6, Lemma 3.7] supplies\nconstants q, r, C and a positive function α(ℓ) at agreement threshold δ2/8, as in the proof of its\nLemma 5.3. Set the thresholds" }, "statement": [ { "blockId": "p31-2563015c96", "page": 31, "quote": "Fix a soundness parameter δ ∈(0, 1).\nThe agreement lemma [6, Lemma 3.7] supplies\nconstants q, r, C and a positive function α(ℓ) at agreement threshold δ2/8, as in the proof of its\nLemma 5.3. Set the thresholds" }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-112", "kind": "external", "label": "[6] Lemma 3.12", "source": { "blockId": "p31-39ad0ac56d", "page": 31, "quote": "Both vectors are fixed before choosing ℓ; the function α(ℓ) is positive once ℓis fixed. We choose\nℓlarge enough for that agreement lemma and for the ordinary list bound [6, Lemma 3.12] on\n(ℓ−q)-spaces with thresholds (ζi). We also require q ≤ℓ−1 and" }, "statement": [ { "blockId": "p31-39ad0ac56d", "page": 31, "quote": "Both vectors are fixed before choosing ℓ; the function α(ℓ) is positive once ℓis fixed. We choose\nℓlarge enough for that agreement lemma and for the ordinary list bound [6, Lemma 3.12] on\n(ℓ−q)-spaces with thresholds (ζi). We also require q ≤ℓ−1 and" }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-113", "kind": "external", "label": "[6] Lemma 5.3", "source": { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, "statement": [ { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-114", "kind": "external", "label": "[6] Claim 5.6", "source": { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, "statement": [ { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-115", "kind": "external", "label": "[6] Lemma 4.11", "source": { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, "statement": [ { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-116", "kind": "external", "label": "External context: Lemma 3.7", "source": { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, "statement": [ { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." } ], "proof": [], "guide": [], "externalStatus": "The prose identifies an external source, but its work identity, statement, and proof have not been retrieved." }, { "id": "external-117", "kind": "external", "label": "[6] Lemma 5.5", "source": { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, "statement": [ { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." }, { "id": "external-118", "kind": "external", "label": "[6] Lemma 5.4", "source": { "blockId": "p34-380ad96976", "page": 34, "quote": "The overall success probability is at least δα(ℓ)c∗/8, a positive constant independent of k0. The\nupper bound in [6, Lemma 5.4] tends to zero as βk0 →∞. Choosing k0 sufficiently large gives\nthe same contradiction as in the source and proves that raw valid acceptance is at most δ." }, "statement": [ { "blockId": "p34-380ad96976", "page": 34, "quote": "The overall success probability is at least δα(ℓ)c∗/8, a positive constant independent of k0. The\nupper bound in [6, Lemma 5.4] tends to zero as βk0 →∞. Choosing k0 sufficiently large gives\nthe same contradiction as in the source and proves that raw valid acceptance is at most δ." }, { "blockId": "p37-da425ed08d", "page": 37, "quote": "[6] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of\nthe 2-to-1 games conjecture? Theory of Computing, 21(11):1–50, 2025. Preliminary version\nin STOC 2018." } ], "proof": [], "guide": [], "externalStatus": "External result cited by this paper; statement and proof not retrieved." } ], "edges": [ { "id": "edge-1", "from": "equation-40", "to": "theorem-1", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p1-875921d5b7", "page": 1, "quote": "π arcsin t,\n0 < t < 1.\n(1.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-2", "from": "equation-45", "to": "definition-5", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p7-477ca2f74c", "page": 7, "quote": "Fv(z),\n(−1)zg = Zg,0Zg,1.\n(3.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-3", "from": "equation-46", "to": "lemma-6", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p7-6aee721e9d", "page": 7, "quote": "E\nW(R1, F)(a)W(R2, F ′)(a) | D\n = tr2d.\n(3.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-4", "from": "equation-47", "to": "lemma-6", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p8-4efa298442", "page": 8, "quote": "i∈P\nxi,\n|P| = d,\n(3.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-5", "from": "equation-48", "to": "lemma-6", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p8-7efa825524", "page": 8, "quote": "f(R, F) = ϕ(W(R, F))\n(3.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-6", "from": "equation-50", "to": "lemma-7", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p9-42fcfc6700", "page": 9, "quote": "2\n−tµ −2Ta.\n(4.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-7", "from": "equation-51", "to": "lemma-7", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-faaa210733", "page": 10, "quote": "Ge(R, F) = σo\nx(e)(W(R, F) ◦πe),\n(4.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-8", "from": "equation-52", "to": "lemma-7", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-872453b709", "page": 10, "quote": "Gy(R, F) = Ee|yGe(R, F),\n¯Gy = TrGy.\n(4.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-9", "from": "equation-53", "to": "lemma-8", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-542d9b17d9", "page": 10, "quote": "2E StabFD\nt\n( ¯Ay),\n(4.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-10", "from": "equation-54", "to": "lemma-8", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p10-1301175d92", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy).\n(4.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-11", "from": "equation-56", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p11-826b929997", "page": 11, "quote": "qv,g0 + qv,g1 = qv\n(g ∈[s]).\n(5.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-12", "from": "equation-57", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p11-fe2c2859c8", "page": 11, "quote": "bfC(F v,−) = (−1)qv(1) bfC(F),\n(5.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-13", "from": "equation-58", "to": "lemma-9", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p11-49471e5bc7", "page": 11, "quote": "g zgqv,g1(1) bfC(F).\n(5.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-14", "from": "equation-59", "to": "lemma-10", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p12-091c8a397c", "page": 12, "quote": "v∈D\n∥(I −Sv) ¯f∥2\n2,R,F ≤\ns\n2emµ.\n(5.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-15", "from": "equation-60", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p14-b98783d5c4", "page": 14, "quote": "z∈Fs\n2\n∥U(z)∥2 = M.\n(6.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-16", "from": "equation-61", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-17", "from": "equation-62", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-670fa7848e", "page": 15, "quote": ".\n(6.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-18", "from": "equation-63", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-e5cedc4c6c", "page": 15, "quote": "Fourier inversion in z gives\nU(z) = Ep(−1)p·zAp(Sp).\n(6.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-19", "from": "equation-64", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p15-21f25791dc", "page": 15, "quote": "q\n∥αq(p)∥2,\nEpmp = M,\nEp∥αq(p)∥2 = wq.\n(6.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-20", "from": "equation-65", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p16-21fe70f778", "page": 16, "quote": "D2\n.\n(6.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-21", "from": "equation-66", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p16-f67cba1c4e", "page": 16, "quote": ".\n(6.7)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-22", "from": "equation-67", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p16-961cce19bb", "page": 16, "quote": "= Ep1,p2,p3E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n,\np4 = p1 + p2 + p3.\n(6.8)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-23", "from": "equation-68", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p16-6e934d6f5f", "page": 16, "quote": "q\n⟨α′\nq(p1), α′\nq(p2)⟩⟨α′\nq(p3), α′\nq(p4)⟩.\n(6.9)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-24", "from": "equation-69", "to": "lemma-12", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p17-a5d7732e54", "page": 17, "quote": "h .\n(6.10)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-25", "from": "equation-72", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-26", "from": "equation-73", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-27", "from": "equation-74", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p19-2048fcebc1", "page": 19, "quote": "8 .\n(7.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-28", "from": "equation-75", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p19-2bd4dfb0c5", "page": 19, "quote": "8 =: Λ.\n(7.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-29", "from": "equation-76", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p19-1fcabf9810", "page": 19, "quote": "z\nInfv,z(Hv) ≤K.\n(7.7)" }, "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": "p19-5abdbe5cbd", "page": 19, "quote": "2s .\n(7.8)" }, "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": "p20-3e08f51158", "page": 20, "quote": "q∈A\nχq(LΣ)Ψq\nFv(· + J(q)), FD\\{v}\n.\n(7.9)" }, "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-dbdc944b36", "page": 20, "quote": "I\n∥aq,I∥2\n2,FD ≤mq( ¯fω; Γ, Fout).\n(7.10)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-33", "from": "equation-80", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p20-ef57cb6081", "page": 20, "quote": "2\n.\n(7.11)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-34", "from": "equation-81", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-35", "from": "equation-82", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p21-e512194a97", "page": 21, "quote": "z\nInfv,z(Hodd\nv\n) ≤K.\n(7.13)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-36", "from": "equation-83", "to": "proposition-13", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p21-b8e4cf0492", "page": 21, "quote": "K ,\np∗=\nΛ\n4K 2m .\n(7.14)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-37", "from": "equation-84", "to": "lemma-14", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p22-470d569f6e", "page": 22, "quote": "mq( ¯Gy; Γ, Fout) ≤EE, e|ymq(Ge; Γ + EΓ, Fout).\n(8.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-38", "from": "equation-85", "to": "lemma-14", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p22-7eded40296", "page": 22, "quote": "q∗= q∗(y, D, v, Γ, Fout) ∈arg max\nq∈A\nmq( ¯Gy; Γ, Fout)\n(8.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-39", "from": "equation-86", "to": "lemma-14", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-40", "from": "equation-88", "to": "lemma-15", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p23-266720f7f6", "page": 23, "quote": "Tp0\n≤\n1\n2√Tp0\n.\n(8.5)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-41", "from": "equation-89", "to": "lemma-15", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p23-d3c148ea4c", "page": 23, "quote": "T ≥\n1\np0(p∗θ)2 ,\n(8.6)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-42", "from": "equation-90", "to": "lemma-15", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p23-555402b059", "page": 23, "quote": "2 .\n(8.7)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-43", "from": "equation-91", "to": "lemma-16", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p24-242a8fe97e", "page": 24, "quote": "emeq( ef).\n(8.8)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-44", "from": "equation-92", "to": "proposition-17", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p24-37148e8ed5", "page": 24, "quote": "2 .\n(8.9)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-45", "from": "equation-93", "to": "theorem-1", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p27-c007b94b4a", "page": 27, "quote": "1 + t\n,\n(9.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-46", "from": "equation-94", "to": "theorem-1", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p27-f5698fca36", "page": 27, "quote": "2\n+ ε + ϵ′.\n(9.2)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-47", "from": "equation-101", "to": "lemma-18", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p32-85bfea9df7", "page": 32, "quote": "2k0+2f0 −1\n≤2q−k0.\n(A.7)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-48", "from": "equation-102", "to": "lemma-18", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-49", "from": "equation-103", "to": "lemma-19", "relation": "occurs-in", "basis": "inferred", "evidence": { "blockId": "p36-4579cd2c8e", "page": 36, "quote": "p2 .\n(B.1)" }, "note": "Placement in a statement or nearby proof; not a verified logical dependency." }, { "id": "edge-50", "from": "external-26", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-51", "from": "external-21", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-52", "from": "external-20", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-53", "from": "external-29", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-54", "from": "external-27", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-55", "from": "external-32", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-56", "from": "external-35", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-57", "from": "external-24", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-58", "from": "external-28", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-59", "from": "external-36", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-60", "from": "external-37", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-61", "from": "external-25", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-62", "from": "external-34", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-63", "from": "external-22", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-64", "from": "external-33", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-65", "from": "proposition-4", "to": "theorem-2", "relation": "mentioned-in", "basis": "inferred", "evidence": { "blockId": "p3-5f5b709dd7", "page": 3, "quote": "the completeness error is made arbitrarily small. Theorem 2.2 states this input. Appendix A\nderives it from the published construction, specifying both the valid-subspace conditioning\nand the auxiliary candidate lists used in the soundness proof.\nTogether with Majority Is\nStablest, that construction supplies the external foundation of the argument. The Unique Games\nConjecture enters only as historical context. The tree Fourier analysis, hashing bound, and\nhidden-coordinate decoding are proved here." }, "note": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch." }, { "id": "edge-66", "from": "external-23", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p3-2347d0bc2e", "page": 3, "quote": "As in long-code reductions [4, 10], a label a is represented by the dictator w 7→w(a), where w\nranges over Boolean words indexed by labels. The graph has one such word domain for each\ntuple of left vertices of the source game. Thus an arbitrary cut specifies a Boolean function\non each domain. To sample an edge, the test chooses two left tuples through a common right\ntuple, generates two words on the right label space, and pulls them back through the sampled\naffine projections. It negates the second word. When the sampled constraints are satisfied,\ndictator labels give a cut probability of one half of one plus the correlation between the two\nwords evaluated at a common label.\nThe word generator is a finite rooted tree. Every internal node has s pairs of children and a\nrandom table from Fs\n2 to {−1, 1}; the leaves carry affine forms on the label space. At a label a,\nthese forms give leaf signs, and the tables recursively combine the child signs into the root sign\nW(a). The gate rule is chosen so that each monomial of its Walsh expansion keeps one child\nfrom every pair. A second exploration keeps both children of one randomly selected pair at each\nreached node. At any fixed depth, the two explorations have exactly one node in common. The\ntest couples the tables reached by this second exploration at a random depth, using entrywise\ncorrelation t, and leaves all other tables unchanged. Every monomial therefore acquires precisely\none factor of t. A small independent resampling of the affine rows gives the smoothing needed\nfor soundness and only a small completeness loss.\nFor an arbitrary cut, we first average its odd part over the source-game edges. The resulting\nfunction is balanced in the selected table entries, and a cut above the desired NO threshold\nmakes its expected noise stability exceed B(t). Majority Is Stablest then detects an influential\nentry. That entry is indexed by a gate input, not by a source-game label. The remaining task is\nto connect its influence to labels that the two endpoints of a source edge can choose consistently.\nThree steps address this task." }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-67", "from": "external-29", "to": "theorem-2", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p3-2347d0bc2e", "page": 3, "quote": "As in long-code reductions [4, 10], a label a is represented by the dictator w 7→w(a), where w\nranges over Boolean words indexed by labels. The graph has one such word domain for each\ntuple of left vertices of the source game. Thus an arbitrary cut specifies a Boolean function\non each domain. To sample an edge, the test chooses two left tuples through a common right\ntuple, generates two words on the right label space, and pulls them back through the sampled\naffine projections. It negates the second word. When the sampled constraints are satisfied,\ndictator labels give a cut probability of one half of one plus the correlation between the two\nwords evaluated at a common label.\nThe word generator is a finite rooted tree. Every internal node has s pairs of children and a\nrandom table from Fs\n2 to {−1, 1}; the leaves carry affine forms on the label space. At a label a,\nthese forms give leaf signs, and the tables recursively combine the child signs into the root sign\nW(a). The gate rule is chosen so that each monomial of its Walsh expansion keeps one child\nfrom every pair. A second exploration keeps both children of one randomly selected pair at each\nreached node. At any fixed depth, the two explorations have exactly one node in common. The\ntest couples the tables reached by this second exploration at a random depth, using entrywise\ncorrelation t, and leaves all other tables unchanged. Every monomial therefore acquires precisely\none factor of t. A small independent resampling of the affine rows gives the smoothing needed\nfor soundness and only a small completeness loss.\nFor an arbitrary cut, we first average its odd part over the source-game edges. The resulting\nfunction is balanced in the selected table entries, and a cut above the desired NO threshold\nmakes its expected noise stability exceed B(t). Majority Is Stablest then detects an influential\nentry. That entry is indexed by a gate input, not by a source-game label. The remaining task is\nto connect its influence to labels that the two endpoints of a source edge can choose consistently.\nThree steps address this task." }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-68", "from": "proposition-4", "to": "theorem-2", "relation": "mentioned-in", "basis": "inferred", "evidence": { "blockId": "p4-2adef6757d", "page": 4, "quote": "The resulting decoding loss is independent of the source alphabet. This is essential: the\nalphabet dimension depends on the soundness we request, so a dimension-dependent loss could\nmake the soundness choice circular. We first fix that loss and the source soundness, then the\nalphabet and the number of product factors, and finally the source completeness error. The\nquantifier order in Theorem 2.2 permits exactly this sequence." }, "note": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch." }, { "id": "edge-69", "from": "external-104", "to": "lemma-3", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-70", "from": "external-105", "to": "lemma-3", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-71", "from": "external-106", "to": "lemma-3", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-72", "from": "external-24", "to": "lemma-3", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-73", "from": "lemma-19", "to": "proposition-4", "relation": "mentioned-in", "basis": "inferred", "evidence": { "blockId": "p6-7b93d7ee44", "page": 6, "quote": "A weighted graph is likewise specified by a rational distribution on unordered edges, allowing\nloops. A cut assigns signs to its vertices and satisfies an edge when its signs differ. Its value is\nits satisfaction probability. Loops have zero cut weight. The conversion in Theorem B.1 keeps a\ncommon scale even when loops are deleted." }, "note": "Text says Theorem B.1; candidate heading is Lemma B.1. Check the mismatch." }, { "id": "edge-74", "from": "equation-45", "to": "lemma-6", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p8-64d8f6ec74", "page": 8, "quote": "g∈I zg.\nSubstituting into Equation (3.1), the term indexed by I is" }, "note": "" }, { "id": "edge-75", "from": "equation-47", "to": "lemma-6", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p8-bcf0ef5bfe", "page": 8, "quote": "where P is the family of leaf sets produced by choosing one child per pair at each reached internal\nnode. The leaf set determines all these choices: a selected child has a nonempty intersection\nwith that leaf set, whereas an unselected child has none. Thus different traversals give different\nmonomials, and aP (F) is the product of one Walsh coefficient of Fv for each node reached by\nthe traversal.\nAt the root both a monomial traversal and the slice exploration are present. At any common\nnode above the slice depth, exactly one child is selected by both explorations. Induction gives\nexactly one common node at the slice depth. Hence each aP contains exactly one factor from a\ntable on D. Negating every such table negates each coefficient in Equation (3.3) and therefore\nnegates the word. Moreover, each Walsh coefficient of a slice table satisfies" }, "note": "" }, { "id": "edge-76", "from": "equation-47", "to": "lemma-6", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, "note": "" }, { "id": "edge-77", "from": "equation-46", "to": "lemma-6", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, "note": "" }, { "id": "edge-78", "from": "lemma-6", "to": "lemma-7", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p10-0e07627ed1", "page": 10, "quote": "Each coordinate of either sampled edge tuple has the original edge law.\nA union bound\nshows that all 2T sampled constraints are satisfied except with probability at most 2Ta.\nWhen all these constraints are satisfied, the two queried tuples project to the same label\nl = (ℓY (y1), . . . , ℓY (yT )) ∈A. The cut signs then have product −W(R1, F)(l)W(R2, F ′)(l). The\nword-generation data are independent of the game edges, so Theorem 3.2 gives conditional cut\nprobability (1 + tr2d)/2. On the remaining event the cut probability is nonnegative. This proves\nthe first bound when 2Ta ≤1; if 2Ta > 1 that bound is negative and is automatic. Finally,\nr2d ≥1 −2µ, and the loss due to the prefactor is at most 2Ta, giving the second bound." }, "note": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch." }, { "id": "edge-79", "from": "lemma-6", "to": "lemma-7", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p10-12c4ff12f9", "page": 10, "quote": "where ϕy is odd and depends on no slice or row data. Thus ¯Gy has mean zero in the slice entries\nfor every fixed y, D, R, Fout, by the observation following Theorem 3.2. The later extraction\nargument, Theorem 7.1, applies to precisely such bounded odd functions of the generated word." }, "note": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch." }, { "id": "edge-80", "from": "proposition-13", "to": "lemma-7", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p10-12c4ff12f9", "page": 10, "quote": "where ϕy is odd and depends on no slice or row data. Thus ¯Gy has mean zero in the slice entries\nfor every fixed y, D, R, Fout, by the observation following Theorem 3.2. The later extraction\nargument, Theorem 7.1, applies to precisely such bounded odd functions of the generated word." }, "note": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch." }, { "id": "edge-81", "from": "proposition-13", "to": "lemma-8", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p11-164121ec4c", "page": 11, "quote": "If a cut has value at least (1 + B(t))/2 + δ/2, then its odd average has expected slice stability\nat least B(t) + δ. The next three sections show that this excess stability yields a label carrying\nsubstantial Fourier mass, with constants independent of the game dimension and the number of\nrepetitions; the formal statement is Theorem 7.1. Section 8 then turns that mass into a labeling\nof a single copy of the original game." }, "note": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch." }, { "id": "edge-82", "from": "equation-56", "to": "lemma-9", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p12-6c5c7c7ebd", "page": 12, "quote": "Distinct characters are distinct functions, proving Equation (5.2).\nNegating Fv has the same effect on the word as adding 1 to all rows in P(v). This proves\nEquation (5.3). To translate the input of Fv by z, add 1 to the generator of its second child\nin each pair with zg = 1. This changes precisely those gate input bits and leaves its prefactor\nunchanged. Fourier transformation gives Equation (5.4).\nRow smoothing and row Fourier projections act diagonally on C, so preserve all three\nidentities. On the cube of table-entry signs, a domain translation permutes coordinates, and\nnegating a gate table multiplies each monomial by its parity sign. Both operations preserve\ntotal degree in any specified collection of whole gate tables and therefore commute with the\nstated degree projections." }, "note": "" }, { "id": "edge-83", "from": "equation-57", "to": "lemma-9", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p12-6c5c7c7ebd", "page": 12, "quote": "Distinct characters are distinct functions, proving Equation (5.2).\nNegating Fv has the same effect on the word as adding 1 to all rows in P(v). This proves\nEquation (5.3). To translate the input of Fv by z, add 1 to the generator of its second child\nin each pair with zg = 1. This changes precisely those gate input bits and leaves its prefactor\nunchanged. Fourier transformation gives Equation (5.4).\nRow smoothing and row Fourier projections act diagonally on C, so preserve all three\nidentities. On the cube of table-entry signs, a domain translation permutes coordinates, and\nnegating a gate table multiplies each monomial by its parity sign. Both operations preserve\ntotal degree in any specified collection of whole gate tables and therefore commute with the\nstated degree projections." }, "note": "" }, { "id": "edge-84", "from": "equation-58", "to": "lemma-9", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p12-6c5c7c7ebd", "page": 12, "quote": "Distinct characters are distinct functions, proving Equation (5.2).\nNegating Fv has the same effect on the word as adding 1 to all rows in P(v). This proves\nEquation (5.3). To translate the input of Fv by z, add 1 to the generator of its second child\nin each pair with zg = 1. This changes precisely those gate input bits and leaves its prefactor\nunchanged. Fourier transformation gives Equation (5.4).\nRow smoothing and row Fourier projections act diagonally on C, so preserve all three\nidentities. On the cube of table-entry signs, a domain translation permutes coordinates, and\nnegating a gate table multiplies each monomial by its parity sign. Both operations preserve\ntotal degree in any specified collection of whole gate tables and therefore commute with the\nstated degree projections." }, "note": "" }, { "id": "edge-85", "from": "equation-56", "to": "lemma-10", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p12-20009794ba", "page": 12, "quote": "Proof. Fix an index C satisfying Equation (5.2), and let Nj be the number of nodes at depth\nj with nonzero index qv. If qv ̸= 0, each child pair contains at least one nonzero index. A\nfailure of the unsplit condition at such a node supplies at least one additional nonzero child. If\nqv = 0, a failure supplies at least two nonzero children, whereas its minimum contribution is\nzero. Therefore the number Bj of failures at depth j satisfies" }, "note": "" }, { "id": "edge-86", "from": "equation-55", "to": "lemma-10", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p13-7a47c97d6f", "page": 13, "quote": "Here Nm = |C|. By Equation (5.1) and Parseval, the left side of Equation (5.5) is at most" }, "note": "" }, { "id": "edge-87", "from": "equation-59", "to": "lemma-10", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p13-7a47c97d6f", "page": 13, "quote": "Here Nm = |C|. By Equation (5.1) and Parseval, the left side of Equation (5.5) is at most" }, "note": "" }, { "id": "edge-88", "from": "equation-63", "to": "lemma-12", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p16-29e6db68c4", "page": 16, "quote": "Equation (6.4) and the triangle inequality imply maxz∥U(z) −U′(z)∥≤X. The contribution to\nEX from the indices with mp > CM is at most" }, "note": "" }, { "id": "edge-89", "from": "lemma-3", "to": "proposition-13", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, "note": "Text says Theorem 2.1; candidate heading is Lemma 2.1. Check the mismatch." }, { "id": "edge-90", "from": "lemma-10", "to": "proposition-13", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p19-52d05b19a0", "page": 19, "quote": "By Theorem 5.2, the required bound on the projection loss is" }, "note": "Text says Theorem 5.2; candidate heading is Lemma 5.2. Check the mismatch." }, { "id": "edge-91", "from": "equation-43", "to": "proposition-13", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p19-d7a06cebf5", "page": 19, "quote": "To verify it, integrate over all rows. Each Sv is an orthogonal projection in the rows and\ncommutes with the slice-bit Fourier projections. It can therefore only decrease the integrated\ninfluence at its entries. After dropping each Sv separately, the left side is bounded by the total\ninfluence of P FD\n≤K ¯fω, averaged over the remaining variables. Equation (2.3) and | ¯fω| ≤1 give\n(7.7). No pointwise boundedness of Hv is asserted or needed." }, "note": "" }, { "id": "edge-92", "from": "lemma-9", "to": "proposition-13", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p19-20ff4ebca5", "page": 19, "quote": "By the sign action in Theorem 5.1, the even part has qv(1) = 0 on row support. Since the\nindex is unsplit, all its child indices are either 0 or qv, so they too take value zero on 1. The\ndomain-translation action in that lemma shows that the even part is invariant under translations\nof the domain of Fv, with the rows and outside tables fixed. These translations act transitively\non its entries. All entry influences at v are therefore equal. By (7.7)," }, "note": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch." }, { "id": "edge-93", "from": "equation-78", "to": "proposition-13", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p20-b92cb4dec5", "page": 20, "quote": "The inequality follows because the unsplit row projection is determined by the free shift index,\nand the degree and parity projections are orthogonal projections in the slice bits at each fixed\ncontext. Every Ψq has degree at most K.\nEquation (7.9) has converted the free-shift characters into translations of a gate table. We\nnext collect its Boolean Fourier coefficients into vectors whose squared norms are entry influences.\nThese Fourier coefficients are indexed by sets of table entries; the earlier index I ⊆[s] instead\nspecified a character of the table’s domain Fs\n2.\nDenote by bΨq(B, D) the slice Fourier coefficient with entry set B ⊆Fs\n2 at gate v and entry\nset D on the other slice gates. For z ∈Fs\n2 define a real vector c(q, z), with coordinates indexed\nby pairs (A, D) where 0 ∈A ⊆Fs\n2, by" }, "note": "" }, { "id": "edge-94", "from": "lemma-12", "to": "proposition-13", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p21-2a433dcc94", "page": 21, "quote": "The factor |B| counts the entries whose influences contain that Fourier monomial; it is at most\nthe monomial’s total degree, which is at most K. Thus Mv ≤K at every context. The total-mass\nidentity in Theorem 6.1, together with (7.11), also gives" }, "note": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch." }, { "id": "edge-95", "from": "lemma-12", "to": "proposition-13", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, "note": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch." }, { "id": "edge-96", "from": "equation-77", "to": "proposition-13", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p21-b276b21d53", "page": 21, "quote": "On class (i), the hashing lemma bounds the expected maximum odd influence by ζMv. Summing\nand using (7.13) costs at most Λ/4. On class (ii), the maximum is bounded in expectation\nby the total influence, namely Mv < M0. Since |D| ≤2m, these contexts cost at most Λ/4.\nOn class (iii), the same total-mass bound is at most K per node. Equation (7.8) bounds all\neven contributions by Λ/4. Let Niii count the nodes whose induced contexts lie in class (iii).\nComparing all four contributions with (7.6) gives" }, "note": "" }, { "id": "edge-97", "from": "proposition-13", "to": "lemma-14", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p22-62d9b730ba", "page": 22, "quote": "using that ordering to break ties. This definition uses the original conditional edge average and\nthe original row-smoothing operator. It will remain unchanged when we modify the distribution\nof the row data below.\nSample y, the slice, its uniform node, the context, and the outside tables as in Proposition 7.1,\nand then sample Rnew, B, and e | y. Combining that Proposition with Lemma 8.1 gives" }, "note": "" }, { "id": "edge-98", "from": "equation-86", "to": "lemma-15", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p23-11ce6c955c", "page": 23, "quote": "The score in Equation (8.3) is a [0, 1]-valued function of the displayed data without the mark.\nIn particular, q∗is the fixed function defined in Equation (8.2); it does not use the mark, the\nsampled incident edges beyond their Y -endpoints, or the sampled noise. Its statistical dependence\non zero slots causes no difficulty. Expectations of this score under the two laws differ by at most\ntheir total variation distance. Equations (8.3) and (8.6) therefore imply Equation (8.7)." }, "note": "" }, { "id": "edge-99", "from": "equation-85", "to": "lemma-15", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p23-11ce6c955c", "page": 23, "quote": "The score in Equation (8.3) is a [0, 1]-valued function of the displayed data without the mark.\nIn particular, q∗is the fixed function defined in Equation (8.2); it does not use the mark, the\nsampled incident edges beyond their Y -endpoints, or the sampled noise. Its statistical dependence\non zero slots causes no difficulty. Expectations of this score under the two laws differ by at most\ntheir total variation distance. Equations (8.3) and (8.6) therefore imply Equation (8.7)." }, "note": "" }, { "id": "edge-100", "from": "equation-90", "to": "lemma-15", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p23-11ce6c955c", "page": 23, "quote": "The score in Equation (8.3) is a [0, 1]-valued function of the displayed data without the mark.\nIn particular, q∗is the fixed function defined in Equation (8.2); it does not use the mark, the\nsampled incident edges beyond their Y -endpoints, or the sampled noise. Its statistical dependence\non zero slots causes no difficulty. Expectations of this score under the two laws differ by at most\ntheir total variation distance. Equations (8.3) and (8.6) therefore imply Equation (8.7)." }, "note": "" }, { "id": "edge-101", "from": "equation-70", "to": "lemma-16", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p24-ff91954647", "page": 24, "quote": "The masses on both sides use the common-pair-sum definition of Equation (7.1), with the\nrespective affine row spaces." }, "note": "" }, { "id": "edge-102", "from": "equation-91", "to": "lemma-16", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p24-e0bbb65da1", "page": 24, "quote": "so eq is an evaluation functional, hence a label in e\nA, and ρ(eq) = q. Each lifted index is counted\nonly once. Averaging over the slice tables proves Equation (8.8)." }, "note": "" }, { "id": "edge-103", "from": "proposition-13", "to": "proposition-17", "relation": "mentioned-in", "basis": "explicit-reference", "evidence": { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" }, "note": "" }, { "id": "edge-104", "from": "equation-89", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" }, "note": "" }, { "id": "edge-105", "from": "equation-90", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p25-03898445eb", "page": 25, "quote": "Shared data and the Y -labeler.\nGiven a random input edge e = (x, y) from the original\nedge law, choose a uniform public mark a∗∈[T]. At every other slot, publicly sample an\nindependent edge, including its endpoints and projection. Insert the input edge at the marked\nslot. Its projection is used only in the analysis; it is not revealed to either labeler. Conditional on\nany mark, the resulting full edge tuple has the product edge law; hence the tuple is independent\nof the mark. Each labeler knows its own endpoint tuple, and both know all edges outside the\nmarked slot.\nPublicly sample the depth, slice, node, outside tables, context, latent rows, and masks\naccording to the planted experiment above. The row sampling uses the fixed alphabet dimensions,\nthe public tree coordinates, and the mark, and is independent of the input edge. Together with\nthe edge tuple, these data have exactly the planted law used in Equation (8.7). The Y -labeler\nforms its endpoint tuple y, computes the label q∗from Equation (8.2), and outputs its marked\ncomponent q∗\na∗. All quantities used in that computation are available to this labeler: in particular\nthe conditional average defining ¯Gy can be computed from the game and cut as fixed background\ndata. It does not use the actual marked edge or its projection." }, "note": "" }, { "id": "edge-106", "from": "equation-85", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p25-03898445eb", "page": 25, "quote": "Shared data and the Y -labeler.\nGiven a random input edge e = (x, y) from the original\nedge law, choose a uniform public mark a∗∈[T]. At every other slot, publicly sample an\nindependent edge, including its endpoints and projection. Insert the input edge at the marked\nslot. Its projection is used only in the analysis; it is not revealed to either labeler. Conditional on\nany mark, the resulting full edge tuple has the product edge law; hence the tuple is independent\nof the mark. Each labeler knows its own endpoint tuple, and both know all edges outside the\nmarked slot.\nPublicly sample the depth, slice, node, outside tables, context, latent rows, and masks\naccording to the planted experiment above. The row sampling uses the fixed alphabet dimensions,\nthe public tree coordinates, and the mark, and is independent of the input edge. Together with\nthe edge tuple, these data have exactly the planted law used in Equation (8.7). The Y -labeler\nforms its endpoint tuple y, computes the label q∗from Equation (8.2), and outputs its marked\ncomponent q∗\na∗. All quantities used in that computation are available to this labeler: in particular\nthe conditional average defining ¯Gy can be computed from the game and cut as fixed background\ndata. It does not use the actual marked edge or its projection." }, "note": "" }, { "id": "edge-107", "from": "lemma-16", "to": "proposition-17", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p26-3843e9cd2b", "page": 26, "quote": "h multiplies the output of gate v by (−1)h(el) at every label el. Applying this shift in two distinct\npairs leaves that output, and hence the whole word, unchanged. These operations only translate\nthe free shifts and preserve the context. Character orthogonality therefore shows that every\nshift Fourier index of eG has a common pair sum. This argument holds for each fixed choice of\nall gate tables.\nWord generation commutes with pullback of rows, pointwise in the label. Since πe = ρ ◦κ,\nrestricting each free shift of eG to ρ∗H yields exactly Ge with context Γ′ and the same tables.\nThe map ρ has two-element fibers, so Lemma 8.3 applies and gives\nX" }, "note": "" }, { "id": "edge-108", "from": "equation-90", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p26-1de248c808", "page": 26, "quote": "Conditional on all the planted data, the left side lower-bounds the probability that the sampled\nfull label projects to q∗. That event in particular implies satisfaction of the marked input edge.\nAveraging and using Equation (8.7) gives success probability at least 2−2sp∗θ/2.\nFinally, fix all shared randomness, including the uniform number used to sample the X-label.\nThe same fixed data define the two strategies at every endpoint, so they give a deterministic\nlabeling of the original game. Their average value over the shared randomness has the lower\nbound just proved, so some fixed choice attains it. This proves Equation (8.9)." }, "note": "" }, { "id": "edge-109", "from": "equation-92", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p26-1de248c808", "page": 26, "quote": "Conditional on all the planted data, the left side lower-bounds the probability that the sampled\nfull label projects to q∗. That event in particular implies satisfaction of the marked input edge.\nAveraging and using Equation (8.7) gives success probability at least 2−2sp∗θ/2.\nFinally, fix all shared randomness, including the uniform number used to sample the X-label.\nThe same fixed data define the two strategies at every endpoint, so they give a deterministic\nlabeling of the original game. Their average value over the shared randomness has the lower\nbound just proved, so some fixed choice attains it. This proves Equation (8.9)." }, "note": "" }, { "id": "edge-110", "from": "equation-92", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p26-852017d85d", "page": 26, "quote": "The decoding loss depends on s, p∗, θ, and hence on the target cut gap, but not on the\nalphabet dimension or the repetition length. After choosing a game soundness smaller than the\nright side of Equation (8.9), one may therefore fix the alphabet dimension and then choose T by\nEquation (8.6)." }, "note": "" }, { "id": "edge-111", "from": "equation-89", "to": "proposition-17", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p26-852017d85d", "page": 26, "quote": "The decoding loss depends on s, p∗, θ, and hence on the target cut gap, but not on the\nalphabet dimension or the repetition length. After choosing a game soundness smaller than the\nright side of Equation (8.9), one may therefore fix the alphabet dimension and then choose T by\nEquation (8.6)." }, "note": "" }, { "id": "edge-112", "from": "proposition-13", "to": "theorem-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, "note": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch." }, { "id": "edge-113", "from": "proposition-4", "to": "theorem-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p26-842530a02d", "page": 26, "quote": "By Theorem 2.2, this fixes an alphabet dimension n for which the game has arbitrarily small\ncompleteness error. Choose an integer T satisfying" }, "note": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch." }, { "id": "edge-114", "from": "lemma-7", "to": "theorem-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "note": "Text says Theorem 4.1; candidate heading is Lemma 4.1. Check the mismatch." }, { "id": "edge-115", "from": "lemma-8", "to": "theorem-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "note": "Text says Theorem 4.2; candidate heading is Lemma 4.2. Check the mismatch." }, { "id": "edge-116", "from": "proposition-13", "to": "theorem-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "note": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch." }, { "id": "edge-117", "from": "proposition-17", "to": "theorem-2", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "note": "Text says Theorem 8.4; candidate heading is Proposition 8.4. Check the mismatch." }, { "id": "edge-118", "from": "theorem-2", "to": "theorem-1", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p27-7abf2de4cc", "page": 27, "quote": "Apply Theorem 1.2. Delete loops while retaining the original weight scale, so that the remaining\nweights sum to at most one. Apply Theorem B.1 with error ϵ′, producing a simple unweighted\ngraph whose maximum cut, on the common scale Q, differs from the original maximum cut\nweight by at most ϵ′. An α-approximation algorithm on that graph returns, in the YES case, a\ncut of scaled value at least the left side of (9.2). In the NO case, no cut has scaled value greater\nthan its right side. A fixed rational threshold strictly between them therefore distinguishes the\nNP-hard promise cases. This proves the theorem." }, "note": "" }, { "id": "edge-119", "from": "lemma-19", "to": "theorem-1", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p27-7abf2de4cc", "page": 27, "quote": "Apply Theorem 1.2. Delete loops while retaining the original weight scale, so that the remaining\nweights sum to at most one. Apply Theorem B.1 with error ϵ′, producing a simple unweighted\ngraph whose maximum cut, on the common scale Q, differs from the original maximum cut\nweight by at most ϵ′. An α-approximation algorithm on that graph returns, in the YES case, a\ncut of scaled value at least the left side of (9.2). In the NO case, no cut has scaled value greater\nthan its right side. A fixed rational threshold strictly between them therefore distinguishes the\nNP-hard promise cases. This proves the theorem." }, "note": "Text says Theorem B.1; candidate heading is Lemma B.1. Check the mismatch." }, { "id": "edge-120", "from": "external-25", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p32-c85fcfca35", "page": 32, "quote": "The advice distribution.\nThe outer game in [6, Section 5.2] first samples (U, V ), then takes\nQ uniformly from the q-spaces of VV . It sends (U, Q) and (V, Q) to its two players and asks\nfor assignments on var(U) and V , respectively. It accepts when those assignments agree on V\nand the first satisfies every equation of U. If f0 equations contributed all three variables, then\ndim VV = k0 + 2f0 and dim H(V ) = f0. Consequently, under this actual distribution, uniformly\nin U, V ," }, "note": "Citation establishes a textual reference, not a verified dependency on the external theorem." }, { "id": "edge-121", "from": "external-113", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-122", "from": "external-114", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-123", "from": "external-115", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-124", "from": "external-116", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, "note": "The source is explicitly external. Do not resolve this number to a local theorem." }, { "id": "edge-125", "from": "equation-51", "to": "lemma-18", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, "note": "" }, { "id": "edge-126", "from": "external-112", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-127", "from": "external-115", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-128", "from": "external-117", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-129", "from": "external-113", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-130", "from": "external-118", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p34-380ad96976", "page": 34, "quote": "The overall success probability is at least δα(ℓ)c∗/8, a positive constant independent of k0. The\nupper bound in [6, Lemma 5.4] tends to zero as βk0 →∞. Choosing k0 sufficiently large gives\nthe same contradiction as in the source and proves that raw valid acceptance is at most δ." }, "note": "External result named in a citation. Its source has not been checked." }, { "id": "edge-131", "from": "proposition-4", "to": "lemma-18", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p34-80a9431745", "page": 34, "quote": "Fixing soundness before completeness.\nFor the game in Theorem 2.2, take δ = b/4 and\nretain the valid raw weights. Their total mass qedge is at least 1/2 by the preceding rejection\nestimates. Normalization therefore gives" }, "note": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch." }, { "id": "edge-132", "from": "lemma-8", "to": "lemma-18", "relation": "cited-in-proof", "basis": "explicit-reference", "evidence": { "blockId": "p35-5bbe01a21e", "page": 35, "quote": "Effectivity and size.\nAll spaces in an individual question have bounded dimension. There\nare polynomially many k0-tuples of input equations, and each tuple has only constantly many\nsubspaces, bases, and local sampling outcomes. The folding equivalence relation is explicitly\ntestable by linear algebra. To find RC effectively, start from any (U, L) ∈C and enumerate\nthe ℓ-spaces inside L + HU, whose dimension is the fixed constant ℓ+ k0; test each candidate\nagainst (A.1) for every class member. Lemma 4.2 guarantees a successful candidate. Gaussian\nelimination then computes bases and the affine edge maps. The finite sampling distributions,\ntheir conditioning, and the aggregation of parallel edges have rational weights of polynomial\nbit length. These observations give the deterministic polynomial reduction claimed in the\nproposition. This completes the proof of Theorem 2.2." }, "note": "" }, { "id": "edge-133", "from": "proposition-4", "to": "lemma-18", "relation": "cited-in-proof", "basis": "inferred", "evidence": { "blockId": "p35-5bbe01a21e", "page": 35, "quote": "Effectivity and size.\nAll spaces in an individual question have bounded dimension. There\nare polynomially many k0-tuples of input equations, and each tuple has only constantly many\nsubspaces, bases, and local sampling outcomes. The folding equivalence relation is explicitly\ntestable by linear algebra. To find RC effectively, start from any (U, L) ∈C and enumerate\nthe ℓ-spaces inside L + HU, whose dimension is the fixed constant ℓ+ k0; test each candidate\nagainst (A.1) for every class member. Lemma 4.2 guarantees a successful candidate. Gaussian\nelimination then computes bases and the affine edge maps. The finite sampling distributions,\ntheir conditioning, and the aggregation of parallel edges have rational weights of polynomial\nbit length. These observations give the deterministic polynomial reduction claimed in the\nproposition. This completes the proof of Theorem 2.2." }, "note": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch." }, { "id": "edge-134", "from": "equation-103", "to": "lemma-19", "relation": "equation-reference", "basis": "explicit-reference", "evidence": { "blockId": "p36-82d91c1daf", "page": 36, "quote": "Now fix an arbitrary cut of G′, and let au ∈[0, 1] be the fraction of cluster u on its positive\nside. Its crossing edges between clusters u, v form two rectangles. Equation (B.1) therefore\ncompares their scaled count with" }, "note": "" }, { "id": "edge-135", "from": "theorem-2", "to": "theorem-1", "relation": "supports", "basis": "agent-inferred", "evidence": { "blockId": "p27-a71d41604b", "page": 27, "quote": "Proof of Theorem 1.1. Substituting ρ = −t into (1.1) gives" }, "note": "Agent-inferred connection from the final proof and weighted gap statement; inspect subsequent proof blocks." }, { "id": "edge-136", "from": "lemma-19", "to": "theorem-1", "relation": "supports", "basis": "agent-inferred", "evidence": { "blockId": "p35-be0e319110", "page": 35, "quote": "Lemma B.1. Let G be a graph on N vertices with nonnegative rational weights wuv on unordered\ndistinct vertex pairs, and P\nu 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" }, "note": "Agent-inferred role of the simple-unweighted-graph conversion; compare the final theorem proof." } ], "notation": [ { "symbol": "s, m", "meaning": "Tree parameters introduced by Definition 3.1.", "status": "source-located", "scope": "Integers s ≥ 2 and m ≥ 1.", "source": { "blockId": "p6-f809474603", "page": 6, "quote": "Definition 3.1 (Tree code). Fix integers s ≥2 and m ≥1, and put" }, "occurrences": [ { "blockId": "p6-f809474603", "page": 6, "quote": "Definition 3.1 (Tree code). Fix integers s ≥2 and m ≥1, and put" } ], "occurrenceCount": 1 }, { "symbol": "hu", "meaning": "Tu h, smoothed using ordinary bit noise.", "status": "source-located", "scope": "Local degree-cutoff argument in Lemma 2.1.", "source": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "occurrences": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," } ], "occurrenceCount": 1, "latex": "h_{u}", "latexBasis": "pdf-script-position" }, { "symbol": "g", "meaning": "(h+1)/2, the rescaled Boolean function.", "status": "source-located", "scope": "Local to the proof of Lemma 2.1; not a global symbol binding.", "source": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "occurrences": [ { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," } ], "occurrenceCount": 1 }, { "symbol": "B(t)", "meaning": "2 arcsin(t)/π.", "status": "source-located", "scope": "Used in the gap formulation; explicitly extended to negative t in the final proof.", "source": { "blockId": "p27-ee4fdccd3a", "page": 27, "quote": "where the same formula defines B(t) = 2 arcsin(t)/π for negative t. For −1 < t ≤0 the ratio is\nat least one, whereas for 0 < t < 1 it is less than one. At t = 1 it equals one, and as t ↓−1 it\ndiverges. Thus its minimum occurs in (0, 1).\nFix αGW < α ≤1. By continuity and the density of the rationals, choose a rational t ∈(0, 1)\nsuch that" }, "occurrences": [ { "blockId": "p27-ee4fdccd3a", "page": 27, "quote": "where the same formula defines B(t) = 2 arcsin(t)/π for negative t. For −1 < t ≤0 the ratio is\nat least one, whereas for 0 < t < 1 it is less than one. At t = 1 it equals one, and as t ↓−1 it\ndiverges. Thus its minimum occurs in (0, 1).\nFix αGW < α ≤1. By continuity and the density of the rationals, choose a rational t ∈(0, 1)\nsuch that" } ], "occurrenceCount": 1 }, { "symbol": "αGW", "meaning": "Goemans–Williamson rounding threshold.", "status": "source-located", "scope": "Main theorem; its defining formula spans adjacent PDF blocks.", "source": { "blockId": "p1-66e58fef7e", "page": 1, "quote": "αGW =\nmin\n−1≤ρ<1\n2 arccos ρ" }, "occurrences": [ { "blockId": "p1-66e58fef7e", "page": 1, "quote": "αGW =\nmin\n−1≤ρ<1\n2 arccos ρ" } ], "occurrenceCount": 1, "latex": "α_{GW}", "latexBasis": "pdf-script-position" }, { "symbol": "A", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-0787686067", "page": 1, "quote": "For a graph with nonnegative edge weights, Max-Cut asks for a partition of the vertices\nmaximizing the weight of edges crossing the partition. A cut is encoded by signs on vertices,\nand an edge contributes exactly when its signs differ. For a maximization problem, an α-\napproximation returns a feasible solution worth at least α times the optimum. Unless stated\notherwise, the graph and its rational weights are given explicitly. We write MaxCut(G) for the\nmaximum crossing weight and, when the edge weights sum to one, also denote this value by\nVal(G). The decision problem of whether a weighted graph has a cut of at least a given weight\nappears among Karp’s original NP-complete problems [11]. Approximation asks how much of\nthis optimum can still be guaranteed in polynomial time. Goemans and Williamson [9] gave\na semidefinite programming algorithm with random-hyperplane rounding for Max-Cut, with\nexpected approximation ratio approaching" }, "occurrences": [ { "blockId": "p1-9a9db91009", "page": 1, "quote": "A Direct Proof of Optimal Max-Cut Hardness" }, { "blockId": "p1-0787686067", "page": 1, "quote": "For a graph with nonnegative edge weights, Max-Cut asks for a partition of the vertices\nmaximizing the weight of edges crossing the partition. A cut is encoded by signs on vertices,\nand an edge contributes exactly when its signs differ. For a maximization problem, an α-\napproximation returns a feasible solution worth at least α times the optimum. Unless stated\notherwise, the graph and its rational weights are given explicitly. We write MaxCut(G) for the\nmaximum crossing weight and, when the edge weights sum to one, also denote this value by\nVal(G). The decision problem of whether a weighted graph has a cut of at least a given weight\nappears among Karp’s original NP-complete problems [11]. Approximation asks how much of\nthis optimum can still be guaranteed in polynomial time. Goemans and Williamson [9] gave\na semidefinite programming algorithm with random-hyperplane rounding for Max-Cut, with\nexpected approximation ratio approaching" }, { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, { "blockId": "p3-5f5b709dd7", "page": 3, "quote": "the completeness error is made arbitrarily small. Theorem 2.2 states this input. Appendix A\nderives it from the published construction, specifying both the valid-subspace conditioning\nand the auxiliary candidate lists used in the soundness proof.\nTogether with Majority Is\nStablest, that construction supplies the external foundation of the argument. The Unique Games\nConjecture enters only as historical context. The tree Fourier analysis, hashing bound, and\nhidden-coordinate decoding are proved here." }, { "blockId": "p3-2347d0bc2e", "page": 3, "quote": "As in long-code reductions [4, 10], a label a is represented by the dictator w 7→w(a), where w\nranges over Boolean words indexed by labels. The graph has one such word domain for each\ntuple of left vertices of the source game. Thus an arbitrary cut specifies a Boolean function\non each domain. To sample an edge, the test chooses two left tuples through a common right\ntuple, generates two words on the right label space, and pulls them back through the sampled\naffine projections. It negates the second word. When the sampled constraints are satisfied,\ndictator labels give a cut probability of one half of one plus the correlation between the two\nwords evaluated at a common label.\nThe word generator is a finite rooted tree. Every internal node has s pairs of children and a\nrandom table from Fs\n2 to {−1, 1}; the leaves carry affine forms on the label space. At a label a,\nthese forms give leaf signs, and the tables recursively combine the child signs into the root sign\nW(a). The gate rule is chosen so that each monomial of its Walsh expansion keeps one child\nfrom every pair. A second exploration keeps both children of one randomly selected pair at each\nreached node. At any fixed depth, the two explorations have exactly one node in common. The\ntest couples the tables reached by this second exploration at a random depth, using entrywise\ncorrelation t, and leaves all other tables unchanged. Every monomial therefore acquires precisely\none factor of t. A small independent resampling of the affine rows gives the smoothing needed\nfor soundness and only a small completeness loss.\nFor an arbitrary cut, we first average its odd part over the source-game edges. The resulting\nfunction is balanced in the selected table entries, and a cut above the desired NO threshold\nmakes its expected noise stability exceed B(t). Majority Is Stablest then detects an influential\nentry. That entry is indexed by a gate input, not by a source-game label. The remaining task is\nto connect its influence to labels that the two endpoints of a source edge can choose consistently.\nThree steps address this task." }, { "blockId": "p3-82cb3aa717", "page": 3, "quote": "2. A vector-valued affine hashing lemma. For independent affine shifts, a diffuse distribution\nof Fourier mass cannot create a large entry influence on average. The bound is uniform\nin the dimension of the row space. Its proof uses two truncations and a fourth-moment\nidentity; it may be useful whenever translated coefficient arrays must be decoded without\nan alphabet-size loss." }, { "blockId": "p3-21146313d0", "page": 3, "quote": "3. Decoding with a hidden coordinate. The mass concentration is obtained after fixing many\nrow coordinates. A left endpoint cannot usually construct the pullback of this context" }, { "blockId": "p4-629a2e57f3", "page": 4, "quote": "Section 2 fixes the Fourier conventions. Sections 3 and 4 define the code and the Max-Cut\ntest. Sections 5 to 7 prove the analytic ingredients. Sections 8 and 9 give the decoding and\nfinish the proof. Appendix A derives the precise starting game from the published construction,\nand Appendix B gives a deterministic conversion to simple unweighted graphs." }, { "blockId": "p4-b9e0878f20", "page": 4, "quote": "Let A be a nonempty affine space over F2, with a fixed affine coordinate system, and set" }, { "blockId": "p4-20be6b6373", "page": 4, "quote": "H = Aff(A, F2)." }, { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" }, { "blockId": "p6-13601d3979", "page": 6, "quote": "A weighted bipartite projection game consists of finite sets X, Y , a rational probability distribu-\ntion λ on a finite edge set E with endpoint maps to X and Y (parallel edges are allowed), finite\nalphabets, and a projection πe for each edge. Its value is" }, { "blockId": "p6-5948153d4c", "page": 6, "quote": "The precise promise is given below; its derivation from the published construction is in Section A." }, { "blockId": "p6-7b93d7ee44", "page": 6, "quote": "A weighted graph is likewise specified by a rational distribution on unordered edges, allowing\nloops. A cut assigns signs to its vertices and satisfies an edge when its signs differ. Its value is\nits satisfaction probability. Loops have zero cut weight. The conversion in Theorem B.1 keeps a\ncommon scale even when loops are deleted." }, { "blockId": "p6-5544a8e1fa", "page": 6, "quote": "3\nA tree code with complementary traversals" }, { "blockId": "p6-58466c653a", "page": 6, "quote": "We use the affine label space A and row space H = Aff(A, F2) of Section 2. The code below\nturns affine rows and Boolean gate tables into a word indexed by labels. Its two complementary\ntraversals control the correlation seen by a cut that evaluates the word at a single label." }, { "blockId": "p6-552df38005", "page": 6, "quote": "Let T be the complete rooted 2s-ary tree of depth m. The children of each internal node v are\nindexed by (g, b) ∈[s] × {0, 1}. Its leaves are indexed by [k]. To each internal node assign a\ntable Fv : Fs\n2 →{−1, 1}, and to each leaf assign a row Ri ∈H. The word W(R, F) ∈{−1, 1}A" }, { "blockId": "p7-6027f8900c", "page": 7, "quote": "is defined pointwise as follows. At a ∈A, leaf i has sign (−1)Ri(a). An internal node with child\nsigns (Zg,b) outputs\n" }, { "blockId": "p7-7bd60fd28f", "page": 7, "quote": "We choose the tables to be noised by a second traversal of the tree. A random slice is\nobtained by first choosing J0 uniformly from {0, . . . , m−1}. Starting at the root, at each reached\nnode of depth less than J0 choose one of its s child pairs uniformly and independently, and\ncontinue through both children of that pair. The set D of reached nodes at depth J0 is the slice.\nConditionally on J0 = j, the slice has 2j nodes, and every fixed node of depth j belongs to it\nwith probability s−j. We write Fout = (Fu)u/∈D for the tables outside the slice.\nThe two traversals of the tree used below complement one another. The slice construction\nchooses one pair and keeps both children. The Walsh expansion of a gate keeps one child from\nevery pair. Their intersection therefore follows a single path. Figure 1 shows the local rule;\niterating it gives one common node at every slice depth." }, { "blockId": "p7-6239856c19", "page": 7, "quote": "Figure 1: Complementary choices at one gate, illustrated for s = 3. A monomial traversal (solid)\nselects one child from each pair; the slice exploration (dashed) selects both children of one pair.\nExactly one child is shared, shown in purple. Only this local selection is depicted; the same rule\nis repeated above the chosen slice." } ], "occurrenceCount": 84 }, { "symbol": "B", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-9516123ba0", "page": 1, "quote": "The statement means that a polynomial-time algorithm which always returns a cut of weight\nat least α times the maximum would imply P = NP. We prove the following more precise gap\ntheorem. Write\nB(t) = 2" }, "occurrences": [ { "blockId": "p1-9516123ba0", "page": 1, "quote": "The statement means that a polynomial-time algorithm which always returns a cut of weight\nat least α times the maximum would imply P = NP. We prove the following more precise gap\ntheorem. Write\nB(t) = 2" }, { "blockId": "p2-2780d2e59a", "page": 2, "quote": "Theorem 1.2. Fix a rational t ∈(0, 1) and a constant 0 < ε < (t−B(t))/4. For graphs equipped\nwith a rational probability distribution on edges, it is NP-hard to distinguish" }, { "blockId": "p2-9875f02b11", "page": 2, "quote": "NO:\nVal ≤1 + B(t)" }, { "blockId": "p3-2347d0bc2e", "page": 3, "quote": "As in long-code reductions [4, 10], a label a is represented by the dictator w 7→w(a), where w\nranges over Boolean words indexed by labels. The graph has one such word domain for each\ntuple of left vertices of the source game. Thus an arbitrary cut specifies a Boolean function\non each domain. To sample an edge, the test chooses two left tuples through a common right\ntuple, generates two words on the right label space, and pulls them back through the sampled\naffine projections. It negates the second word. When the sampled constraints are satisfied,\ndictator labels give a cut probability of one half of one plus the correlation between the two\nwords evaluated at a common label.\nThe word generator is a finite rooted tree. Every internal node has s pairs of children and a\nrandom table from Fs\n2 to {−1, 1}; the leaves carry affine forms on the label space. At a label a,\nthese forms give leaf signs, and the tables recursively combine the child signs into the root sign\nW(a). The gate rule is chosen so that each monomial of its Walsh expansion keeps one child\nfrom every pair. A second exploration keeps both children of one randomly selected pair at each\nreached node. At any fixed depth, the two explorations have exactly one node in common. The\ntest couples the tables reached by this second exploration at a random depth, using entrywise\ncorrelation t, and leaves all other tables unchanged. Every monomial therefore acquires precisely\none factor of t. A small independent resampling of the affine rows gives the smoothing needed\nfor soundness and only a small completeness loss.\nFor an arbitrary cut, we first average its odd part over the source-game edges. The resulting\nfunction is balanced in the selected table entries, and a cut above the desired NO threshold\nmakes its expected noise stability exceed B(t). Majority Is Stablest then detects an influential\nentry. That entry is indexed by a gate input, not by a source-game label. The remaining task is\nto connect its influence to labels that the two endpoints of a source edge can choose consistently.\nThree steps address this task." }, { "blockId": "p4-629a2e57f3", "page": 4, "quote": "Section 2 fixes the Fourier conventions. Sections 3 and 4 define the code and the Max-Cut\ntest. Sections 5 to 7 prove the analytic ingredients. Sections 8 and 9 give the decoding and\nfinish the proof. Appendix A derives the precise starting game from the published construction,\nand Appendix B gives a deterministic conversion to simple unweighted graphs." }, { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." }, { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p6-7b93d7ee44", "page": 6, "quote": "A weighted graph is likewise specified by a rational distribution on unordered edges, allowing\nloops. A cut assigns signs to its vertices and satisfies an edge when its signs differ. Its value is\nits satisfaction probability. Loops have zero cut weight. The conversion in Theorem B.1 keeps a\ncommon scale even when loops are deleted." }, { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, { "blockId": "p9-14533a682c", "page": 9, "quote": "For each x ∈XT , make one graph vertex (x, w) for every word w ∈{−1, 1}B. This is the\nstandard long-code word domain [4, 10]; the tree construction specifies how the test samples its\nqueries. Thus an arbitrary cut is exactly a family of functions" }, { "blockId": "p9-c8bdd19d46", "page": 9, "quote": "σx : {−1, 1}B −→{−1, 1}." }, { "blockId": "p9-9b03894ad3", "page": 9, "quote": "1. Draw y ∈Y T from the product right-endpoint marginal of λ. Independently draw two\nedge tuples e, e′ from the product conditional edge law given y. Write x, x′ for their left\nendpoints and πe, πe′ : B →A for their coordinatewise projections." }, { "blockId": "p11-164121ec4c", "page": 11, "quote": "If a cut has value at least (1 + B(t))/2 + δ/2, then its odd average has expected slice stability\nat least B(t) + δ. The next three sections show that this excess stability yields a label carrying\nsubstantial Fourier mass, with constants independent of the game dimension and the number of\nrepetitions; the formal statement is Theorem 7.1. Section 8 then turns that mass into a labeling\nof a single copy of the original game." }, { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, { "blockId": "p18-65f0ea897b", "page": 18, "quote": "some entry z ∈Fs\n2 of some v ∈D has Inf≤K\nv,z ( ¯fω) ≥τ. Indeed, if the probability of this event is\np, the expected stability is at most B(t) + δ/2 + p." }, { "blockId": "p20-b92cb4dec5", "page": 20, "quote": "The inequality follows because the unsplit row projection is determined by the free shift index,\nand the degree and parity projections are orthogonal projections in the slice bits at each fixed\ncontext. Every Ψq has degree at most K.\nEquation (7.9) has converted the free-shift characters into translations of a gate table. We\nnext collect its Boolean Fourier coefficients into vectors whose squared norms are entry influences.\nThese Fourier coefficients are indexed by sets of table entries; the earlier index I ⊆[s] instead\nspecified a character of the table’s domain Fs\n2.\nDenote by bΨq(B, D) the slice Fourier coefficient with entry set B ⊆Fs\n2 at gate v and entry\nset D on the other slice gates. For z ∈Fs\n2 define a real vector c(q, z), with coordinates indexed\nby pairs (A, D) where 0 ∈A ⊆Fs\n2, by" }, { "blockId": "p20-5019de2b8d", "page": 20, "quote": "Using entry sets relative to z makes this vector space the same for every z and every label q.\nSet c(q, z) = 0 for functionals q which are not labels. For the influence at z, a slice Fourier\nset B containing z has the unique form B = z + A with 0 ∈A. Translation by J(q) sends its\ncoefficient to bΨq(z + A + J(q), D). Hence, exactly," }, { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" }, { "blockId": "p21-2a433dcc94", "page": 21, "quote": "The factor |B| counts the entries whose influences contain that Fourier monomial; it is at most\nthe monomial’s total degree, which is at most K. Thus Mv ≤K at every context. The total-mass\nidentity in Theorem 6.1, together with (7.11), also gives" } ], "occurrenceCount": 37 }, { "symbol": "Bp", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p16-06d5e24e4d", "page": 16, "quote": "The fourth moment.\nFor brevity write Bp = A′\np(Sp). Expanding the fourth power of the\nnorm and summing the characters in z gives the exact identity" }, "occurrences": [ { "blockId": "p16-06d5e24e4d", "page": 16, "quote": "The fourth moment.\nFor brevity write Bp = A′\np(Sp). Expanding the fourth power of the\nnorm and summing the characters in z gives the exact identity" }, { "blockId": "p16-3b57fa6b54", "page": 16, "quote": "p1,p2,p3\nE\n⟨Bp1, Bp2⟩⟨Bp3, Bp1+p2+p3⟩\n" }, { "blockId": "p16-961cce19bb", "page": 16, "quote": "= Ep1,p2,p3E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n,\np4 = p1 + p2 + p3.\n(6.8)" }, { "blockId": "p16-73cffa56cb", "page": 16, "quote": "Here p1, p2, p3 are independent uniform indices in the second line. The factor h−3 in the first\nline comes from four factors h−1 in Fourier inversion and the sum of h characters over z.\nIf any two of p1, p2, p3, p4 coincide, then some two of p1, p2, p3 coincide. This event has\nprobability at most 3/h. Since ∥Bp∥≤D\n√" }, { "blockId": "p16-35bfa1fe78", "page": 16, "quote": "E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n =\nX" } ], "occurrenceCount": 5, "latex": "B_{p}", "latexBasis": "pdf-script-position" }, { "symbol": "E", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p6-13601d3979", "page": 6, "quote": "A weighted bipartite projection game consists of finite sets X, Y , a rational probability distribu-\ntion λ on a finite edge set E with endpoint maps to X and Y (parallel edges are allowed), finite\nalphabets, and a projection πe for each edge. Its value is" }, "occurrences": [ { "blockId": "p5-cd21ecba28", "page": 5, "quote": "This operator preserves the range [−1, 1]. It is also convolution by an independent increment\nE ∈Hk, whose coordinates are zero with probability r and independent uniform rows with\nprobability 1 −r. Convolution commutes with every fixed translation of the rows." }, { "blockId": "p5-cda18a336e", "page": 5, "quote": "Stabt(h) = E[h(x)h(y)] =\nX" }, { "blockId": "p6-13601d3979", "page": 6, "quote": "A weighted bipartite projection game consists of finite sets X, Y , a rational probability distribu-\ntion λ on a finite edge set E with endpoint maps to X and Y (parallel edges are allowed), finite\nalphabets, and a projection πe for each edge. Its value is" }, { "blockId": "p7-6aee721e9d", "page": 7, "quote": "E\nW(R1, F)(a)W(R2, F ′)(a) | D\n = tr2d.\n(3.2)" }, { "blockId": "p8-1d56f60d7e", "page": 8, "quote": "E[ bF ′\nv(I) | Fv] = t bFv(I)." }, { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, { "blockId": "p10-84cd1fc5fb", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy) −1" }, { "blockId": "p10-542d9b17d9", "page": 10, "quote": "2E StabFD\nt\n( ¯Ay),\n(4.5)" }, { "blockId": "p10-1301175d92", "page": 10, "quote": "2E StabFD\nt\n( ¯Gy).\n(4.6)" }, { "blockId": "p14-6293ddf1d4", "page": 14, "quote": "Then\nE\nX" }, { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" }, { "blockId": "p15-afd7da3e46", "page": 15, "quote": "E max\nz ∥U(z)∥2 ≤2M" }, { "blockId": "p16-9f579ae464", "page": 16, "quote": "E max\nz ∥U(z) −U′(z)∥2 ≤M" }, { "blockId": "p16-34f6aa8d11", "page": 16, "quote": "E\nX" }, { "blockId": "p16-3b57fa6b54", "page": 16, "quote": "p1,p2,p3\nE\n⟨Bp1, Bp2⟩⟨Bp3, Bp1+p2+p3⟩\n" }, { "blockId": "p16-961cce19bb", "page": 16, "quote": "= Ep1,p2,p3E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n,\np4 = p1 + p2 + p3.\n(6.8)" }, { "blockId": "p16-35bfa1fe78", "page": 16, "quote": "E\n⟨Bp1, Bp2⟩⟨Bp3, Bp4⟩\n =\nX" }, { "blockId": "p17-c22ad44175", "page": 17, "quote": "i=1\n∥α′\nq(pi)∥≤\n\nE∥α′\nq(p1)∥2∥α′\nq(p2)∥21/2 \nE∥α′\nq(p3)∥2∥α′\nq(p4)∥21/2" }, { "blockId": "p17-a89add6fb0", "page": 17, "quote": "E\nX" }, { "blockId": "p17-d3200f345b", "page": 17, "quote": "E\nX" } ], "occurrenceCount": 44 }, { "symbol": "E1", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, "occurrences": [ { "blockId": "p22-bc1e47a7de", "page": 22, "quote": "Ei = BiRnew\ni\n,\nE = (E1, . . . , Ek)." }, { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p34-56088078a9", "page": 34, "quote": "occurs for Tγ and hence has a maximal occurring extension in that auxiliary table. Lifting any\nsuch extension by the same pγ gives a function agreeing with f on all of S0. This conclusion\nconcerns function extension and is unaffected by the extra agreements that zero completion may\nintroduce.\nThe second player selects this tag and one such maximal extension with probability at\nleast 1/M, where M := 2qM0 is independent of k0. Its eventual answer then agrees with f\non W ∩VV . The first player’s uniform extension agrees with this answer on all of VV with\nprobability 2−i ≥2−r: the quotient VV /(W ∩VV ) has dimension i. Thus, conditionally on\nE1 ∩E2 ∩E3 ∩E4, the strategy wins with probability at least" } ], "occurrenceCount": 5, "latex": "E_{1}", "latexBasis": "pdf-script-position" }, { "symbol": "E2", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, "occurrences": [ { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p34-56088078a9", "page": 34, "quote": "occurs for Tγ and hence has a maximal occurring extension in that auxiliary table. Lifting any\nsuch extension by the same pγ gives a function agreeing with f on all of S0. This conclusion\nconcerns function extension and is unaffected by the extra agreements that zero completion may\nintroduce.\nThe second player selects this tag and one such maximal extension with probability at\nleast 1/M, where M := 2qM0 is independent of k0. Its eventual answer then agrees with f\non W ∩VV . The first player’s uniform extension agrees with this answer on all of VV with\nprobability 2−i ≥2−r: the quotient VV /(W ∩VV ) has dimension i. Thus, conditionally on\nE1 ∩E2 ∩E3 ∩E4, the strategy wins with probability at least" } ], "occurrenceCount": 4, "latex": "E_{2}", "latexBasis": "pdf-script-position" }, { "symbol": "E3", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, "occurrences": [ { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p34-56088078a9", "page": 34, "quote": "occurs for Tγ and hence has a maximal occurring extension in that auxiliary table. Lifting any\nsuch extension by the same pγ gives a function agreeing with f on all of S0. This conclusion\nconcerns function extension and is unaffected by the extra agreements that zero completion may\nintroduce.\nThe second player selects this tag and one such maximal extension with probability at\nleast 1/M, where M := 2qM0 is independent of k0. Its eventual answer then agrees with f\non W ∩VV . The first player’s uniform extension agrees with this answer on all of VV with\nprobability 2−i ≥2−r: the quotient VV /(W ∩VV ) has dimension i. Thus, conditionally on\nE1 ∩E2 ∩E3 ∩E4, the strategy wins with probability at least" } ], "occurrenceCount": 3, "latex": "E_{3}", "latexBasis": "pdf-script-position" }, { "symbol": "Fout", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p7-7bd60fd28f", "page": 7, "quote": "We choose the tables to be noised by a second traversal of the tree. A random slice is\nobtained by first choosing J0 uniformly from {0, . . . , m−1}. Starting at the root, at each reached\nnode of depth less than J0 choose one of its s child pairs uniformly and independently, and\ncontinue through both children of that pair. The set D of reached nodes at depth J0 is the slice.\nConditionally on J0 = j, the slice has 2j nodes, and every fixed node of depth j belongs to it\nwith probability s−j. We write Fout = (Fu)u/∈D for the tables outside the slice.\nThe two traversals of the tree used below complement one another. The slice construction\nchooses one pair and keeps both children. The Walsh expansion of a gate keeps one child from\nevery pair. Their intersection therefore follows a single path. Figure 1 shows the local rule;\niterating it gives one common node at every slice depth." }, "occurrences": [ { "blockId": "p7-7bd60fd28f", "page": 7, "quote": "We choose the tables to be noised by a second traversal of the tree. A random slice is\nobtained by first choosing J0 uniformly from {0, . . . , m−1}. Starting at the root, at each reached\nnode of depth less than J0 choose one of its s child pairs uniformly and independently, and\ncontinue through both children of that pair. The set D of reached nodes at depth J0 is the slice.\nConditionally on J0 = j, the slice has 2j nodes, and every fixed node of depth j belongs to it\nwith probability s−j. We write Fout = (Fu)u/∈D for the tables outside the slice.\nThe two traversals of the tree used below complement one another. The slice construction\nchooses one pair and keeps both children. The Walsh expansion of a gate keeps one child from\nevery pair. Their intersection therefore follows a single path. Figure 1 shows the local rule;\niterating it gives one common node at every slice depth." }, { "blockId": "p10-12c4ff12f9", "page": 10, "quote": "where ϕy is odd and depends on no slice or row data. Thus ¯Gy has mean zero in the slice entries\nfor every fixed y, D, R, Fout, by the observation following Theorem 3.2. The later extraction\nargument, Theorem 7.1, applies to precisely such bounded odd functions of the generated word." }, { "blockId": "p10-4dd6fad46a", "page": 10, "quote": "where both expectations are over y, D, R, Fout. In particular," }, { "blockId": "p10-c4b16ddf34", "page": 10, "quote": "Proof. Condition on y, D, R, Fout. The sampled edge tuples and the two row resamplings are\nindependent between copies. After averaging over them, the expected first sign as a function\nof FD is ¯Ay + ¯Gy. The minus sign in the second queried word makes the expected second sign\n¯Ay −¯Gy at F ′\nD. The two slice inputs are ordinary t-correlated uniform cube inputs. The rows\nin these functions are the common latent array R; the two independent row resamplings have\nalready been absorbed into Tr. Negating every slice entry leaves ¯Ay unchanged and negates ¯Gy.\nTheir Fourier supports therefore have opposite total parities, so the cross terms vanish. The\nexpected sign product is Stabt( ¯Ay) −Stabt( ¯Gy). An edge is cut with probability one half of one\nminus its sign product, giving (4.5). Nonnegativity of stability for t > 0 gives (4.6)." }, { "blockId": "p17-01d2e48829", "page": 17, "quote": "The word, tree, and random slice D are those of Section 3. Write Fout for the tables outside D.\nOnce ω, D, R, Fout are fixed, ¯fω is a function of the independent slice-table bits." }, { "blockId": "p18-af115d7cb4", "page": 18, "quote": "mq(f; Γ, Fout) = EFD\nX" }, { "blockId": "p18-1a521f670f", "page": 18, "quote": "q∈A\nmq(f; Γ, Fout) ≤1.\n(7.2)" }, { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, { "blockId": "p18-01097b485f", "page": 18, "quote": "Here we sample ω, the random slice D, and a uniform node v ∈D; conditional on these, we\nsample Fout and Γ independently and uniformly." }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p19-d0b22816e9", "page": 19, "quote": "For any fixed s, the last inequality can be ensured by increasing m. The same expectation is the\nexpectation over ω, D, R, Fout of the sum of the corresponding conditional squared norms in FD.\nMarkov’s inequality shows that this sum is at most τ/4 except on an event of probability η/2.\nOn the intersection with the influential-entry event, the triangle inequality for the projection\ndefining that influence gives" }, { "blockId": "p20-56ef8758e5", "page": 20, "quote": "Step 3: express odd influences as affine hashes.\nFix ω, D, v, Γ, Fout. The remaining row\ncoordinates Lgb are independent uniform elements of H. On the odd part, the common pair\nsum q is a label, by (2.1), and hence is nonzero. An unsplit shift index with common sum q\nputs q in the second child on a subset I ⊆[s] and in the first child elsewhere. Let aq,I(FD) be\nits coefficient in Hodd\nv\n.\nSet" }, { "blockId": "p20-dbdc944b36", "page": 20, "quote": "I\n∥aq,I∥2\n2,FD ≤mq( ¯fω; Γ, Fout).\n(7.10)" }, { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" }, { "blockId": "p21-a65ea046f4", "page": 21, "quote": "For each node, condition on ω, D, v, Γ, Fout and integrate the free shifts. Partition these\ncontexts into three classes:" }, { "blockId": "p22-955a28fdcb", "page": 22, "quote": "Lemma 8.1 (Removing smoothing). For every fixed y, D, v, Γ, Fout, and label q ∈A," }, { "blockId": "p22-470d569f6e", "page": 22, "quote": "mq( ¯Gy; Γ, Fout) ≤EE, e|ymq(Ge; Γ + EΓ, Fout).\n(8.1)" }, { "blockId": "p22-7eded40296", "page": 22, "quote": "q∗= q∗(y, D, v, Γ, Fout) ∈arg max\nq∈A\nmq( ¯Gy; Γ, Fout)\n(8.2)" }, { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" } ], "occurrenceCount": 24, "latex": "F_{out}", "latexBasis": "pdf-script-position" }, { "symbol": "G", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-0787686067", "page": 1, "quote": "For a graph with nonnegative edge weights, Max-Cut asks for a partition of the vertices\nmaximizing the weight of edges crossing the partition. A cut is encoded by signs on vertices,\nand an edge contributes exactly when its signs differ. For a maximization problem, an α-\napproximation returns a feasible solution worth at least α times the optimum. Unless stated\notherwise, the graph and its rational weights are given explicitly. We write MaxCut(G) for the\nmaximum crossing weight and, when the edge weights sum to one, also denote this value by\nVal(G). The decision problem of whether a weighted graph has a cut of at least a given weight\nappears among Karp’s original NP-complete problems [11]. Approximation asks how much of\nthis optimum can still be guaranteed in polynomial time. Goemans and Williamson [9] gave\na semidefinite programming algorithm with random-hyperplane rounding for Max-Cut, with\nexpected approximation ratio approaching" }, "occurrences": [ { "blockId": "p1-0787686067", "page": 1, "quote": "For a graph with nonnegative edge weights, Max-Cut asks for a partition of the vertices\nmaximizing the weight of edges crossing the partition. A cut is encoded by signs on vertices,\nand an edge contributes exactly when its signs differ. For a maximization problem, an α-\napproximation returns a feasible solution worth at least α times the optimum. Unless stated\notherwise, the graph and its rational weights are given explicitly. We write MaxCut(G) for the\nmaximum crossing weight and, when the edge weights sum to one, also denote this value by\nVal(G). The decision problem of whether a weighted graph has a cut of at least a given weight\nappears among Karp’s original NP-complete problems [11]. Approximation asks how much of\nthis optimum can still be guaranteed in polynomial time. Goemans and Williamson [9] gave\na semidefinite programming algorithm with random-hyperplane rounding for Max-Cut, with\nexpected approximation ratio approaching" }, { "blockId": "p6-f7f78a9e2b", "page": 6, "quote": "Val(G) = max\nℓX,ℓY\nPr\ne=(x,y)∼λ[πe(ℓX(x)) = ℓY (y)]." }, { "blockId": "p6-cadeb61e43", "page": 6, "quote": "YES :\nVal(G) ≥1 −a,\nNO :\nVal(G) ≤b." }, { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, { "blockId": "p9-a96718c718", "page": 9, "quote": "Lemma 4.1 (Completeness). If Val(G) ≥1 −a, the graph has a cut of value at least" }, { "blockId": "p24-44276ea803", "page": 24, "quote": "Val(G) ≥2−2s p∗θ" }, { "blockId": "p26-8659c32a43", "page": 26, "quote": "Finally choose a positive rational a < 1 −b small enough that 2Ta ≤ε/4, and start with the\nNP-hard game gap Val(G) ≥1 −a versus Val(G) ≤b." }, { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, { "blockId": "p35-be0e319110", "page": 35, "quote": "Lemma B.1. Let G be a graph on N vertices with nonnegative rational weights wuv on unordered\ndistinct vertex pairs, and P\nu 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" }, { "blockId": "p35-b21848e281", "page": 35, "quote": "Q\n−MaxCut(G)\n ≤ϵ′." }, { "blockId": "p36-82d91c1daf", "page": 36, "quote": "Now fix an arbitrary cut of G′, and let au ∈[0, 1] be the fraction of cluster u on its positive\nside. Its crossing edges between clusters u, v form two rectangles. Equation (B.1) therefore\ncompares their scaled count with" }, { "blockId": "p36-301ecd5146", "page": 36, "quote": "to error at most 2/p + 2/p2. Summing over the at most N(N −1)/2 pairs costs at most\nN(N −1)(1/p + 1/p2). The displayed weighted expression, summed over pairs, is the expected\ncut weight obtained by independently assigning vertex u a positive sign with probability au. It\nis at most MaxCut(G). This proves the desired upper bound for every cut of G′.\nConversely, give all vertices in each cluster the sign of a maximum cut of G. The same\nrectangle bound compares its scaled weight with MaxCut(G) to at most the same error, giving\nthe reverse inequality. Our choice of p ensures N(N −1)(1/p + 1/p2) < ϵ′, which proves the\nassertion. Loops contribute zero to every cut and can be deleted at the outset." } ], "occurrenceCount": 12 }, { "symbol": "H", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" }, "occurrences": [ { "blockId": "p1-4155ac1062", "page": 1, "quote": "The expression compares two contributions of an edge whose endpoint vectors have inner product\nρ: its semidefinite objective value is (1 −ρ)/2, whereas a random hyperplane separates the\nvectors with probability arccos(ρ)/π. The minimum ratio over ρ is the uniform guarantee of\nthis rounding rule. Determining whether this ratio is optimal under ordinary NP-hardness has\nbeen a central question in approximation algorithms. Håstad’s hardness theorem [10], combined\nwith the gadgets of Trevisan, Sorkin, Sudan, and Williamson [19], rules out ratios greater than\n16/17. Khot, Kindler, Mossel, and O’Donnell [13] established optimality of αGW assuming the\nUnique Games Conjecture [12]; the analytic input is the Majority Is Stablest theorem of Mossel,\nO’Donnell, and Oleszkiewicz [16].\nWe resolve the unconditional optimality question for Max-Cut." }, { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, { "blockId": "p4-20be6b6373", "page": 4, "quote": "H = Aff(A, F2)." }, { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" }, { "blockId": "p4-767850a4b2", "page": 4, "quote": "Indeed, if h(a) = h0 + P\ni hiai, a functional taking value one on 1 has the unique form\nq(h) = h0 + P\ni hiai. We call precisely these functionals labels; a general nonzero functional in\nH∗need not be a label.\nFor f : Hk →R and C = (q1, . . . , qk) ∈(H∗)k, put" }, { "blockId": "p6-58466c653a", "page": 6, "quote": "We use the affine label space A and row space H = Aff(A, F2) of Section 2. The code below\nturns affine rows and Boolean gate tables into a word indexed by labels. Its two complementary\ntraversals control the correlation seen by a cut that evaluates the word at a single label." }, { "blockId": "p6-552df38005", "page": 6, "quote": "Let T be the complete rooted 2s-ary tree of depth m. The children of each internal node v are\nindexed by (g, b) ∈[s] × {0, 1}. Its leaves are indexed by [k]. To each internal node assign a\ntable Fv : Fs\n2 →{−1, 1}, and to each leaf assign a row Ri ∈H. The word W(R, F) ∈{−1, 1}A" }, { "blockId": "p9-bc40310c57", "page": 9, "quote": "and the row space H = Aff(A, F2)." }, { "blockId": "p11-0801c4df00", "page": 11, "quote": "For each node v, define its generator P(v), a set of leaves, recursively: P(v) = {v} at a leaf, and\nP(v) = P(v1,0) ∪P(v1,1) at an internal node. Thus a generator always follows both children\nof the first pair; unlike the slice, it involves no random choices. Adding the same h ∈H to all\nrows in P(v) multiplies the subtree output at a by (−1)h(a). This follows by induction from\nEquation (3.1): multiplying both children of one pair by the same sign leaves all gate inputs zg\nunchanged and multiplies the prefactor by that sign.\nFor a row Fourier index C = (qi)k\ni=1 ∈(H∗)k, put" }, { "blockId": "p12-b13f87e64f", "page": 12, "quote": "χqv,g0+qv,g1(h) bfC(F) = χqv(h) bfC(F)\n(h ∈H)." }, { "blockId": "p13-10cb301361", "page": 13, "quote": "Hk −→H2s × Hk−2s,\nR 7−→((Lg,b), Γ)" }, { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." }, { "blockId": "p14-edbd4f63d7", "page": 14, "quote": "q∈H∗\nwq." }, { "blockId": "p14-0e6f947b09", "page": 14, "quote": "Choose L, J1, . . . , Js independently and uniformly in H, and define" }, { "blockId": "p14-a0193972bf", "page": 14, "quote": "q∈H∗\nχq(L)c(q, z + J(q))." }, { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" }, { "blockId": "p14-445c75ee85", "page": 14, "quote": "Proof. All expectations over H and Fs\n2 below use uniform probability measure; the displayed\nsums defining wq and M are unnormalized. Conditional on the Jg, orthogonality of the characters\nof L gives\nEL∥U(z)∥2 =\nX" }, { "blockId": "p15-960ace62e8", "page": 15, "quote": "Fourier representation.\nFor p ∈Fs\n2 and u ∈H, define" }, { "blockId": "p15-3b1a12fcb4", "page": 15, "quote": "q∈H∗\nχq(u)αq(p),\nSp = L +" }, { "blockId": "p15-167b2af07b", "page": 15, "quote": "For distinct p, p′, the variables Sp, Sp′ are independent and uniform in H: the coefficient\nvectors (1, p), (1, p′) are linearly independent over F2. Similarly, any three distinct p1, p2, p3 give\nthree independent uniform variables Sp1, Sp2, Sp3. Indeed, any nonempty subcollection of their\ncoefficient vectors whose sum is zero must have even cardinality, and hence would consist of two\nequal vectors." } ], "occurrenceCount": 47 }, { "symbol": "Nj", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p12-20009794ba", "page": 12, "quote": "Proof. Fix an index C satisfying Equation (5.2), and let Nj be the number of nodes at depth\nj with nonzero index qv. If qv ̸= 0, each child pair contains at least one nonzero index. A\nfailure of the unsplit condition at such a node supplies at least one additional nonzero child. If\nqv = 0, a failure supplies at least two nonzero children, whereas its minimum contribution is\nzero. Therefore the number Bj of failures at depth j satisfies" }, "occurrences": [ { "blockId": "p12-20009794ba", "page": 12, "quote": "Proof. Fix an index C satisfying Equation (5.2), and let Nj be the number of nodes at depth\nj with nonzero index qv. If qv ̸= 0, each child pair contains at least one nonzero index. A\nfailure of the unsplit condition at such a node supplies at least one additional nonzero child. If\nqv = 0, a failure supplies at least two nonzero children, whereas its minimum contribution is\nzero. Therefore the number Bj of failures at depth j satisfies" }, { "blockId": "p12-eb5659cc8e", "page": 12, "quote": "Bj ≤Nj+1 −sNj." }, { "blockId": "p13-1d94de894b", "page": 13, "quote": "Nj+1 −sNj" } ], "occurrenceCount": 3, "latex": "N_{j}", "latexBasis": "pdf-script-position" }, { "symbol": "P", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-9516123ba0", "page": 1, "quote": "The statement means that a polynomial-time algorithm which always returns a cut of weight\nat least α times the maximum would imply P = NP. We prove the following more precise gap\ntheorem. Write\nB(t) = 2" }, "occurrences": [ { "blockId": "p1-9516123ba0", "page": 1, "quote": "The statement means that a polynomial-time algorithm which always returns a cut of weight\nat least α times the maximum would imply P = NP. We prove the following more precise gap\ntheorem. Write\nB(t) = 2" }, { "blockId": "p4-767850a4b2", "page": 4, "quote": "Indeed, if h(a) = h0 + P\ni hiai, a functional taking value one on 1 has the unique form\nq(h) = h0 + P\ni hiai. We call precisely these functionals labels; a general nonzero functional in\nH∗need not be a label.\nFor f : Hk →R and C = (q1, . . . , qk) ∈(H∗)k, put" }, { "blockId": "p4-ff7ddf6ede", "page": 4, "quote": "Parseval gives P\nC bf(C)2 = ERf(R)2. The same notation is used for functions with additional\nvariables, which remain fixed during the indicated Fourier expansion." }, { "blockId": "p5-7639a16026", "page": 5, "quote": "Let P≤K be the orthogonal projection onto Fourier degree at most K. In particular,\nX" }, { "blockId": "p5-54e1c6b4a5", "page": 5, "quote": "a∈Ω\nInfa(P≤Kh) ≤K∥h∥2\n2.\n(2.3)" }, { "blockId": "p8-5a286b54c0", "page": 8, "quote": "Proof. For a fixed gate table, expand in its argument z: Fv(z) = P\nI⊆[s] bFv(I)(−1)\nP" }, { "blockId": "p8-4efa298442", "page": 8, "quote": "i∈P\nxi,\n|P| = d,\n(3.3)" }, { "blockId": "p8-6af3299821", "page": 8, "quote": "P∈P\naP (F)\nY" }, { "blockId": "p8-bcf0ef5bfe", "page": 8, "quote": "where P is the family of leaf sets produced by choosing one child per pair at each reached internal\nnode. The leaf set determines all these choices: a selected child has a nonempty intersection\nwith that leaf set, whereas an unselected child has none. Thus different traversals give different\nmonomials, and aP (F) is the product of one Walsh coefficient of Fv for each node reached by\nthe traversal.\nAt the root both a monomial traversal and the slice exploration are present. At any common\nnode above the slice depth, exactly one child is selected by both explorations. Induction gives\nexactly one common node at the slice depth. Hence each aP contains exactly one factor from a\ntable on D. Negating every such table negates each coefficient in Equation (3.3) and therefore\nnegates the word. Moreover, each Walsh coefficient of a slice table satisfies" }, { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, { "blockId": "p11-0801c4df00", "page": 11, "quote": "For each node v, define its generator P(v), a set of leaves, recursively: P(v) = {v} at a leaf, and\nP(v) = P(v1,0) ∪P(v1,1) at an internal node. Thus a generator always follows both children\nof the first pair; unlike the slice, it involves no random choices. Adding the same h ∈H to all\nrows in P(v) multiplies the subtree output at a by (−1)h(a). This follows by induction from\nEquation (3.1): multiplying both children of one pair by the same sign leaves all gate inputs zg\nunchanged and multiplies the prefactor by that sign.\nFor a row Fourier index C = (qi)k\ni=1 ∈(H∗)k, put" }, { "blockId": "p11-9b076a7bc4", "page": 11, "quote": "i∈P(v)\nqi." }, { "blockId": "p11-ef6839a7ad", "page": 11, "quote": "bfC(F v,z) = (−1)\nP" }, { "blockId": "p12-197aaf8a57", "page": 12, "quote": "Proof. For each g, shifting the generators of both children in pair g by h has exactly the same\neffect on the subtree at v as shifting P(v) by h. Consequently the two row translations have the\nsame effect on the whole word. Taking row Fourier coefficients yields" }, { "blockId": "p12-6c5c7c7ebd", "page": 12, "quote": "Distinct characters are distinct functions, proving Equation (5.2).\nNegating Fv has the same effect on the word as adding 1 to all rows in P(v). This proves\nEquation (5.3). To translate the input of Fv by z, add 1 to the generator of its second child\nin each pair with zg = 1. This changes precisely those gate input bits and leaves its prefactor\nunchanged. Fourier transformation gives Equation (5.4).\nRow smoothing and row Fourier projections act diagonally on C, so preserve all three\nidentities. On the cube of table-entry signs, a domain translation permutes coordinates, and\nnegating a gate table multiplies each monomial by its parity sign. Both operations preserve\ntotal degree in any specified collection of whole gate tables and therefore commute with the\nstated degree projections." }, { "blockId": "p13-7c81722455", "page": 13, "quote": "For x ≥0, xe−2µx ≤(2eµ)−1, and log r ≤−µ/d. Hence every parenthesized factor is at most\n(2eµ)−1, while P\nC EF | bfC(F)|2 = ∥f∥2\n2 ≤1. This proves the bound. The proof is pointwise in\nthe background data, so its average satisfies the same inequality." }, { "blockId": "p13-e6707ed124", "page": 13, "quote": "The extraction argument must use the common index after fixing most of the rows. We therefore\nchoose coordinates in which translating a child generator changes one free row and leaves all\nother coordinates fixed. The unsplit projection will then act separately at each fixed context,\nand its retained characters will have an explicit form.\nFix a slice D and a node v ∈D. The 2s sets P(vg,b) are nonempty and disjoint. Choose\none representative leaf ig,b from each of them, using a rule depending only on the tree, and set\nLg,b = Rig,b. For every other leaf i in that generator, replace Ri by Ri + Lg,b. Keep all rows\noutside these generators as they are. Denote the resulting k −2s remaining coordinates by\nΓ ∈Hk−2s." }, { "blockId": "p13-a07f9ca181", "page": 13, "quote": "Proof. The inverse map sets the representative row equal to Lg,b and recovers every other row in\nits generator by adding Lg,b to its recorded difference. This proves bijectivity. In a row character,\nthe coefficient of Lg,b is exactly P\ni∈P(vg,b) qi = qv,gb. Thus the condition defining Sv depends\nonly on the shift index Q, not on any Fourier index of Γ. Grouping the full Fourier expansion\nby Q proves its asserted action on each fixed fiber. The same grouping proves the support and\ntable-action identities: all full row coefficients with a given Q satisfy the same identities and\ncarry the same signs." }, { "blockId": "p14-443399ff79", "page": 14, "quote": "If q = 0, there is only the all-zero unsplit index, with character 1. Negating Fv multiplies the\ncorresponding coefficient by (−1)q(1). On the part odd in Fv, q is therefore a label and hence\nnonzero, so the description by q and I is unique. For such a label, translating the argument of\nFv by z ∈Fs\n2 multiplies its coefficient by (−1)\nP" }, { "blockId": "p15-afbe23d3c0", "page": 15, "quote": "Pointwise truncation does not increase the total squared norm, so Parseval gives P\nq w′\nq ≤M. If\nmp ≤CM, then" } ], "occurrenceCount": 36 }, { "symbol": "S0", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p34-be24debe49", "page": 34, "quote": "Here L has dimension ℓ. Set S0 = W ∩VV and γ = f|Q. Every equality FV [L] = f|L in this\ndisplay is retained in Tγ. Since ζi ≤τ0/2 ≤τi/2, the quotient pair" }, "occurrences": [ { "blockId": "p34-be24debe49", "page": 34, "quote": "Here L has dimension ℓ. Set S0 = W ∩VV and γ = f|Q. Every equality FV [L] = f|L in this\ndisplay is retained in Tγ. Since ζi ≤τ0/2 ≤τi/2, the quotient pair" }, { "blockId": "p34-78524b1e91", "page": 34, "quote": "f|S0 −pγ|S0, S0/Q\n" }, { "blockId": "p34-56088078a9", "page": 34, "quote": "occurs for Tγ and hence has a maximal occurring extension in that auxiliary table. Lifting any\nsuch extension by the same pγ gives a function agreeing with f on all of S0. This conclusion\nconcerns function extension and is unaffected by the extra agreements that zero completion may\nintroduce.\nThe second player selects this tag and one such maximal extension with probability at\nleast 1/M, where M := 2qM0 is independent of k0. Its eventual answer then agrees with f\non W ∩VV . The first player’s uniform extension agrees with this answer on all of VV with\nprobability 2−i ≥2−r: the quotient VV /(W ∩VV ) has dimension i. Thus, conditionally on\nE1 ∩E2 ∩E3 ∩E4, the strategy wins with probability at least" } ], "occurrenceCount": 3, "latex": "S_{0}", "latexBasis": "pdf-script-position" }, { "symbol": "Sv", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p12-f744950367", "page": 12, "quote": "Equation (5.2) assigns one common sum qv to every child pair. Equation (5.3) then has a useful\ninterpretation: coefficients odd under negating Fv have qv(1) = 1, so their common index is a\nlabel by (2.1). The individual child indices, however, can still be arbitrary functionals whose\nsum is qv.\nWe simplify this last freedom at one gate at a time. Keeping qv in one child of each pair\nleaves only the choice of that child, rather than two arbitrary functionals with prescribed sum.\nCall C unsplit at v if, for every g, the ordered pair (qv,g0, qv,g1) is either (qv, 0) or (0, qv). Let Sv\nbe the orthogonal row Fourier projection onto such indices. When qv = 0, this requires every\nchild index to be zero. The next bound averages the cost over the slice depth. It does not\nrequire an index to be unsplit at every gate of the tree." }, "occurrences": [ { "blockId": "p12-f744950367", "page": 12, "quote": "Equation (5.2) assigns one common sum qv to every child pair. Equation (5.3) then has a useful\ninterpretation: coefficients odd under negating Fv have qv(1) = 1, so their common index is a\nlabel by (2.1). The individual child indices, however, can still be arbitrary functionals whose\nsum is qv.\nWe simplify this last freedom at one gate at a time. Keeping qv in one child of each pair\nleaves only the choice of that child, rather than two arbitrary functionals with prescribed sum.\nCall C unsplit at v if, for every g, the ordered pair (qv,g0, qv,g1) is either (qv, 0) or (0, qv). Let Sv\nbe the orthogonal row Fourier projection onto such indices. When qv = 0, this requires every\nchild index to be zero. The next bound averages the cost over the slice depth. It does not\nrequire an index to be unsplit at every gate of the tree." }, { "blockId": "p12-091c8a397c", "page": 12, "quote": "v∈D\n∥(I −Sv) ¯f∥2\n2,R,F ≤\ns\n2emµ.\n(5.5)" }, { "blockId": "p13-bbf043b794", "page": 13, "quote": "is a linear bijection. In particular, under uniform rows the free shifts are independent uniforms\nconditional on Γ. The Fourier index of Lg,b corresponding to a full row index C is Qg,b = qv,gb.\nConsequently, at every fixed Γ, the operator Sv acts as the Fourier projection in the 2s shifts\nonto precisely the unsplit shift indices. For a function of the code or its row smoothing, the shift\ncoefficients vanish unless the sums Qg,0 + Qg,1 are the same for every g. The table actions in\nEquations (5.3) and (5.4) hold for each shift coefficient at this fixed context, with qv = Q1,0 +Q1,1\nand qv,g1 = Qg,1." }, { "blockId": "p13-a07f9ca181", "page": 13, "quote": "Proof. The inverse map sets the representative row equal to Lg,b and recovers every other row in\nits generator by adding Lg,b to its recorded difference. This proves bijectivity. In a row character,\nthe coefficient of Lg,b is exactly P\ni∈P(vg,b) qi = qv,gb. Thus the condition defining Sv depends\nonly on the shift index Q, not on any Fourier index of Γ. Grouping the full Fourier expansion\nby Q proves its asserted action on each fixed fiber. The same grouping proves the support and\ntable-action identities: all full row coefficients with a given Q satisfy the same identities and\ncarry the same signs." }, { "blockId": "p19-e5b0903627", "page": 19, "quote": "Let Sv be the orthogonal row projection onto indices unsplit at v. Put" }, { "blockId": "p19-f2ddd11aec", "page": 19, "quote": "v∈D\n∥(1 −Sv) ¯fω∥2\n2,R,F ≤\ns\n2emµ ≤ητ" }, { "blockId": "p19-d3d4a81378", "page": 19, "quote": "Inf≤K\nv,z ( ¯fω) −∥(1 −Sv) ¯fω∥2,FD ≥\n√τ" }, { "blockId": "p19-d7a06cebf5", "page": 19, "quote": "To verify it, integrate over all rows. Each Sv is an orthogonal projection in the rows and\ncommutes with the slice-bit Fourier projections. It can therefore only decrease the integrated\ninfluence at its entries. After dropping each Sv separately, the left side is bounded by the total\ninfluence of P FD\n≤K ¯fω, averaged over the remaining variables. Equation (2.3) and | ¯fω| ≤1 give\n(7.7). No pointwise boundedness of Hv is asserted or needed." } ], "occurrenceCount": 8, "latex": "S_{v}", "latexBasis": "pdf-script-position" }, { "symbol": "T", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p6-552df38005", "page": 6, "quote": "Let T be the complete rooted 2s-ary tree of depth m. The children of each internal node v are\nindexed by (g, b) ∈[s] × {0, 1}. Its leaves are indexed by [k]. To each internal node assign a\ntable Fv : Fs\n2 →{−1, 1}, and to each leaf assign a row Ri ∈H. The word W(R, F) ∈{−1, 1}A" }, "occurrences": [ { "blockId": "p6-552df38005", "page": 6, "quote": "Let T be the complete rooted 2s-ary tree of depth m. The children of each internal node v are\nindexed by (g, b) ∈[s] × {0, 1}. Its leaves are indexed by [k]. To each internal node assign a\ntable Fv : Fs\n2 →{−1, 1}, and to each leaf assign a row Ri ∈H. The word W(R, F) ∈{−1, 1}A" }, { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, { "blockId": "p9-9b03894ad3", "page": 9, "quote": "1. Draw y ∈Y T from the product right-endpoint marginal of λ. Independently draw two\nedge tuples e, e′ from the product conditional edge law given y. Write x, x′ for their left\nendpoints and πe, πe′ : B →A for their coordinatewise projections." }, { "blockId": "p10-0e07627ed1", "page": 10, "quote": "Each coordinate of either sampled edge tuple has the original edge law.\nA union bound\nshows that all 2T sampled constraints are satisfied except with probability at most 2Ta.\nWhen all these constraints are satisfied, the two queried tuples project to the same label\nl = (ℓY (y1), . . . , ℓY (yT )) ∈A. The cut signs then have product −W(R1, F)(l)W(R2, F ′)(l). The\nword-generation data are independent of the game edges, so Theorem 3.2 gives conditional cut\nprobability (1 + tr2d)/2. On the remaining event the cut probability is nonnegative. This proves\nthe first bound when 2Ta ≤1; if 2Ta > 1 that bound is negative and is automatic. Finally,\nr2d ≥1 −2µ, and the loss due to the prefactor is at most 2Ta, giving the second bound." }, { "blockId": "p22-0472ef3c19", "page": 22, "quote": "coefficients. Hiding the planted coordinate makes the resulting distribution close to the original\none. The comparison across the unknown projection then involves only the 2s free shifts, so its\nloss is independent of n and T.\nThroughout this section, the cut tables are fixed. We use the tree parameters and the row\ncoordinates of the preceding sections. For each choice of a slice D and a node v ∈D, fix the\ncoordinate change" }, { "blockId": "p23-d402bf53a5", "page": 23, "quote": "Write each affine row in coordinates on A = (Fn\n2)T . Apart from its single constant coefficient, it\nhas one block of n linear coefficients for each of the T slots. For a slot a ∈[T], let Za be the\nevent that the slot-a blocks vanish in all k −2s context rows and all k latent rows Rnew\ni\n. No\ncondition is imposed on any constant coefficient. Under the original, unplanted law," }, { "blockId": "p23-c1cbe53520", "page": 23, "quote": "Conditional on the depth, slice, and node, the events Za are independent: they concern disjoint\nblocks of independent uniform coefficients.\nThe planted experiment chooses a uniform mark a∗∈[T] and sets these blocks to zero at a∗.\nAll other row coefficients, the noise masks, the outside tables, and the game edges retain their\noriginal distributions. Equivalently, conditional on a∗= a, this is the unplanted law conditioned\non Za." }, { "blockId": "p23-d3c148ea4c", "page": 23, "quote": "T ≥\n1\np0(p∗θ)2 ,\n(8.6)" }, { "blockId": "p23-4d47b40518", "page": 23, "quote": "Conditional on the tree data, Z has the binomial distribution with parameters T, p0; these\nparameters do not depend on the tree data. Thus Cauchy–Schwarz gives" }, { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" }, { "blockId": "p25-03898445eb", "page": 25, "quote": "Shared data and the Y -labeler.\nGiven a random input edge e = (x, y) from the original\nedge law, choose a uniform public mark a∗∈[T]. At every other slot, publicly sample an\nindependent edge, including its endpoints and projection. Insert the input edge at the marked\nslot. Its projection is used only in the analysis; it is not revealed to either labeler. Conditional on\nany mark, the resulting full edge tuple has the product edge law; hence the tuple is independent\nof the mark. Each labeler knows its own endpoint tuple, and both know all edges outside the\nmarked slot.\nPublicly sample the depth, slice, node, outside tables, context, latent rows, and masks\naccording to the planted experiment above. The row sampling uses the fixed alphabet dimensions,\nthe public tree coordinates, and the mark, and is independent of the input edge. Together with\nthe edge tuple, these data have exactly the planted law used in Equation (8.7). The Y -labeler\nforms its endpoint tuple y, computes the label q∗from Equation (8.2), and outputs its marked\ncomponent q∗\na∗. All quantities used in that computation are available to this labeler: in particular\nthe conditional average defining ¯Gy can be computed from the game and cut as fixed background\ndata. It does not use the actual marked edge or its projection." }, { "blockId": "p25-dd8d66dbbc", "page": 25, "quote": "with factors in their original slot order. Let B = (Fn+1\n2\n)T . The full projection factors as" }, { "blockId": "p26-852017d85d", "page": 26, "quote": "The decoding loss depends on s, p∗, θ, and hence on the target cut gap, but not on the\nalphabet dimension or the repetition length. After choosing a game soundness smaller than the\nright side of Equation (8.9), one may therefore fix the alphabet dimension and then choose T by\nEquation (8.6)." }, { "blockId": "p26-e1770fd031", "page": 26, "quote": "(t, ε, µ) −→(s, m, p∗, θ) −→b −→n −→T −→a." }, { "blockId": "p26-842530a02d", "page": 26, "quote": "By Theorem 2.2, this fixes an alphabet dimension n for which the game has arbitrarily small\ncompleteness error. Choose an integer T satisfying" }, { "blockId": "p26-7a6c9e8358", "page": 26, "quote": "T ≥2n(2k−2s)" }, { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, { "blockId": "p33-6794ebf1cb", "page": 33, "quote": "Tγ[L/Q] =" }, { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." } ], "occurrenceCount": 28 }, { "symbol": "Tr", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-e8d52610e2", "page": 5, "quote": "For 0 ≤r ≤1, let Tr be the row-noise operator: independently in each coordinate, retain the\nrow with probability r and otherwise replace it by an independent uniform row. Then" }, "occurrences": [ { "blockId": "p5-e8d52610e2", "page": 5, "quote": "For 0 ≤r ≤1, let Tr be the row-noise operator: independently in each coordinate, retain the\nrow with probability r and otherwise replace it by an independent uniform row. Then" }, { "blockId": "p8-4481b3a5f3", "page": 8, "quote": "for a fixed function ϕ on words. This class is closed under averaging over fixed choices of ϕ,\nincluding choices that first pull a word back through a fixed affine map. For r ∈[0, 1], let\nTr denote independent row resampling with retention probability r, and write ¯f = Trf. The\noperator preserves bounds on the range of f. If ϕ is odd, Lemma 3.2 shows that both f and ¯f\nare odd under simultaneous negation of the slice tables. In particular, they have mean zero in\nthose table entries when all rows and outside tables are fixed.\nAt these fixed data, the remaining random input is the cube of |D|2s independent entries,\nindexed by (v, z) ∈D × Fs\n2. Fourier degree and influence on this cube refer to subsets of\nentries. This differs from the Walsh expansion of a fixed gate in its s-bit argument z used in the\ncorrelation proof." }, { "blockId": "p10-c4b16ddf34", "page": 10, "quote": "Proof. Condition on y, D, R, Fout. The sampled edge tuples and the two row resamplings are\nindependent between copies. After averaging over them, the expected first sign as a function\nof FD is ¯Ay + ¯Gy. The minus sign in the second queried word makes the expected second sign\n¯Ay −¯Gy at F ′\nD. The two slice inputs are ordinary t-correlated uniform cube inputs. The rows\nin these functions are the common latent array R; the two independent row resamplings have\nalready been absorbed into Tr. Negating every slice entry leaves ¯Ay unchanged and negates ¯Gy.\nTheir Fourier supports therefore have opposite total parities, so the cross terms vanish. The\nexpected sign product is Stabt( ¯Ay) −Stabt( ¯Gy). An edge is cut with probability one half of one\nminus its sign product, giving (4.5). Nonnegativity of stability for t > 0 gives (4.6)." } ], "occurrenceCount": 3, "latex": "T_{r}", "latexBasis": "pdf-script-position" }, { "symbol": "U", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p23-e015ef835b", "page": 23, "quote": "Lemma 8.2 (Hiding the planted coordinate). Let U be the unplanted law and P the planted law\nafter forgetting the mark. Both laws are taken on the full data" }, "occurrences": [ { "blockId": "p14-8e87797cb0", "page": 14, "quote": "J(q) = (q(J1), . . . , q(Js)),\nU(z) =\nX" }, { "blockId": "p14-b98783d5c4", "page": 14, "quote": "z∈Fs\n2\n∥U(z)∥2 = M.\n(6.1)" }, { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" }, { "blockId": "p14-445c75ee85", "page": 14, "quote": "Proof. All expectations over H and Fs\n2 below use uniform probability measure; the displayed\nsums defining wq and M are unnormalized. Conditional on the Jg, orthogonality of the characters\nof L gives\nEL∥U(z)∥2 =\nX" }, { "blockId": "p15-afd7da3e46", "page": 15, "quote": "E max\nz ∥U(z)∥2 ≤2M" }, { "blockId": "p15-e5cedc4c6c", "page": 15, "quote": "Fourier inversion in z gives\nU(z) = Ep(−1)p·zAp(Sp).\n(6.4)" }, { "blockId": "p15-192503b724", "page": 15, "quote": "α′\nq(p) = Euχq(u)A′\np(u),\nw′\nq = Ep∥α′\nq(p)∥2,\nU′(z) = Ep(−1)p·zA′\np(Sp)." }, { "blockId": "p16-29e6db68c4", "page": 16, "quote": "Equation (6.4) and the triangle inequality imply maxz∥U(z) −U′(z)∥≤X. The contribution to\nEX from the indices with mp > CM is at most" }, { "blockId": "p16-9f579ae464", "page": 16, "quote": "E max\nz ∥U(z) −U′(z)∥2 ≤M" }, { "blockId": "p16-611205ecdb", "page": 16, "quote": "z\n∥U′(z)∥4 = h−3\nX" }, { "blockId": "p17-fd58efeff6", "page": 17, "quote": "z\n∥U′(z)∥4 ≤\nX" }, { "blockId": "p17-e09a1b19d1", "page": 17, "quote": "E max\nz ∥U′(z)∥2 ≤" }, { "blockId": "p17-c9db8d03d2", "page": 17, "quote": "z\n∥U′(z)∥4\n!1/2\n≤M" }, { "blockId": "p23-e015ef835b", "page": 23, "quote": "Lemma 8.2 (Hiding the planted coordinate). Let U be the unplanted law and P the planted law\nafter forgetting the mark. Both laws are taken on the full data" }, { "blockId": "p23-04c4b640a3", "page": 23, "quote": "TV(P, U) ≤1" }, { "blockId": "p23-21e0eefd1c", "page": 23, "quote": "TV(P, U) = 1" }, { "blockId": "p28-e7894e9b58", "page": 28, "quote": "where xe has its three variable coordinates equal to one. A legitimate tuple U consists of k0\ndisjoint equations with the additional separation condition imposed in [6, Section 4.2]: variables\nfrom different equations of U do not occur together in any input equation. Put" }, { "blockId": "p28-0e18b8e432", "page": 28, "quote": "VU = Fvar(U)\n2\n,\nHU = span{xe : e ∈U}." }, { "blockId": "p28-59fb5928e1", "page": 28, "quote": "The vectors xe, e ∈U, are independent, so dim VU = 3k0, dim HU = k0, and there is a unique\nlinear function hU : HU →F2 with hU(xe) = be.\nBefore folding, a left question is (U, L), where" }, { "blockId": "p28-182d329c3c", "page": 28, "quote": "For a variable subset V ⊆Z, write VV = FV\n2 ≤V0; the rule for sampling V is given below. A\nright question is (V, L′), with L′ ≤VV and dim L′ = ℓ−1. On an incident pair, L′ ≤L. The\nrespective answers are linear functions on L and L′, and the constraint is restriction. Choosing\nbases therefore identifies the unfolded alphabets with Fℓ\n2 and Fℓ−1\n2\n.\nThe folding of [6, Section 4.2] puts (U, L) and (U1, L1) in the same class when" } ], "occurrenceCount": 39 }, { "symbol": "V", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." }, "occurrences": [ { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." }, { "blockId": "p28-e64092dca6", "page": 28, "quote": "Spaces, labels, and folding.\nLet Z be the variables of the 3Lin instance, let V0 = FZ\n2 , and\nview each equation e as\n⟨xe, α⟩= be,\nxe ∈V0,\nbe ∈F2," }, { "blockId": "p28-182d329c3c", "page": 28, "quote": "For a variable subset V ⊆Z, write VV = FV\n2 ≤V0; the rule for sampling V is given below. A\nright question is (V, L′), with L′ ≤VV and dim L′ = ℓ−1. On an incident pair, L′ ≤L. The\nrespective answers are linear functions on L and L′, and the constraint is restriction. Choosing\nbases therefore identifies the unfolded alphabets with Fℓ\n2 and Fℓ−1\n2\n.\nThe folding of [6, Section 4.2] puts (U, L) and (U1, L1) in the same class when" }, { "blockId": "p29-4c304fab40", "page": 29, "quote": "Validity of the edge sampling.\nWe specify the treatment of ill-formed samples omitted from\nthe displayed sampler. Sampling k0 equations independently produces a nonlegitimate tuple\nwith probability O(k2\n0/|Z|), by bounded occurrence. For fixed k0 this can be made arbitrarily\nsmall. If necessary, take sufficiently many disjoint copies of the input instance; this preserves its\nvalue and regularity.\nGiven a legitimate U, independently for each equation, the right variable set V contains\nall three variables with probability 1 −β and otherwise one uniformly chosen variable. If f0\nequations contribute all three, then" }, { "blockId": "p30-5d1d5ad667", "page": 30, "quote": "Completing the tables.\nFor a right variable set V occurring with a legitimate U, define" }, { "blockId": "p30-6582b8c640", "page": 30, "quote": "H(V ) = span{xe : e is an input equation and var(e) ⊆V }." }, { "blockId": "p30-5c2cf5223d", "page": 30, "quote": "HU ∩VV = H(V ).\n(A.4)" }, { "blockId": "p30-00283762e6", "page": 30, "quote": "In particular, for L ⊆VV , validity of (U, L) is independent of the containing legitimate tuple U.\nFix a folded labeling. Let FU[L] be its unfolded answer at each valid (U, L), and the zero\nlinear function whenever L ∩HU ̸= {0}. For each V , define FV [L] using any containing U when\nL ∩H(V ) = {0}, and by zero otherwise. For valid L, all vertices (U, L) with that same L lie in\none folding class; their unfolded answers agree by [6, Claim 4.4]. Equation (A.4) gives the same\nindependence for invalid L. Hence the completed tables satisfy" }, { "blockId": "p32-c85fcfca35", "page": 32, "quote": "The advice distribution.\nThe outer game in [6, Section 5.2] first samples (U, V ), then takes\nQ uniformly from the q-spaces of VV . It sends (U, Q) and (V, Q) to its two players and asks\nfor assignments on var(U) and V , respectively. It accepts when those assignments agree on V\nand the first satisfies every equation of U. If f0 equations contributed all three variables, then\ndim VV = k0 + 2f0 and dim H(V ) = f0. Consequently, under this actual distribution, uniformly\nin U, V ," }, { "blockId": "p32-413730e086", "page": 32, "quote": "Pr[Q ∩HU ̸= {0} | U, V ] ≤(2f0 −1)(2q −1)" }, { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p33-f5108712f2", "page": 33, "quote": "Pr[Q ⊆L | V ] =\nNq(ℓ)\nNq(dim VV )." }, { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." }, { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p35-4807687585", "page": 35, "quote": "Completeness with the alphabet fixed.\nLet α : Z →F2 satisfy all but an ε fraction\nof the input equations, and also write α : V0 →F2 for its linear extension. Give the folded\nleft question C the answer α|RC and give (V, L′) the answer α|L′. Whenever every equation of\nsampled U is satisfied, hU = α|HU , so (A.2) is exactly α|RC+HU . The edge is therefore satisfied.\nThis argument concerns the sampled edge; it does not require every member of its folding class\nto involve satisfied equations.\nBefore either conditioning step, a union bound gives probability at most k0ε of sampling an\nunsatisfied equation. Since the total retained probability qall is at least 1/2, the conditioned\ngame’s value is at least 1 −k0ε/qall ≥1 −2k0ε. After ℓ, k0 have been fixed for soundness, choose\nε ≤a/(2k0) in the initial 3Lin gap. This gives the asserted completeness without altering n." } ], "occurrenceCount": 18 }, { "symbol": "V0", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p28-e64092dca6", "page": 28, "quote": "Spaces, labels, and folding.\nLet Z be the variables of the 3Lin instance, let V0 = FZ\n2 , and\nview each equation e as\n⟨xe, α⟩= be,\nxe ∈V0,\nbe ∈F2," }, "occurrences": [ { "blockId": "p28-e64092dca6", "page": 28, "quote": "Spaces, labels, and folding.\nLet Z be the variables of the 3Lin instance, let V0 = FZ\n2 , and\nview each equation e as\n⟨xe, α⟩= be,\nxe ∈V0,\nbe ∈F2," }, { "blockId": "p28-182d329c3c", "page": 28, "quote": "For a variable subset V ⊆Z, write VV = FV\n2 ≤V0; the rule for sampling V is given below. A\nright question is (V, L′), with L′ ≤VV and dim L′ = ℓ−1. On an incident pair, L′ ≤L. The\nrespective answers are linear functions on L and L′, and the constraint is restriction. Choosing\nbases therefore identifies the unfolded alphabets with Fℓ\n2 and Fℓ−1\n2\n.\nThe folding of [6, Section 4.2] puts (U, L) and (U1, L1) in the same class when" }, { "blockId": "p35-4807687585", "page": 35, "quote": "Completeness with the alphabet fixed.\nLet α : Z →F2 satisfy all but an ε fraction\nof the input equations, and also write α : V0 →F2 for its linear extension. Give the folded\nleft question C the answer α|RC and give (V, L′) the answer α|L′. Whenever every equation of\nsampled U is satisfied, hU = α|HU , so (A.2) is exactly α|RC+HU . The edge is therefore satisfied.\nThis argument concerns the sampled edge; it does not require every member of its folding class\nto involve satisfied equations.\nBefore either conditioning step, a union bound gives probability at most k0ε of sampling an\nunsatisfied equation. Since the total retained probability qall is at least 1/2, the conditioned\ngame’s value is at least 1 −k0ε/qall ≥1 −2k0ε. After ℓ, k0 have been fixed for soundness, choose\nε ≤a/(2k0) in the initial 3Lin gap. This gives the asserted completeness without altering n." } ], "occurrenceCount": 3, "latex": "V_{0}", "latexBasis": "pdf-script-position" }, { "symbol": "VV", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p28-182d329c3c", "page": 28, "quote": "For a variable subset V ⊆Z, write VV = FV\n2 ≤V0; the rule for sampling V is given below. A\nright question is (V, L′), with L′ ≤VV and dim L′ = ℓ−1. On an incident pair, L′ ≤L. The\nrespective answers are linear functions on L and L′, and the constraint is restriction. Choosing\nbases therefore identifies the unfolded alphabets with Fℓ\n2 and Fℓ−1\n2\n.\nThe folding of [6, Section 4.2] puts (U, L) and (U1, L1) in the same class when" }, "occurrences": [ { "blockId": "p28-182d329c3c", "page": 28, "quote": "For a variable subset V ⊆Z, write VV = FV\n2 ≤V0; the rule for sampling V is given below. A\nright question is (V, L′), with L′ ≤VV and dim L′ = ℓ−1. On an incident pair, L′ ≤L. The\nrespective answers are linear functions on L and L′, and the constraint is restriction. Choosing\nbases therefore identifies the unfolded alphabets with Fℓ\n2 and Fℓ−1\n2\n.\nThe folding of [6, Section 4.2] puts (U, L) and (U1, L1) in the same class when" }, { "blockId": "p29-2d8014adae", "page": 29, "quote": "dim VV = k0 + 2f0,\ndim(VV ∩HU) = f0." }, { "blockId": "p29-3506d73b43", "page": 29, "quote": "For a uniformly chosen (ℓ−1)-space L′ ≤VV , counting nonzero vectors gives" }, { "blockId": "p30-793f5a5b9f", "page": 30, "quote": "The soundness proof uses tables on every ℓ-dimensional subspace, although a folded labeling\nspecifies answers only at valid vertices. We first extend those answers, preserving the consistency\nbetween the tables on VU and VV . We then show that the linear functions decoded from these\ntables still satisfy the equations in U." }, { "blockId": "p30-5c2cf5223d", "page": 30, "quote": "HU ∩VV = H(V ).\n(A.4)" }, { "blockId": "p30-00283762e6", "page": 30, "quote": "In particular, for L ⊆VV , validity of (U, L) is independent of the containing legitimate tuple U.\nFix a folded labeling. Let FU[L] be its unfolded answer at each valid (U, L), and the zero\nlinear function whenever L ∩HU ̸= {0}. For each V , define FV [L] using any containing U when\nL ∩H(V ) = {0}, and by zero otherwise. For valid L, all vertices (U, L) with that same L lie in\none folding class; their unfolded answers agree by [6, Claim 4.4]. Equation (A.4) gives the same\nindependence for invalid L. Hence the completed tables satisfy" }, { "blockId": "p30-9a2aac86b9", "page": 30, "quote": "FU[L] = FV [L]\n(L ⊆VV , dim L = ℓ).\n(A.5)" }, { "blockId": "p32-c85fcfca35", "page": 32, "quote": "The advice distribution.\nThe outer game in [6, Section 5.2] first samples (U, V ), then takes\nQ uniformly from the q-spaces of VV . It sends (U, Q) and (V, Q) to its two players and asks\nfor assignments on var(U) and V , respectively. It accepts when those assignments agree on V\nand the first satisfies every equation of U. If f0 equations contributed all three variables, then\ndim VV = k0 + 2f0 and dim H(V ) = f0. Consequently, under this actual distribution, uniformly\nin U, V ," }, { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p33-f5108712f2", "page": 33, "quote": "Pr[Q ⊆L | V ] =\nNq(ℓ)\nNq(dim VV )." }, { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." }, { "blockId": "p34-7df6b3632a", "page": 34, "quote": "codimVV (W ∩VV ) = i,\nPr\nQ⊆L⊆W∩VV [FV [L] = f|L] > τi/2." }, { "blockId": "p34-be24debe49", "page": 34, "quote": "Here L has dimension ℓ. Set S0 = W ∩VV and γ = f|Q. Every equality FV [L] = f|L in this\ndisplay is retained in Tγ. Since ζi ≤τ0/2 ≤τi/2, the quotient pair" }, { "blockId": "p34-56088078a9", "page": 34, "quote": "occurs for Tγ and hence has a maximal occurring extension in that auxiliary table. Lifting any\nsuch extension by the same pγ gives a function agreeing with f on all of S0. This conclusion\nconcerns function extension and is unaffected by the extra agreements that zero completion may\nintroduce.\nThe second player selects this tag and one such maximal extension with probability at\nleast 1/M, where M := 2qM0 is independent of k0. Its eventual answer then agrees with f\non W ∩VV . The first player’s uniform extension agrees with this answer on all of VV with\nprobability 2−i ≥2−r: the quotient VV /(W ∩VV ) has dimension i. Thus, conditionally on\nE1 ∩E2 ∩E3 ∩E4, the strategy wins with probability at least" } ], "occurrenceCount": 16, "latex": "V_{V}", "latexBasis": "pdf-script-position" }, { "symbol": "Xp", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p16-07b05b513e", "page": 16, "quote": "Let\nXp = ∥(Ap −A′\np)(Sp)∥,\nX = EpXp." }, "occurrences": [ { "blockId": "p16-07b05b513e", "page": 16, "quote": "Let\nXp = ∥(Ap −A′\np)(Sp)∥,\nX = EpXp." }, { "blockId": "p16-1d632d1379", "page": 16, "quote": "M/D. Moreover,\nthe Xp are pairwise independent, by the pairwise independence of the Sp. Since Ap −A′\np is\neither Ap or zero at each point," }, { "blockId": "p16-256d7952e5", "page": 16, "quote": "p\nVar(Xp) ≤h−2 X" } ], "occurrenceCount": 3, "latex": "X_{p}", "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": "p23-38adbae09d", "page": 23, "quote": "Proof. Let Z = PT\na=1 1Za. Averaging the conditional densities over the uniform mark gives" }, "occurrences": [ { "blockId": "p23-38adbae09d", "page": 23, "quote": "Proof. Let Z = PT\na=1 1Za. Averaging the conditional densities over the uniform mark gives" }, { "blockId": "p23-a7c4fa153b", "page": 23, "quote": "dP\ndU =\nZ\nTp0\n." }, { "blockId": "p23-4d47b40518", "page": 23, "quote": "Conditional on the tree data, Z has the binomial distribution with parameters T, p0; these\nparameters do not depend on the tree data. Thus Cauchy–Schwarz gives" }, { "blockId": "p23-195866d846", "page": 23, "quote": "2EU| Z" }, { "blockId": "p28-e64092dca6", "page": 28, "quote": "Spaces, labels, and folding.\nLet Z be the variables of the 3Lin instance, let V0 = FZ\n2 , and\nview each equation e as\n⟨xe, α⟩= be,\nxe ∈V0,\nbe ∈F2," }, { "blockId": "p28-182d329c3c", "page": 28, "quote": "For a variable subset V ⊆Z, write VV = FV\n2 ≤V0; the rule for sampling V is given below. A\nright question is (V, L′), with L′ ≤VV and dim L′ = ℓ−1. On an incident pair, L′ ≤L. The\nrespective answers are linear functions on L and L′, and the constraint is restriction. Choosing\nbases therefore identifies the unfolded alphabets with Fℓ\n2 and Fℓ−1\n2\n.\nThe folding of [6, Section 4.2] puts (U, L) and (U1, L1) in the same class when" }, { "blockId": "p29-4c304fab40", "page": 29, "quote": "Validity of the edge sampling.\nWe specify the treatment of ill-formed samples omitted from\nthe displayed sampler. Sampling k0 equations independently produces a nonlegitimate tuple\nwith probability O(k2\n0/|Z|), by bounded occurrence. For fixed k0 this can be made arbitrarily\nsmall. If necessary, take sufficiently many disjoint copies of the input instance; this preserves its\nvalue and regularity.\nGiven a legitimate U, independently for each equation, the right variable set V contains\nall three variables with probability 1 −β and otherwise one uniformly chosen variable. If f0\nequations contribute all three, then" }, { "blockId": "p29-2422708018", "page": 29, "quote": "We condition the sampling on legitimacy and L ∩HU = {0}. The discarded probability is at\nmost\nκ = O(k2\n0/|Z|) + 2ℓ−k0 + 2ℓ−2k0.\n(A.3)" }, { "blockId": "p35-4807687585", "page": 35, "quote": "Completeness with the alphabet fixed.\nLet α : Z →F2 satisfy all but an ε fraction\nof the input equations, and also write α : V0 →F2 for its linear extension. Give the folded\nleft question C the answer α|RC and give (V, L′) the answer α|L′. Whenever every equation of\nsampled U is satisfied, hU = α|HU , so (A.2) is exactly α|RC+HU . The edge is therefore satisfied.\nThis argument concerns the sampled edge; it does not require every member of its folding class\nto involve satisfied equations.\nBefore either conditioning step, a union bound gives probability at most k0ε of sampling an\nunsatisfied equation. Since the total retained probability qall is at least 1/2, the conditioned\ngame’s value is at least 1 −k0ε/qall ≥1 −2k0ε. After ℓ, k0 have been fixed for soundness, choose\nε ≤a/(2k0) in the initial 3Lin gap. This gives the asserted completeness without altering n." } ], "occurrenceCount": 9 }, { "symbol": "Za", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p23-d402bf53a5", "page": 23, "quote": "Write each affine row in coordinates on A = (Fn\n2)T . Apart from its single constant coefficient, it\nhas one block of n linear coefficients for each of the T slots. For a slot a ∈[T], let Za be the\nevent that the slot-a blocks vanish in all k −2s context rows and all k latent rows Rnew\ni\n. No\ncondition is imposed on any constant coefficient. Under the original, unplanted law," }, "occurrences": [ { "blockId": "p23-d402bf53a5", "page": 23, "quote": "Write each affine row in coordinates on A = (Fn\n2)T . Apart from its single constant coefficient, it\nhas one block of n linear coefficients for each of the T slots. For a slot a ∈[T], let Za be the\nevent that the slot-a blocks vanish in all k −2s context rows and all k latent rows Rnew\ni\n. No\ncondition is imposed on any constant coefficient. Under the original, unplanted law," }, { "blockId": "p23-8d3bd8ebde", "page": 23, "quote": "p0 := Pr(Za) = 2−n(2k−2s).\n(8.4)" }, { "blockId": "p23-c1cbe53520", "page": 23, "quote": "Conditional on the depth, slice, and node, the events Za are independent: they concern disjoint\nblocks of independent uniform coefficients.\nThe planted experiment chooses a uniform mark a∗∈[T] and sets these blocks to zero at a∗.\nAll other row coefficients, the noise masks, the outside tables, and the game edges retain their\noriginal distributions. Equivalently, conditional on a∗= a, this is the unplanted law conditioned\non Za." }, { "blockId": "p23-38adbae09d", "page": 23, "quote": "Proof. Let Z = PT\na=1 1Za. Averaging the conditional densities over the uniform mark gives" } ], "occurrenceCount": 4, "latex": "Z_{a}", "latexBasis": "pdf-script-position" }, { "symbol": "au", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p36-82d91c1daf", "page": 36, "quote": "Now fix an arbitrary cut of G′, and let au ∈[0, 1] be the fraction of cluster u on its positive\nside. Its crossing edges between clusters u, v form two rectangles. Equation (B.1) therefore\ncompares their scaled count with" }, "occurrences": [ { "blockId": "p36-82d91c1daf", "page": 36, "quote": "Now fix an arbitrary cut of G′, and let au ∈[0, 1] be the fraction of cluster u on its positive\nside. Its crossing edges between clusters u, v form two rectangles. Equation (B.1) therefore\ncompares their scaled count with" }, { "blockId": "p36-cbf457085c", "page": 36, "quote": "wuv\nau(1 −av) + (1 −au)av\n" }, { "blockId": "p36-301ecd5146", "page": 36, "quote": "to error at most 2/p + 2/p2. Summing over the at most N(N −1)/2 pairs costs at most\nN(N −1)(1/p + 1/p2). The displayed weighted expression, summed over pairs, is the expected\ncut weight obtained by independently assigning vertex u a positive sign with probability au. It\nis at most MaxCut(G). This proves the desired upper bound for every cut of G′.\nConversely, give all vertices in each cluster the sign of a maximum cut of G. The same\nrectangle bound compares its scaled weight with MaxCut(G) to at most the same error, giving\nthe reverse inequality. Our choice of p ensures N(N −1)(1/p + 1/p2) < ϵ′, which proves the\nassertion. Loops contribute zero to every cut and can be deleted at the outset." } ], "occurrenceCount": 3, "latex": "a_{u}", "latexBasis": "pdf-script-position" }, { "symbol": "emeq", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p25-b66d45fd9b", "page": 25, "quote": "Fix its context to eΓ′ and its outside tables to the public Fout, leaving all 2s shifts free in eH. Let\nemeq be its masses. They are nonnegative and their sum over labels is at most one, by Parseval and\n| eG| ≤1. The labeler samples a label eq with probability at least emeq for each label, assigning any\nleftover probability to a fixed label. It uses an additional independent public uniform random\nnumber to sample from this distribution and outputs the marked component of eq. This sampling\ndoes not use q∗." }, "occurrences": [ { "blockId": "p24-242a8fe97e", "page": 24, "quote": "emeq( ef).\n(8.8)" }, { "blockId": "p25-b66d45fd9b", "page": 25, "quote": "Fix its context to eΓ′ and its outside tables to the public Fout, leaving all 2s shifts free in eH. Let\nemeq be its masses. They are nonnegative and their sum over labels is at most one, by Parseval and\n| eG| ≤1. The labeler samples a label eq with probability at least emeq for each label, assigning any\nleftover probability to a fixed label. It uses an additional independent public uniform random\nnumber to sample from this distribution and outputs the marked component of eq. This sampling\ndoes not use q∗." }, { "blockId": "p26-5741bc9606", "page": 26, "quote": "ρ(eq)=q∗\nemeq ≥2−2smq∗(Ge; Γ′, Fout)." } ], "occurrenceCount": 3 }, { "symbol": "f", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p4-767850a4b2", "page": 4, "quote": "Indeed, if h(a) = h0 + P\ni hiai, a functional taking value one on 1 has the unique form\nq(h) = h0 + P\ni hiai. We call precisely these functionals labels; a general nonzero functional in\nH∗need not be a label.\nFor f : Hk →R and C = (q1, . . . , qk) ∈(H∗)k, put" }, "occurrences": [ { "blockId": "p4-767850a4b2", "page": 4, "quote": "Indeed, if h(a) = h0 + P\ni hiai, a functional taking value one on 1 has the unique form\nq(h) = h0 + P\ni hiai. We call precisely these functionals labels; a general nonzero functional in\nH∗need not be a label.\nFor f : Hk →R and C = (q1, . . . , qk) ∈(H∗)k, put" }, { "blockId": "p8-74ab4e8894", "page": 8, "quote": "The correlation identity describes a cut that evaluates the word at one label. To analyze\narbitrary cuts, we also need functions that can depend on the entire word but do not see its\ngenerating rows or tables. We call f a function of the code if" }, { "blockId": "p8-7efa825524", "page": 8, "quote": "f(R, F) = ϕ(W(R, F))\n(3.4)" }, { "blockId": "p8-4481b3a5f3", "page": 8, "quote": "for a fixed function ϕ on words. This class is closed under averaging over fixed choices of ϕ,\nincluding choices that first pull a word back through a fixed affine map. For r ∈[0, 1], let\nTr denote independent row resampling with retention probability r, and write ¯f = Trf. The\noperator preserves bounds on the range of f. If ϕ is odd, Lemma 3.2 shows that both f and ¯f\nare odd under simultaneous negation of the slice tables. In particular, they have mean zero in\nthose table entries when all rows and outside tables are fixed.\nAt these fixed data, the remaining random input is the cube of |D|2s independent entries,\nindexed by (v, z) ∈D × Fs\n2. Fourier degree and influence on this cube refer to subsets of\nentries. This differs from the Walsh expansion of a fixed gate in its s-bit argument z used in the\ncorrelation proof." }, { "blockId": "p11-6189896bf3", "page": 11, "quote": "The cut depends on the generated word, so every transformation that preserves that word also\nconstrains the cut’s row Fourier coefficients. At each gate these constraints give a common\nFourier index shared by all child pairs. Negating the gate table distinguishes whether this index\nis an evaluation label. We then show that, at a random slice, little smoothed Fourier mass is\nlost by keeping the entire common index in just one child of each pair. Finally we express these\nretained indices in row coordinates that remain free after the other rows are fixed. This is the\nlocal form used in Section 7. Throughout the section, f is a function of the code in the sense of\n(3.4)." }, { "blockId": "p11-848fc46d9b", "page": 11, "quote": "f(R, F) =\nX" }, { "blockId": "p11-9c0d842b01", "page": 11, "quote": "Lemma 5.1 (Row symmetries). Let f be a function of the code. Every nonzero row Fourier\ncoefficient of f satisfies, at each internal node v," }, { "blockId": "p12-77d0658e91", "page": 12, "quote": "Lemma 5.2 (Average cost of the unsplit projection). Let f be a function of the code with |f| ≤1,\nlet r = 1 −µ/d with 0 < µ < 1/2, and let ¯f = Trf. Choose the random slice independently of f.\nThen\nEJ0,D\nX" }, { "blockId": "p12-091c8a397c", "page": 12, "quote": "v∈D\n∥(I −Sv) ¯f∥2\n2,R,F ≤\ns\n2emµ.\n(5.5)" }, { "blockId": "p12-986ee343c1", "page": 12, "quote": "The same bound holds after averaging over additional background data determining f, provided\nthe slice is chosen independently of those data." }, { "blockId": "p13-7c81722455", "page": 13, "quote": "For x ≥0, xe−2µx ≤(2eµ)−1, and log r ≤−µ/d. Hence every parenthesized factor is at most\n(2eµ)−1, while P\nC EF | bfC(F)|2 = ∥f∥2\n2 ≤1. This proves the bound. The proof is pointwise in\nthe background data, so its average satisfies the same inequality." }, { "blockId": "p17-c44d5d344e", "page": 17, "quote": "fω(R, F) = ϕω(W(R, F)),\n¯fω = Trfω,\nr = 1 −µ/d." }, { "blockId": "p17-01d2e48829", "page": 17, "quote": "The word, tree, and random slice D are those of Section 3. Write Fout for the tables outside D.\nOnce ω, D, R, Fout are fixed, ¯fω is a function of the independent slice-table bits." }, { "blockId": "p18-af115d7cb4", "page": 18, "quote": "mq(f; Γ, Fout) = EFD\nX" }, { "blockId": "p18-ea72a11348", "page": 18, "quote": "The dependence on v, D, and ω is suppressed. The sets of indices in this definition are disjoint\nas q varies. For a fixed q, the mass includes every way to distribute that common sum between\nthe two children of each pair. We seek such mass for the original smoothed function ¯fω. The\nunsplit projection and a table-degree cutoff will serve only to find a label carrying this mass;\nneither projection is part of the conclusion. Keeping all the indices with a given common sum\nwill let us compare masses under affine pullback in Section 8.\nIn particular, if |f| ≤1, then at every fixed context" }, { "blockId": "p18-1a521f670f", "page": 18, "quote": "q∈A\nmq(f; Γ, Fout) ≤1.\n(7.2)" }, { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p18-65f0ea897b", "page": 18, "quote": "some entry z ∈Fs\n2 of some v ∈D has Inf≤K\nv,z ( ¯fω) ≥τ. Indeed, if the probability of this event is\np, the expected stability is at most B(t) + δ/2 + p." } ], "occurrenceCount": 44 }, { "symbol": "h", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" }, "occurrences": [ { "blockId": "p4-c4f50f238b", "page": 4, "quote": "This is a vector space, containing the constant form 1. For q ∈H∗define χq(h) = (−1)q(h).\nEvaluation identifies A with\n{q ∈H∗: q(1) = 1}.\n(2.1)" }, { "blockId": "p4-767850a4b2", "page": 4, "quote": "Indeed, if h(a) = h0 + P\ni hiai, a functional taking value one on 1 has the unique form\nq(h) = h0 + P\ni hiai. We call precisely these functionals labels; a general nonzero functional in\nH∗need not be a label.\nFor f : Hk →R and C = (q1, . . . , qk) ∈(H∗)k, put" }, { "blockId": "p5-9fcae3c41f", "page": 5, "quote": "For a finite set Ωand h : {−1, 1}Ω→R, write" }, { "blockId": "p5-bc0ad88201", "page": 5, "quote": "h(x) =\nX" }, { "blockId": "p5-981ed37655", "page": 5, "quote": "Infa(h) =\nX" }, { "blockId": "p5-3fd71f479c", "page": 5, "quote": "bh(S)2,\nInf≤K\na\n(h) =\nX" }, { "blockId": "p5-54e1c6b4a5", "page": 5, "quote": "a∈Ω\nInfa(P≤Kh) ≤K∥h∥2\n2.\n(2.3)" }, { "blockId": "p5-cda18a336e", "page": 5, "quote": "Stabt(h) = E[h(x)h(y)] =\nX" }, { "blockId": "p5-85b0c0a68a", "page": 5, "quote": "Thus stability is nonnegative for t ≥0. If h ∈[−1, 1], it is at most one." }, { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." }, { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p5-ac48b29911", "page": 5, "quote": "0 ≤Stabt(h) −Stabt(hu) ≤sup\nj≥0\ntj(1 −u2j) −→0\n(u ↑1)." }, { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, { "blockId": "p11-0801c4df00", "page": 11, "quote": "For each node v, define its generator P(v), a set of leaves, recursively: P(v) = {v} at a leaf, and\nP(v) = P(v1,0) ∪P(v1,1) at an internal node. Thus a generator always follows both children\nof the first pair; unlike the slice, it involves no random choices. Adding the same h ∈H to all\nrows in P(v) multiplies the subtree output at a by (−1)h(a). This follows by induction from\nEquation (3.1): multiplying both children of one pair by the same sign leaves all gate inputs zg\nunchanged and multiplies the prefactor by that sign.\nFor a row Fourier index C = (qi)k\ni=1 ∈(H∗)k, put" }, { "blockId": "p12-197aaf8a57", "page": 12, "quote": "Proof. For each g, shifting the generators of both children in pair g by h has exactly the same\neffect on the subtree at v as shifting P(v) by h. Consequently the two row translations have the\nsame effect on the whole word. Taking row Fourier coefficients yields" }, { "blockId": "p12-b13f87e64f", "page": 12, "quote": "χqv,g0+qv,g1(h) bfC(F) = χqv(h) bfC(F)\n(h ∈H)." }, { "blockId": "p15-3a6aba3253", "page": 15, "quote": "Summing over z proves (6.1). If M = 0, every vector c(q, z) is zero and (6.2) is immediate.\nHenceforth assume M > 0.\nSet h = 2s and ρ = maxq wq/M. We will prove the quantitative bound below for every\nC, D > 0, and then choose its parameters. The main estimate bounds the maximum squared\nnorm by the square root of the sum of fourth powers. After a Fourier change of variables,\nmost terms in that fourth moment force the same character q in all four factors; their total is\ncontrolled by the dispersion of the character masses. Two truncations will control the remaining\nterms and the cost of this change to the array:" }, { "blockId": "p15-0e40b33fca", "page": 15, "quote": "h\n+\n 1\n√" }, { "blockId": "p15-3d26a9f225", "page": 15, "quote": "h" }, { "blockId": "p16-762981b84e", "page": 16, "quote": "Var(X) = h−2 X" } ], "occurrenceCount": 43 }, { "symbol": "r", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-e8d52610e2", "page": 5, "quote": "For 0 ≤r ≤1, let Tr be the row-noise operator: independently in each coordinate, retain the\nrow with probability r and otherwise replace it by an independent uniform row. Then" }, "occurrences": [ { "blockId": "p5-e8d52610e2", "page": 5, "quote": "For 0 ≤r ≤1, let Tr be the row-noise operator: independently in each coordinate, retain the\nrow with probability r and otherwise replace it by an independent uniform row. Then" }, { "blockId": "p5-537b4ee14b", "page": 5, "quote": "d\nTrf(C) = r|C| bf(C).\n(2.2)" }, { "blockId": "p5-cd21ecba28", "page": 5, "quote": "This operator preserves the range [−1, 1]. It is also convolution by an independent increment\nE ∈Hk, whose coordinates are zero with probability r and independent uniform rows with\nprobability 1 −r. Convolution commutes with every fixed translation of the rows." }, { "blockId": "p7-aa2f88a795", "page": 7, "quote": "Lemma 3.2 (Complementary traversal and correlation). For every fixed collection of tables\nF, the root output as a multilinear polynomial in the k leaf signs has only degree-d monomials.\nEach such monomial chooses one child from every pair at every node that it reaches. Every\nresulting traversal meets every possible slice D at exactly one node. Consequently, negating all\nthe tables on a slice negates W.\nLet 0 < t < 1 and 0 < r < 1. Draw F uniformly, and construct F ′ by leaving the tables\noutside D unchanged and coupling each pair of corresponding table entries on D as uniform signs\nof correlation t, independently across entries. Independently draw R ∈Hk uniformly, and form\nR1, R2 by retaining each row of R with probability r and otherwise replacing it by an independent\nuniform row, independently in the two copies. Then, for every a ∈A and every fixed slice," }, { "blockId": "p7-0bbbca177d", "page": 7, "quote": "In particular, if r = 1 −µ/d with 0 < µ < 1/2, then r2d ≥1 −2µ." }, { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, { "blockId": "p8-4481b3a5f3", "page": 8, "quote": "for a fixed function ϕ on words. This class is closed under averaging over fixed choices of ϕ,\nincluding choices that first pull a word back through a fixed affine map. For r ∈[0, 1], let\nTr denote independent row resampling with retention probability r, and write ¯f = Trf. The\noperator preserves bounds on the range of f. If ϕ is odd, Lemma 3.2 shows that both f and ¯f\nare odd under simultaneous negation of the slice tables. In particular, they have mean zero in\nthose table entries when all rows and outside tables are fixed.\nAt these fixed data, the remaining random input is the cube of |D|2s independent entries,\nindexed by (v, z) ∈D × Fs\n2. Fourier degree and influence on this cube refer to subsets of\nentries. This differs from the Walsh expansion of a fixed gate in its s-bit argument z used in the\ncorrelation proof." }, { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, { "blockId": "p9-cd724517b8", "page": 9, "quote": "3. Draw a uniform common row array R ∈Hk. In each of two copies, independently retain\neach row with probability r and otherwise resample it uniformly, obtaining R1, R2." }, { "blockId": "p10-0e07627ed1", "page": 10, "quote": "Each coordinate of either sampled edge tuple has the original edge law.\nA union bound\nshows that all 2T sampled constraints are satisfied except with probability at most 2Ta.\nWhen all these constraints are satisfied, the two queried tuples project to the same label\nl = (ℓY (y1), . . . , ℓY (yT )) ∈A. The cut signs then have product −W(R1, F)(l)W(R2, F ′)(l). The\nword-generation data are independent of the game edges, so Theorem 3.2 gives conditional cut\nprobability (1 + tr2d)/2. On the remaining event the cut probability is nonnegative. This proves\nthe first bound when 2Ta ≤1; if 2Ta > 1 that bound is negative and is automatic. Finally,\nr2d ≥1 −2µ, and the loss due to the prefactor is at most 2Ta, giving the second bound." }, { "blockId": "p11-f3c7473fca", "page": 11, "quote": "bfC(F)χC(R),\nb¯fC(F) = r|C| bfC(F).\n(5.1)" }, { "blockId": "p12-77d0658e91", "page": 12, "quote": "Lemma 5.2 (Average cost of the unsplit projection). Let f be a function of the code with |f| ≤1,\nlet r = 1 −µ/d with 0 < µ < 1/2, and let ¯f = Trf. Choose the random slice independently of f.\nThen\nEJ0,D\nX" }, { "blockId": "p13-5bdf4ab73d", "page": 13, "quote": "\nr2|C| |C|" }, { "blockId": "p13-7c81722455", "page": 13, "quote": "For x ≥0, xe−2µx ≤(2eµ)−1, and log r ≤−µ/d. Hence every parenthesized factor is at most\n(2eµ)−1, while P\nC EF | bfC(F)|2 = ∥f∥2\n2 ≤1. This proves the bound. The proof is pointwise in\nthe background data, so its average satisfies the same inequality." }, { "blockId": "p17-c44d5d344e", "page": 17, "quote": "fω(R, F) = ϕω(W(R, F)),\n¯fω = Trfω,\nr = 1 −µ/d." }, { "blockId": "p22-6667c81fbb", "page": 22, "quote": "Let Rnew\n1\n, . . . , Rnew\nk\nbe independent uniform elements of H, and independently let B1, . . . , Bk be\nBernoulli variables with Pr(Bi = 1) = 1 −r. Set" }, { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, { "blockId": "p27-74a15e2a80", "page": 27, "quote": "vertices. All factors except |X| are constants for fixed t, ε. There are polynomially many\nendpoint and edge tuples, and all word-generation sample spaces have constant size. Enumerate\nevery test outcome and add its probability to the corresponding edge weight. The initial game\nweights, t, and r are rational, so these weights have polynomial bit complexity. Conditional edge\nprobabilities also have polynomial bit complexity. All alphabets and words can be enumerated\nin constant time relative to the input size. The resulting graph is therefore computable in\ndeterministic polynomial time. It has a rational edge distribution and satisfies the asserted,\nweaker ε-loss bounds." }, { "blockId": "p29-e4deb03dd4", "page": 29, "quote": "Both sides have dimension ℓ+ k0; consequently RC ∩HU = {0} as well. A folded answer is\na linear function σ : RC →F2. Its unfolding on (U, L) is the restriction to L of the unique\nextension\neσU(r + h) = σ(r) + hU(h),\nr ∈RC,\nh ∈HU.\n(A.2)" }, { "blockId": "p31-2563015c96", "page": 31, "quote": "Fix a soundness parameter δ ∈(0, 1).\nThe agreement lemma [6, Lemma 3.7] supplies\nconstants q, r, C and a positive function α(ℓ) at agreement threshold δ2/8, as in the proof of its\nLemma 5.3. Set the thresholds" } ], "occurrenceCount": 30 }, { "symbol": "wq", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p14-bd99ad447a", "page": 14, "quote": "Set\nwq =\nX" }, "occurrences": [ { "blockId": "p14-bd99ad447a", "page": 14, "quote": "Set\nwq =\nX" }, { "blockId": "p14-edbd4f63d7", "page": 14, "quote": "q∈H∗\nwq." }, { "blockId": "p14-5b7ceaf841", "page": 14, "quote": "max\nq∈H∗wq ≤γM\n=⇒\nE max\nz∈Fs\n2\n∥U(z)∥2 ≤ζM.\n(6.2)" }, { "blockId": "p14-445c75ee85", "page": 14, "quote": "Proof. All expectations over H and Fs\n2 below use uniform probability measure; the displayed\nsums defining wq and M are unnormalized. Conditional on the Jg, orthogonality of the characters\nof L gives\nEL∥U(z)∥2 =\nX" }, { "blockId": "p15-3a6aba3253", "page": 15, "quote": "Summing over z proves (6.1). If M = 0, every vector c(q, z) is zero and (6.2) is immediate.\nHenceforth assume M > 0.\nSet h = 2s and ρ = maxq wq/M. We will prove the quantitative bound below for every\nC, D > 0, and then choose its parameters. The main estimate bounds the maximum squared\nnorm by the square root of the sum of fourth powers. After a Fourier change of variables,\nmost terms in that fourth moment force the same character q in all four factors; their total is\ncontrolled by the dispersion of the character masses. Two truncations will control the remaining\nterms and the cost of this change to the array:" }, { "blockId": "p15-21f25791dc", "page": 15, "quote": "q\n∥αq(p)∥2,\nEpmp = M,\nEp∥αq(p)∥2 = wq.\n(6.5)" }, { "blockId": "p16-d08d31fb5a", "page": 16, "quote": "q\nw′\nq ≤M,\nw′\nq ≤2wq + 2C2M" }, { "blockId": "p21-05000909ec", "page": 21, "quote": "Writing wq = P\nz∥c(q, z)∥2 and Mv = P\nq wq, we have" }, { "blockId": "p21-991df0479c", "page": 21, "quote": "wq =\nX" }, { "blockId": "p21-1e104952eb", "page": 21, "quote": "(i) wq ≤γMv for every q;" }, { "blockId": "p21-125cee3bff", "page": 21, "quote": "(iii) Mv ≥M0 and wq > γMv for some q." } ], "occurrenceCount": 11, "latex": "w_{q}", "latexBasis": "pdf-script-position" }, { "symbol": "µ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, "occurrences": [ { "blockId": "p7-0bbbca177d", "page": 7, "quote": "In particular, if r = 1 −µ/d with 0 < µ < 1/2, then r2d ≥1 −2µ." }, { "blockId": "p8-1baa50afdb", "page": 8, "quote": "Conditional independence between tables therefore gives E[aP (F ′) | F, D] = taP (F).\nFor fixed a, the signs xb\ni = (−1)Rb\ni(a) are uniform, independent across i, and E[x1\ni x2\ni ] = r2.\nThus the product expectation of two monomials in Equation (3.3) is zero unless their leaf\nsets agree, and is r2d when they agree. Since WF is Boolean, Parseval gives P\nP aP (F)2 = 1.\nAveraging first over rows and then over F ′ proves Equation (3.2). The last claim is Bernoulli’s\ninequality: (1 −µ/d)2d ≥1 −2µ." }, { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, { "blockId": "p10-0e07627ed1", "page": 10, "quote": "Each coordinate of either sampled edge tuple has the original edge law.\nA union bound\nshows that all 2T sampled constraints are satisfied except with probability at most 2Ta.\nWhen all these constraints are satisfied, the two queried tuples project to the same label\nl = (ℓY (y1), . . . , ℓY (yT )) ∈A. The cut signs then have product −W(R1, F)(l)W(R2, F ′)(l). The\nword-generation data are independent of the game edges, so Theorem 3.2 gives conditional cut\nprobability (1 + tr2d)/2. On the remaining event the cut probability is nonnegative. This proves\nthe first bound when 2Ta ≤1; if 2Ta > 1 that bound is negative and is automatic. Finally,\nr2d ≥1 −2µ, and the loss due to the prefactor is at most 2Ta, giving the second bound." }, { "blockId": "p12-77d0658e91", "page": 12, "quote": "Lemma 5.2 (Average cost of the unsplit projection). Let f be a function of the code with |f| ≤1,\nlet r = 1 −µ/d with 0 < µ < 1/2, and let ¯f = Trf. Choose the random slice independently of f.\nThen\nEJ0,D\nX" }, { "blockId": "p13-7c81722455", "page": 13, "quote": "For x ≥0, xe−2µx ≤(2eµ)−1, and log r ≤−µ/d. Hence every parenthesized factor is at most\n(2eµ)−1, while P\nC EF | bfC(F)|2 = ∥f∥2\n2 ≤1. This proves the bound. The proof is pointwise in\nthe background data, so its average satisfies the same inequality." }, { "blockId": "p17-c44d5d344e", "page": 17, "quote": "fω(R, F) = ϕω(W(R, F)),\n¯fω = Trfω,\nr = 1 −µ/d." }, { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, { "blockId": "p26-e1770fd031", "page": 26, "quote": "(t, ε, µ) −→(s, m, p∗, θ) −→b −→n −→T −→a." }, { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" } ], "occurrenceCount": 10 }, { "symbol": "Γ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p13-e6707ed124", "page": 13, "quote": "The extraction argument must use the common index after fixing most of the rows. We therefore\nchoose coordinates in which translating a child generator changes one free row and leaves all\nother coordinates fixed. The unsplit projection will then act separately at each fixed context,\nand its retained characters will have an explicit form.\nFix a slice D and a node v ∈D. The 2s sets P(vg,b) are nonempty and disjoint. Choose\none representative leaf ig,b from each of them, using a rule depending only on the tree, and set\nLg,b = Rig,b. For every other leaf i in that generator, replace Ri by Ri + Lg,b. Keep all rows\noutside these generators as they are. Denote the resulting k −2s remaining coordinates by\nΓ ∈Hk−2s." }, "occurrences": [ { "blockId": "p13-e6707ed124", "page": 13, "quote": "The extraction argument must use the common index after fixing most of the rows. We therefore\nchoose coordinates in which translating a child generator changes one free row and leaves all\nother coordinates fixed. The unsplit projection will then act separately at each fixed context,\nand its retained characters will have an explicit form.\nFix a slice D and a node v ∈D. The 2s sets P(vg,b) are nonempty and disjoint. Choose\none representative leaf ig,b from each of them, using a rule depending only on the tree, and set\nLg,b = Rig,b. For every other leaf i in that generator, replace Ri by Ri + Lg,b. Keep all rows\noutside these generators as they are. Denote the resulting k −2s remaining coordinates by\nΓ ∈Hk−2s." }, { "blockId": "p13-10cb301361", "page": 13, "quote": "Hk −→H2s × Hk−2s,\nR 7−→((Lg,b), Γ)" }, { "blockId": "p13-bbf043b794", "page": 13, "quote": "is a linear bijection. In particular, under uniform rows the free shifts are independent uniforms\nconditional on Γ. The Fourier index of Lg,b corresponding to a full row index C is Qg,b = qv,gb.\nConsequently, at every fixed Γ, the operator Sv acts as the Fourier projection in the 2s shifts\nonto precisely the unsplit shift indices. For a function of the code or its row smoothing, the shift\ncoefficients vanish unless the sums Qg,0 + Qg,1 are the same for every g. The table actions in\nEquations (5.3) and (5.4) hold for each shift coefficient at this fixed context, with qv = Q1,0 +Q1,1\nand qv,g1 = Qg,1." }, { "blockId": "p13-a07f9ca181", "page": 13, "quote": "Proof. The inverse map sets the representative row equal to Lg,b and recovers every other row in\nits generator by adding Lg,b to its recorded difference. This proves bijectivity. In a row character,\nthe coefficient of Lg,b is exactly P\ni∈P(vg,b) qi = qv,gb. Thus the condition defining Sv depends\nonly on the shift index Q, not on any Fourier index of Γ. Grouping the full Fourier expansion\nby Q proves its asserted action on each fixed fiber. The same grouping proves the support and\ntable-action identities: all full row coefficients with a given Q satisfy the same identities and\ncarry the same signs." }, { "blockId": "p18-8ee1a693e1", "page": 18, "quote": "For a node v ∈D, use the free shift coordinates (Lgb)g∈[s],b∈{0,1} and the remaining context\nΓ ∈Hk−2s from Theorem 5.3. A hat marked “shifts” expands only these 2s free rows. For a\nlabel q ∈A (equivalently, q ∈H∗with q(1) = 1), define" }, { "blockId": "p18-af115d7cb4", "page": 18, "quote": "mq(f; Γ, Fout) = EFD\nX" }, { "blockId": "p18-1a521f670f", "page": 18, "quote": "q∈A\nmq(f; Γ, Fout) ≤1.\n(7.2)" }, { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, { "blockId": "p18-01097b485f", "page": 18, "quote": "Here we sample ω, the random slice D, and a uniform node v ∈D; conditional on these, we\nsample Fout and Γ independently and uniformly." }, { "blockId": "p20-56ef8758e5", "page": 20, "quote": "Step 3: express odd influences as affine hashes.\nFix ω, D, v, Γ, Fout. The remaining row\ncoordinates Lgb are independent uniform elements of H. On the odd part, the common pair\nsum q is a label, by (2.1), and hence is nonzero. An unsplit shift index with common sum q\nputs q in the second child on a subset I ⊆[s] and in the first child elsewhere. Let aq,I(FD) be\nits coefficient in Hodd\nv\n.\nSet" }, { "blockId": "p20-dbdc944b36", "page": 20, "quote": "I\n∥aq,I∥2\n2,FD ≤mq( ¯fω; Γ, Fout).\n(7.10)" }, { "blockId": "p21-5df1859a89", "page": 21, "quote": "B,D\n|B||bΨq(B, D)|2 ≤K∥Ψq∥2\n2 ≤Kmq( ¯fω; Γ, Fout).\n(7.12)" }, { "blockId": "p21-a65ea046f4", "page": 21, "quote": "For each node, condition on ω, D, v, Γ, Fout and integrate the free shifts. Partition these\ncontexts into three classes:" }, { "blockId": "p22-c7a6f2ab8a", "page": 22, "quote": "R ←→(L, Γ),\nL = (Lgb)g∈[s], b∈{0,1} ∈H2s,\nΓ ∈Hk−2s," }, { "blockId": "p22-955a28fdcb", "page": 22, "quote": "Lemma 8.1 (Removing smoothing). For every fixed y, D, v, Γ, Fout, and label q ∈A," }, { "blockId": "p22-470d569f6e", "page": 22, "quote": "mq( ¯Gy; Γ, Fout) ≤EE, e|ymq(Ge; Γ + EΓ, Fout).\n(8.1)" }, { "blockId": "p22-f931f31145", "page": 22, "quote": "¯Gy(L, Γ, F) = EE, e|yGe(L + EL, Γ + EΓ, F)" }, { "blockId": "p22-7eded40296", "page": 22, "quote": "q∗= q∗(y, D, v, Γ, Fout) ∈arg max\nq∈A\nmq( ¯Gy; Γ, Fout)\n(8.2)" }, { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" }, { "blockId": "p23-5cf61811f3", "page": 23, "quote": "e, J0, D, v, Fout, Γ, Rnew, B\n." } ], "occurrenceCount": 24 }, { "symbol": "Λ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, "occurrences": [ { "blockId": "p19-2bd4dfb0c5", "page": 19, "quote": "8 =: Λ.\n(7.6)" }, { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, { "blockId": "p21-de23f92e07", "page": 21, "quote": "M0 =\nΛ\n4 · 2m ,\nθ = γM0" }, { "blockId": "p21-b8e4cf0492", "page": 21, "quote": "K ,\np∗=\nΛ\n4K 2m .\n(7.14)" }, { "blockId": "p21-b276b21d53", "page": 21, "quote": "On class (i), the hashing lemma bounds the expected maximum odd influence by ζMv. Summing\nand using (7.13) costs at most Λ/4. On class (ii), the maximum is bounded in expectation\nby the total influence, namely Mv < M0. Since |D| ≤2m, these contexts cost at most Λ/4.\nOn class (iii), the same total-mass bound is at most K per node. Equation (7.8) bounds all\neven contributions by Λ/4. Let Niii count the nodes whose induced contexts lie in class (iii).\nComparing all four contributions with (7.6) gives" }, { "blockId": "p21-cbfd209c78", "page": 21, "quote": "Λ ≤K2−s + ζK + 2mM0 + KENiii ≤3Λ" }, { "blockId": "p21-ae8861c37e", "page": 21, "quote": "Thus ENiii ≥Λ/(4K). At each such node (7.12) supplies a label with mq > γM0/K = θ.\nChoosing a uniform node from the slice loses at most a factor 2m and gives (7.4). This proves\nthe proposition." } ], "occurrenceCount": 7 }, { "symbol": "α", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p1-0787686067", "page": 1, "quote": "For a graph with nonnegative edge weights, Max-Cut asks for a partition of the vertices\nmaximizing the weight of edges crossing the partition. A cut is encoded by signs on vertices,\nand an edge contributes exactly when its signs differ. For a maximization problem, an α-\napproximation returns a feasible solution worth at least α times the optimum. Unless stated\notherwise, the graph and its rational weights are given explicitly. We write MaxCut(G) for the\nmaximum crossing weight and, when the edge weights sum to one, also denote this value by\nVal(G). The decision problem of whether a weighted graph has a cut of at least a given weight\nappears among Karp’s original NP-complete problems [11]. Approximation asks how much of\nthis optimum can still be guaranteed in polynomial time. Goemans and Williamson [9] gave\na semidefinite programming algorithm with random-hyperplane rounding for Max-Cut, with\nexpected approximation ratio approaching" }, "occurrences": [ { "blockId": "p1-0787686067", "page": 1, "quote": "For a graph with nonnegative edge weights, Max-Cut asks for a partition of the vertices\nmaximizing the weight of edges crossing the partition. A cut is encoded by signs on vertices,\nand an edge contributes exactly when its signs differ. For a maximization problem, an α-\napproximation returns a feasible solution worth at least α times the optimum. Unless stated\notherwise, the graph and its rational weights are given explicitly. We write MaxCut(G) for the\nmaximum crossing weight and, when the edge weights sum to one, also denote this value by\nVal(G). The decision problem of whether a weighted graph has a cut of at least a given weight\nappears among Karp’s original NP-complete problems [11]. Approximation asks how much of\nthis optimum can still be guaranteed in polynomial time. Goemans and Williamson [9] gave\na semidefinite programming algorithm with random-hyperplane rounding for Max-Cut, with\nexpected approximation ratio approaching" }, { "blockId": "p1-9ee5bebb0f", "page": 1, "quote": "Theorem 1.1. For every fixed α with αGW < α ≤1, it is NP-hard to approximate Max-Cut\nwithin a factor of α. This holds for simple unweighted graphs." }, { "blockId": "p1-9516123ba0", "page": 1, "quote": "The statement means that a polynomial-time algorithm which always returns a cut of weight\nat least α times the maximum would imply P = NP. We prove the following more precise gap\ntheorem. Write\nB(t) = 2" }, { "blockId": "p15-192503b724", "page": 15, "quote": "α′\nq(p) = Euχq(u)A′\np(u),\nw′\nq = Ep∥α′\nq(p)∥2,\nU′(z) = Ep(−1)p·zA′\np(Sp)." }, { "blockId": "p15-202173c823", "page": 15, "quote": "∥αq(p) −α′\nq(p)∥≤Eu∥Ap(u)∥1{∥Ap(u)∥>D\n√" }, { "blockId": "p16-4c5b556ff5", "page": 16, "quote": "If mp > CM, then α′\nq(p) = 0. Consequently," }, { "blockId": "p16-6e934d6f5f", "page": 16, "quote": "q\n⟨α′\nq(p1), α′\nq(p2)⟩⟨α′\nq(p3), α′\nq(p4)⟩.\n(6.9)" }, { "blockId": "p17-6b00892076", "page": 17, "quote": "The right side of (6.9) need not be nonnegative. For an upper bound, replace each summand\nby Q4\ni=1∥α′\nq(pi)∥, which is nonnegative, and then drop the restriction that the four indices be\ndistinct. For every fixed q, Cauchy–Schwarz gives" }, { "blockId": "p17-c22ad44175", "page": 17, "quote": "i=1\n∥α′\nq(pi)∥≤\n\nE∥α′\nq(p1)∥2∥α′\nq(p2)∥21/2 \nE∥α′\nq(p3)∥2∥α′\nq(p4)∥21/2" }, { "blockId": "p27-ee4fdccd3a", "page": 27, "quote": "where the same formula defines B(t) = 2 arcsin(t)/π for negative t. For −1 < t ≤0 the ratio is\nat least one, whereas for 0 < t < 1 it is less than one. At t = 1 it equals one, and as t ↓−1 it\ndiverges. Thus its minimum occurs in (0, 1).\nFix αGW < α ≤1. By continuity and the density of the rationals, choose a rational t ∈(0, 1)\nsuch that" }, { "blockId": "p27-bad83a26d4", "page": 27, "quote": "α1 + t" }, { "blockId": "p27-c6f29518ac", "page": 27, "quote": "α\n1 + t" }, { "blockId": "p27-7abf2de4cc", "page": 27, "quote": "Apply Theorem 1.2. Delete loops while retaining the original weight scale, so that the remaining\nweights sum to at most one. Apply Theorem B.1 with error ϵ′, producing a simple unweighted\ngraph whose maximum cut, on the common scale Q, differs from the original maximum cut\nweight by at most ϵ′. An α-approximation algorithm on that graph returns, in the YES case, a\ncut of scaled value at least the left side of (9.2). In the NO case, no cut has scaled value greater\nthan its right side. A fixed rational threshold strictly between them therefore distinguishes the\nNP-hard promise cases. This proves the theorem." }, { "blockId": "p28-e64092dca6", "page": 28, "quote": "Spaces, labels, and folding.\nLet Z be the variables of the 3Lin instance, let V0 = FZ\n2 , and\nview each equation e as\n⟨xe, α⟩= be,\nxe ∈V0,\nbe ∈F2," }, { "blockId": "p31-2563015c96", "page": 31, "quote": "Fix a soundness parameter δ ∈(0, 1).\nThe agreement lemma [6, Lemma 3.7] supplies\nconstants q, r, C and a positive function α(ℓ) at agreement threshold δ2/8, as in the proof of its\nLemma 5.3. Set the thresholds" }, { "blockId": "p31-39ad0ac56d", "page": 31, "quote": "Both vectors are fixed before choosing ℓ; the function α(ℓ) is positive once ℓis fixed. We choose\nℓlarge enough for that agreement lemma and for the ordinary list bound [6, Lemma 3.12] on\n(ℓ−q)-spaces with thresholds (ζi). We also require q ≤ℓ−1 and" }, { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" }, { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p34-380ad96976", "page": 34, "quote": "The overall success probability is at least δα(ℓ)c∗/8, a positive constant independent of k0. The\nupper bound in [6, Lemma 5.4] tends to zero as βk0 →∞. Choosing k0 sufficiently large gives\nthe same contradiction as in the source and proves that raw valid acceptance is at most δ." }, { "blockId": "p35-4807687585", "page": 35, "quote": "Completeness with the alphabet fixed.\nLet α : Z →F2 satisfy all but an ε fraction\nof the input equations, and also write α : V0 →F2 for its linear extension. Give the folded\nleft question C the answer α|RC and give (V, L′) the answer α|L′. Whenever every equation of\nsampled U is satisfied, hU = α|HU , so (A.2) is exactly α|RC+HU . The edge is therefore satisfied.\nThis argument concerns the sampled edge; it does not require every member of its folding class\nto involve satisfied equations.\nBefore either conditioning step, a union bound gives probability at most k0ε of sampling an\nunsatisfied equation. Since the total retained probability qall is at least 1/2, the conditioned\ngame’s value is at least 1 −k0ε/qall ≥1 −2k0ε. After ℓ, k0 have been fixed for soundness, choose\nε ≤a/(2k0) in the initial 3Lin gap. This gives the asserted completeness without altering n." } ], "occurrenceCount": 20 }, { "symbol": "β", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p28-0d80eef515", "page": 28, "quote": "Starting instance.\nTheorem 4.1 of [6], obtained from Håstad’s theorem [10], supplies an\nabsolute constant s∗< 1 such that for every sufficiently small ε > 0 it is NP-hard to distinguish\nregular binary 3Lin instances of value at least 1 −ε from those of value at most s∗. Regularity\nincludes bounded variable occurrence, three distinct variables per equation, and at most one\nvariable shared by two distinct equations.\nFor now let ℓ≥2, k0 ≥ℓ, and β ∈(0, 1) be fixed parameters, with β rational. The two\nalphabet dimensions will be ℓand ℓ−1, so the dimension in Theorem 2.2 is n = ℓ−1. The\nsoundness argument below chooses ℓ, k0, β using only b and the absolute constant s∗." }, "occurrences": [ { "blockId": "p28-0d80eef515", "page": 28, "quote": "Starting instance.\nTheorem 4.1 of [6], obtained from Håstad’s theorem [10], supplies an\nabsolute constant s∗< 1 such that for every sufficiently small ε > 0 it is NP-hard to distinguish\nregular binary 3Lin instances of value at least 1 −ε from those of value at most s∗. Regularity\nincludes bounded variable occurrence, three distinct variables per equation, and at most one\nvariable shared by two distinct equations.\nFor now let ℓ≥2, k0 ≥ℓ, and β ∈(0, 1) be fixed parameters, with β rational. The two\nalphabet dimensions will be ℓand ℓ−1, so the dimension in Theorem 2.2 is n = ℓ−1. The\nsoundness argument below chooses ℓ, k0, β using only b and the absolute constant s∗." }, { "blockId": "p29-4c304fab40", "page": 29, "quote": "Validity of the edge sampling.\nWe specify the treatment of ill-formed samples omitted from\nthe displayed sampler. Sampling k0 equations independently produces a nonlegitimate tuple\nwith probability O(k2\n0/|Z|), by bounded occurrence. For fixed k0 this can be made arbitrarily\nsmall. If necessary, take sufficiently many disjoint copies of the input instance; this preserves its\nvalue and regularity.\nGiven a legitimate U, independently for each equation, the right variable set V contains\nall three variables with probability 1 −β and otherwise one uniformly chosen variable. If f0\nequations contribute all three, then" }, { "blockId": "p33-8960e84e9f", "page": 33, "quote": "β k1/4\n0\n," }, { "blockId": "p34-d3a9576af9", "page": 34, "quote": "Every restriction above concerns only b and parameters fixed from it. After choosing ℓ, take\nk0 = j4 and β = j−3 with j sufficiently large. Then βk0 →∞, β√k0 →0, and √β k1/4\n0\n→0.\nThese limits satisfy the outer-game soundness and covering requirements, including 2ℓβ ≤1/8,\nas well as all the displayed margins. In particular, the alphabet dimension n = ℓ−1 has been\nfixed using only b." } ], "occurrenceCount": 4 }, { "symbol": "γ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, "occurrences": [ { "blockId": "p14-7d2e24e8ce", "page": 14, "quote": "For every ζ > 0, there are γ > 0 and s0 ∈N, depending only on ζ, such that for every s ≥s0," }, { "blockId": "p17-999f55e22e", "page": 17, "quote": "ε ,\nγ = ε2" }, { "blockId": "p17-9e2b210abb", "page": 17, "quote": "If s ≥s0 and ρ ≤γ, the square-root term in (6.3) is at most ε/4. Also D ≥\n√" }, { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, { "blockId": "p33-308d8f1893", "page": 33, "quote": "FV [L] −pγ|L,\nFV [L]|Q = γ,\n0,\nFV [L]|Q ̸= γ." }, { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." }, { "blockId": "p34-be24debe49", "page": 34, "quote": "Here L has dimension ℓ. Set S0 = W ∩VV and γ = f|Q. Every equality FV [L] = f|L in this\ndisplay is retained in Tγ. Since ζi ≤τ0/2 ≤τi/2, the quotient pair" } ], "occurrenceCount": 8 }, { "symbol": "δ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, "occurrences": [ { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, { "blockId": "p11-164121ec4c", "page": 11, "quote": "If a cut has value at least (1 + B(t))/2 + δ/2, then its odd average has expected slice stability\nat least B(t) + δ. The next three sections show that this excess stability yields a label carrying\nsubstantial Fourier mass, with constants independent of the game dimension and the number of\nrepetitions; the formal statement is Theorem 7.1. Section 8 then turns that mass into a labeling\nof a single copy of the original game." }, { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, { "blockId": "p18-6dd3e25df1", "page": 18, "quote": "Eω,D,R,Fout StabFD\nt\n( ¯fω) ≥B(t) + δ,\n(7.3)" }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p18-65f0ea897b", "page": 18, "quote": "some entry z ∈Fs\n2 of some v ∈D has Inf≤K\nv,z ( ¯fω) ≥τ. Indeed, if the probability of this event is\np, the expected stability is at most B(t) + δ/2 + p." }, { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, { "blockId": "p31-2563015c96", "page": 31, "quote": "Fix a soundness parameter δ ∈(0, 1).\nThe agreement lemma [6, Lemma 3.7] supplies\nconstants q, r, C and a positive function α(ℓ) at agreement threshold δ2/8, as in the proof of its\nLemma 5.3. Set the thresholds" }, { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" }, { "blockId": "p33-4e7f10189c", "page": 33, "quote": "From game acceptance to nonempty lists.\nSuppose a folded labeling has raw valid\nacceptance greater than δ under P, the sampler conditioned on legitimate U but not on validity\nof L. Testing the completed FU[L] against the original right answer on (V, L′) can only increase\nthis acceptance: valid entries are unchanged, and invalid outcomes previously had payoff zero.\nWe may therefore apply the agreement/list and covering arguments in [6, Lemma 5.3, Claim 5.6,\nand Equation (5.1)] to the completed tables. Before invoking invariance, these arguments use\ntotal linear tables, their agreement, and (A.5); all are now defined on every sampled subspace.\nFor fixed U, Q, let LU,Q be the uniform law on ℓ-spaces of VU containing Q. Let L′\nU,Q first\nsample V conditional on the outer question (U, Q) and then a uniform ℓ-space of VV containing\nQ. These are the conditional laws in [6, Lemma 4.11]. Indeed, if Nq(d) denotes the number of\nq-spaces in Fd\n2, then for Q ⊆VV a uniform ℓ-space L ≤VV satisfies" }, { "blockId": "p33-961490e74d", "page": 33, "quote": "Thus conditioning the source’s V, L sampler on Q ⊆L weights each V proportionally to\nPr[V | U]/Nq(dim VV ), exactly as in the outer game.\nLet E1 be the event agreement(FU) > δ2/8, let E2 be" }, { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, { "blockId": "p34-380ad96976", "page": 34, "quote": "The overall success probability is at least δα(ℓ)c∗/8, a positive constant independent of k0. The\nupper bound in [6, Lemma 5.4] tends to zero as βk0 →∞. Choosing k0 sufficiently large gives\nthe same contradiction as in the source and proves that raw valid acceptance is at most δ." }, { "blockId": "p34-80a9431745", "page": 34, "quote": "Fixing soundness before completeness.\nFor the game in Theorem 2.2, take δ = b/4 and\nretain the valid raw weights. Their total mass qedge is at least 1/2 by the preceding rejection\nestimates. Normalization therefore gives" }, { "blockId": "p34-20357bc1a9", "page": 34, "quote": "Val ≤δ/qedge ≤b/2 < b." } ], "occurrenceCount": 15 }, { "symbol": "ε", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p17-4bdc769804", "page": 17, "quote": "The inequality (a + b)2 ≤2a2 + 2b2, together with (6.7) and (6.10), proves (6.3).\nFinally, set ε = min{ζ, 1} and choose" }, "occurrences": [ { "blockId": "p2-2780d2e59a", "page": 2, "quote": "Theorem 1.2. Fix a rational t ∈(0, 1) and a constant 0 < ε < (t−B(t))/4. For graphs equipped\nwith a rational probability distribution on edges, it is NP-hard to distinguish" }, { "blockId": "p2-b8ed68c6cc", "page": 2, "quote": "2\n−ε," }, { "blockId": "p2-3f56d83e08", "page": 2, "quote": "2\n+ ε," }, { "blockId": "p17-4bdc769804", "page": 17, "quote": "The inequality (a + b)2 ≤2a2 + 2b2, together with (6.7) and (6.10), proves (6.3).\nFinally, set ε = min{ζ, 1} and choose" }, { "blockId": "p17-812837a320", "page": 17, "quote": "ε ,\nD = 8C" }, { "blockId": "p17-999f55e22e", "page": 17, "quote": "ε ,\nγ = ε2" }, { "blockId": "p17-48fb9381b9", "page": 17, "quote": "ε2\n, 16" }, { "blockId": "p17-0314d0205a", "page": 17, "quote": "ε" }, { "blockId": "p17-9e2b210abb", "page": 17, "quote": "If s ≥s0 and ρ ≤γ, the square-root term in (6.3) is at most ε/4. Also D ≥\n√" }, { "blockId": "p17-c646c8dd58", "page": 17, "quote": "C, so the other two\nterms in its brackets are at most ε/16 each. The resulting bound is 3εM/4 ≤ζM, establishing\n(6.2) with constants depending only on ζ." }, { "blockId": "p26-e1770fd031", "page": 26, "quote": "(t, ε, µ) −→(s, m, p∗, θ) −→b −→n −→T −→a." }, { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, { "blockId": "p26-8659c32a43", "page": 26, "quote": "Finally choose a positive rational a < 1 −b small enough that 2Ta ≤ε/4, and start with the\nNP-hard game gap Val(G) ≥1 −a versus Val(G) ≤b." }, { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, { "blockId": "p27-74a15e2a80", "page": 27, "quote": "vertices. All factors except |X| are constants for fixed t, ε. There are polynomially many\nendpoint and edge tuples, and all word-generation sample spaces have constant size. Enumerate\nevery test outcome and add its probability to the corresponding edge weight. The initial game\nweights, t, and r are rational, so these weights have polynomial bit complexity. Conditional edge\nprobabilities also have polynomial bit complexity. All alphabets and words can be enumerated\nin constant time relative to the input size. The resulting graph is therefore computable in\ndeterministic polynomial time. It has a rational edge distribution and satisfies the asserted,\nweaker ε-loss bounds." }, { "blockId": "p27-fdadd86d20", "page": 27, "quote": "Choose positive rational ε and ϵ′ small enough that ε < (t −B(t))/4 and" }, { "blockId": "p27-193eb5417a", "page": 27, "quote": "2\n−ε −ϵ′\n\n> 1 + B(t)" }, { "blockId": "p27-f5698fca36", "page": 27, "quote": "2\n+ ε + ϵ′.\n(9.2)" }, { "blockId": "p28-0d80eef515", "page": 28, "quote": "Starting instance.\nTheorem 4.1 of [6], obtained from Håstad’s theorem [10], supplies an\nabsolute constant s∗< 1 such that for every sufficiently small ε > 0 it is NP-hard to distinguish\nregular binary 3Lin instances of value at least 1 −ε from those of value at most s∗. Regularity\nincludes bounded variable occurrence, three distinct variables per equation, and at most one\nvariable shared by two distinct equations.\nFor now let ℓ≥2, k0 ≥ℓ, and β ∈(0, 1) be fixed parameters, with β rational. The two\nalphabet dimensions will be ℓand ℓ−1, so the dimension in Theorem 2.2 is n = ℓ−1. The\nsoundness argument below chooses ℓ, k0, β using only b and the absolute constant s∗." }, { "blockId": "p35-4807687585", "page": 35, "quote": "Completeness with the alphabet fixed.\nLet α : Z →F2 satisfy all but an ε fraction\nof the input equations, and also write α : V0 →F2 for its linear extension. Give the folded\nleft question C the answer α|RC and give (V, L′) the answer α|L′. Whenever every equation of\nsampled U is satisfied, hU = α|HU , so (A.2) is exactly α|RC+HU . The edge is therefore satisfied.\nThis argument concerns the sampled edge; it does not require every member of its folding class\nto involve satisfied equations.\nBefore either conditioning step, a union bound gives probability at most k0ε of sampling an\nunsatisfied equation. Since the total retained probability qall is at least 1/2, the conditioned\ngame’s value is at least 1 −k0ε/qall ≥1 −2k0ε. After ℓ, k0 have been fixed for soundness, choose\nε ≤a/(2k0) in the initial 3Lin gap. This gives the asserted completeness without altering n." } ], "occurrenceCount": 20 }, { "symbol": "ζ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p17-4bdc769804", "page": 17, "quote": "The inequality (a + b)2 ≤2a2 + 2b2, together with (6.7) and (6.10), proves (6.3).\nFinally, set ε = min{ζ, 1} and choose" }, "occurrences": [ { "blockId": "p14-7d2e24e8ce", "page": 14, "quote": "For every ζ > 0, there are γ > 0 and s0 ∈N, depending only on ζ, such that for every s ≥s0," }, { "blockId": "p17-4bdc769804", "page": 17, "quote": "The inequality (a + b)2 ≤2a2 + 2b2, together with (6.7) and (6.10), proves (6.3).\nFinally, set ε = min{ζ, 1} and choose" }, { "blockId": "p17-c646c8dd58", "page": 17, "quote": "C, so the other two\nterms in its brackets are at most ε/16 each. The resulting bound is 3εM/4 ≤ζM, establishing\n(6.2) with constants depending only on ζ." }, { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, { "blockId": "p31-b1ccf446e3", "page": 31, "quote": "ζr = τ0/2,\nζi = 10−9ζ12\ni+1\n(0 ≤i < r)." } ], "occurrenceCount": 5 }, { "symbol": "η", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p30-8b64be6ecb", "page": 30, "quote": "Let P be that sampler after conditioning on legitimate U, and put E = {L ∩HU = {0}} and\nη = 2ℓ−k0 + 2ℓ−2k0. Our edge law is R = P( · | E), so TV(P, R) = P(Ec) ≤η. On the source’s\nvalid edge set define the subprobability weights" }, "occurrences": [ { "blockId": "p2-f36095cee9", "page": 2, "quote": "The reduction of proof verification to approximation gaps has its roots in Feige, Goldwasser,\nLovász, Safra, and Szegedy [7] and in the PCP theorems of Arora–Safra and Arora, Lund,\nMotwani, Sudan, and Szegedy [2, 1]. Håstad’s Fourier analysis of proof-verification tests [10]\nmade many of these gaps sharp. For Max-Cut, the Gaussian geometry of the semidefinite\nrelaxation gives a particularly precise target. Feige and Schechtman [8] constructed integrality\ngaps approaching αGW for this relaxation; a limitation of that relaxation alone does not establish\nhardness for arbitrary algorithms. The Unique Games reduction of Khot, Kindler, Mossel, and\nO’Donnell [13] connected the same geometric threshold to computational hardness.\nThe original analysis of Khot, Kindler, Mossel, and O’Donnell proposed Majority Is Stablest\nas the analytic statement needed to obtain this threshold. Mossel, O’Donnell, and Oleszkiewicz\nproved it through a comparison between low-influence Boolean functions and Gaussian func-\ntions [16], transferring Borell’s halfspace noise-stability bound [5] to the Boolean setting. Their\ntheorem leaves the Unique Games Conjecture as the hypothesis in the classical hardness reduc-\ntion [16, Corollary 2.12]. We use Majority Is Stablest to identify an influential coordinate of our\ntest, then prove the additional extraction needed to turn that coordinate into a source-game label.\nThe algorithmic benchmark above is the ideal rounding ratio; finite-precision SDP optimization\nand random-hyperplane rounding give expected ratio αGW −η for any fixed η > 0 [9, Section 2].\nThe comparison between semidefinite approximation and hardness extends beyond a single\nworst-case ratio. O’Donnell and Wu [17] determined the full Max-Cut approximation curve\nunder the Unique Games Conjecture, relating the attainable cut value to the instance’s optimum.\nRaghavendra [18] established a general correspondence between semidefinite relaxations and\nUnique-Games-based hardness for finite constraint satisfaction problems. Our theorem concerns\nthe unconditional worst-case ratio for Max-Cut; the proof below establishes the specific family\nof completeness and soundness gaps in Theorem 1.2.\nIn a projection game, each vertex of a bipartite graph receives a label from a specified\nalphabet, and the edge weights form a probability distribution. An edge accepts when its\nprescribed map sends the left label to the right label. Its value is the maximum accepted edge\nweight. A two-to-one constraint has exactly two preimages for every right label. Completeness\nand soundness are, respectively, the promised lower bound on the YES-case value and upper\nbound on the NO-case value.\nOur starting point is the published 2-to-1 game construction with imperfect completeness [6].\nThe Grassmann-graph approach was introduced by Khot, Minzer, and Safra [15]. Its soundness\nwas completed through the connection between expansion and agreement established by Barak,\nKothari, and Steurer [3] and the expansion theorem of Khot, Minzer, and Safra [14]. We need a\nprecise affine form of the construction: its projections are surjective affine maps between binary\nspaces of dimensions differing by one, and its alphabet can be fixed for a desired soundness before" }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p19-f2ddd11aec", "page": 19, "quote": "v∈D\n∥(1 −Sv) ¯fω∥2\n2,R,F ≤\ns\n2emµ ≤ητ" }, { "blockId": "p19-d0b22816e9", "page": 19, "quote": "For any fixed s, the last inequality can be ensured by increasing m. The same expectation is the\nexpectation over ω, D, R, Fout of the sum of the corresponding conditional squared norms in FD.\nMarkov’s inequality shows that this sum is at most τ/4 except on an event of probability η/2.\nOn the intersection with the influential-entry event, the triangle inequality for the projection\ndefining that influence gives" }, { "blockId": "p19-377627618d", "page": 19, "quote": "v∈D\nmax\nz\nInfv,z(Hv) ≥ητ" }, { "blockId": "p30-8b64be6ecb", "page": 30, "quote": "Let P be that sampler after conditioning on legitimate U, and put E = {L ∩HU = {0}} and\nη = 2ℓ−k0 + 2ℓ−2k0. Our edge law is R = P( · | E), so TV(P, R) = P(Ec) ≤η. On the source’s\nvalid edge set define the subprobability weights" }, { "blockId": "p30-979bdf5c53", "page": 30, "quote": "e\nw(e) = P(E) ≥1 −η." } ], "occurrenceCount": 7 }, { "symbol": "θ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p26-1de248c808", "page": 26, "quote": "Conditional on all the planted data, the left side lower-bounds the probability that the sampled\nfull label projects to q∗. That event in particular implies satisfaction of the marked input edge.\nAveraging and using Equation (8.7) gives success probability at least 2−2sp∗θ/2.\nFinally, fix all shared randomness, including the uniform number used to sample the X-label.\nThe same fixed data define the two strategies at every endpoint, so they give a deterministic\nlabeling of the original game. Their average value over the shared randomness has the lower\nbound just proved, so some fixed choice attains it. This proves Equation (8.9)." }, "occurrences": [ { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, { "blockId": "p18-11d38c1f8f", "page": 18, "quote": "then\nPr\n\nmax\nq∈A mq( ¯fω; Γ, Fout) ≥θ\n\n≥p∗.\n(7.4)" }, { "blockId": "p21-de23f92e07", "page": 21, "quote": "M0 =\nΛ\n4 · 2m ,\nθ = γM0" }, { "blockId": "p21-ae8861c37e", "page": 21, "quote": "Thus ENiii ≥Λ/(4K). At each such node (7.12) supplies a label with mq > γM0/K = θ.\nChoosing a uniform node from the slice loses at most a factor 2m and gives (7.4). This proves\nthe proposition." }, { "blockId": "p22-0e713b71c9", "page": 22, "quote": "E mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ.\n(8.3)" }, { "blockId": "p23-d3c148ea4c", "page": 23, "quote": "T ≥\n1\np0(p∗θ)2 ,\n(8.6)" }, { "blockId": "p23-46c96a798b", "page": 23, "quote": "EP mq∗(Ge; Γ + EΓ, Fout) ≥p∗θ" }, { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" }, { "blockId": "p24-44276ea803", "page": 24, "quote": "Val(G) ≥2−2s p∗θ" }, { "blockId": "p26-1de248c808", "page": 26, "quote": "Conditional on all the planted data, the left side lower-bounds the probability that the sampled\nfull label projects to q∗. That event in particular implies satisfaction of the marked input edge.\nAveraging and using Equation (8.7) gives success probability at least 2−2sp∗θ/2.\nFinally, fix all shared randomness, including the uniform number used to sample the X-label.\nThe same fixed data define the two strategies at every endpoint, so they give a deterministic\nlabeling of the original game. Their average value over the shared randomness has the lower\nbound just proved, so some fixed choice attains it. This proves Equation (8.9)." }, { "blockId": "p26-852017d85d", "page": 26, "quote": "The decoding loss depends on s, p∗, θ, and hence on the target cut gap, but not on the\nalphabet dimension or the repetition length. After choosing a game soundness smaller than the\nright side of Equation (8.9), one may therefore fix the alphabet dimension and then choose T by\nEquation (8.6)." }, { "blockId": "p26-e1770fd031", "page": 26, "quote": "(t, ε, µ) −→(s, m, p∗, θ) −→b −→n −→T −→a." }, { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, { "blockId": "p26-719578cdd9", "page": 26, "quote": "0 < b < 2−2s p∗θ" }, { "blockId": "p26-b35054634f", "page": 26, "quote": "(p∗θ)2 ." }, { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" } ], "occurrenceCount": 16 }, { "symbol": "ι", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p24-e555a043e6", "page": 24, "quote": "Set f(L, FD) = ef(ιL, FD), where ι acts on each shift. For every label q ∈A," }, "occurrences": [ { "blockId": "p24-43eee9121f", "page": 24, "quote": "eH = Aff( e\nA, F2),\nι = ρ∗: H ,→eH." }, { "blockId": "p24-a466781a1e", "page": 24, "quote": "Pullback is linear on the vector spaces of affine forms, even when ρ has a nonzero translation\npart. The image of ι has codimension one, and ι(1) = 1. Its dual restriction map ι∗: eH∗→H∗" }, { "blockId": "p24-e555a043e6", "page": 24, "quote": "Set f(L, FD) = ef(ιL, FD), where ι acts on each shift. For every label q ∈A," }, { "blockId": "p24-ec8f6287b9", "page": 24, "quote": "e\nQ\nι∗e\nQgb=Qgb ∀g,b" }, { "blockId": "p24-dae4f91596", "page": 24, "quote": "eq(1) = eq(ι(1)) = q(1) = 1," } ], "occurrenceCount": 5 }, { "symbol": "κ", "meaning": "No explicit definition located by the extractor.", "status": "not-located", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p25-b92708bed6", "page": 25, "quote": "B\nκ\n−−→e\nA\nρ\n−−→A." }, "occurrences": [ { "blockId": "p25-b92708bed6", "page": 25, "quote": "B\nκ\n−−→e\nA\nρ\n−−→A." }, { "blockId": "p25-c16f205107", "page": 25, "quote": "Here κ is the public edge projection at every unmarked slot and the identity at the marked slot;\nρ is the identity at every unmarked slot and the unknown map πe at the marked slot. Thus κ is\nknown to the X-labeler, while ρ is used only in the analysis.\nThe planted zeros imply that every row in Γ and Rnew has zero linear coefficient block in\nthe marked slot. The same holds for E, EΓ, and Γ′ = Γ + EΓ, since the noise masks and context\nchange only multiply by scalars and add rows. Consequently eΓ′ = ρ∗Γ′ is known to the X-labeler:\nit is obtained by leaving every unmarked coefficient and constant coefficient unchanged and\nsetting all marked coefficients to zero. This description does not require πe, including its affine\ntranslation part.\nThe X-labeler forms the bounded raw function" }, { "blockId": "p25-bc667bad25", "page": 25, "quote": "W e\nA( eR, F) ◦κ\n,\neR ∈eHk." }, { "blockId": "p26-3843e9cd2b", "page": 26, "quote": "h multiplies the output of gate v by (−1)h(el) at every label el. Applying this shift in two distinct\npairs leaves that output, and hence the whole word, unchanged. These operations only translate\nthe free shifts and preserve the context. Character orthogonality therefore shows that every\nshift Fourier index of eG has a common pair sum. This argument holds for each fixed choice of\nall gate tables.\nWord generation commutes with pullback of rows, pointwise in the label. Since πe = ρ ◦κ,\nrestricting each free shift of eG to ρ∗H yields exactly Ge with context Γ′ and the same tables.\nThe map ρ has two-element fibers, so Lemma 8.3 applies and gives\nX" }, { "blockId": "p29-2422708018", "page": 29, "quote": "We condition the sampling on legitimacy and L ∩HU = {0}. The discarded probability is at\nmost\nκ = O(k2\n0/|Z|) + 2ℓ−k0 + 2ℓ−2k0.\n(A.3)" } ], "occurrenceCount": 5 }, { "symbol": "λ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p6-13601d3979", "page": 6, "quote": "A weighted bipartite projection game consists of finite sets X, Y , a rational probability distribu-\ntion λ on a finite edge set E with endpoint maps to X and Y (parallel edges are allowed), finite\nalphabets, and a projection πe for each edge. Its value is" }, "occurrences": [ { "blockId": "p6-13601d3979", "page": 6, "quote": "A weighted bipartite projection game consists of finite sets X, Y , a rational probability distribu-\ntion λ on a finite edge set E with endpoint maps to X and Y (parallel edges are allowed), finite\nalphabets, and a projection πe for each edge. Its value is" }, { "blockId": "p6-f7f78a9e2b", "page": 6, "quote": "Val(G) = max\nℓX,ℓY\nPr\ne=(x,y)∼λ[πe(ℓX(x)) = ℓY (y)]." }, { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, { "blockId": "p9-9b03894ad3", "page": 9, "quote": "1. Draw y ∈Y T from the product right-endpoint marginal of λ. Independently draw two\nedge tuples e, e′ from the product conditional edge law given y. Write x, x′ for their left\nendpoints and πe, πe′ : B →A for their coordinatewise projections." } ], "occurrenceCount": 4 }, { "symbol": "ν", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, "occurrences": [ { "blockId": "p33-5eabaae0c9", "page": 33, "quote": "The second player’s quotient lists.\nFor each question (V, Q) and each linear form γ : Q →\nF2, fix a linear extension pγ : VV →F2. Let ν : VV →VV /Q be the quotient map. Define a total\ntable Tγ on the (ℓ−q)-spaces of VV /Q by" }, { "blockId": "p33-118f470b93", "page": 33, "quote": "Here Q ⊆L ⊆VV , dim L = ℓ, and the overline denotes the induced linear form on L/Q; it is\ndefined in the first case because the difference vanishes on Q.\nFor this total table, use the ordinary maximal list with thresholds (ζi): a pair (g, S) occurs\nif S ≤VV /Q has codimension i ≤r and Tγ agrees with g on more than a ζi fraction of the\n(ℓ−q)-spaces in S. By [6, Lemma 3.12], each such list has size at most M0 = M0(ℓ−q, r, ζr),\nindependent of k0. The second player chooses a uniform γ ∈Q∗, then a uniform pair (g, S)\nfrom this list. It lifts g to the linear form g ◦ν + pγ on ν−1(S), extends it uniformly to VV , and\noutputs the corresponding assignment. If its chosen list is empty, it uses an arbitrary assignment.\nThe entire procedure uses only (V, Q) and the fixed tables.\nThe zero entries may create additional occurring extensions in an auxiliary table. This\ncauses no difficulty: the proof below needs an extension of a specified linear form, rather than\nan identification of these auxiliary maximal pairs with the maximal pairs of FV ." } ], "occurrenceCount": 2 }, { "symbol": "ξ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, "occurrences": [ { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." }, { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," } ], "occurrenceCount": 3 }, { "symbol": "π", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, "occurrences": [ { "blockId": "p1-1700164c77", "page": 1, "quote": "π(1 −ρ) = 0.878567 . . . .\n(1.1)" }, { "blockId": "p1-4155ac1062", "page": 1, "quote": "The expression compares two contributions of an edge whose endpoint vectors have inner product\nρ: its semidefinite objective value is (1 −ρ)/2, whereas a random hyperplane separates the\nvectors with probability arccos(ρ)/π. The minimum ratio over ρ is the uniform guarantee of\nthis rounding rule. Determining whether this ratio is optimal under ordinary NP-hardness has\nbeen a central question in approximation algorithms. Håstad’s hardness theorem [10], combined\nwith the gadgets of Trevisan, Sorkin, Sudan, and Williamson [19], rules out ratios greater than\n16/17. Khot, Kindler, Mossel, and O’Donnell [13] established optimality of αGW assuming the\nUnique Games Conjecture [12]; the analytic input is the Majority Is Stablest theorem of Mossel,\nO’Donnell, and Oleszkiewicz [16].\nWe resolve the unconditional optimality question for Max-Cut." }, { "blockId": "p1-875921d5b7", "page": 1, "quote": "π arcsin t,\n0 < t < 1.\n(1.2)" }, { "blockId": "p5-d5c080fd4b", "page": 5, "quote": "Proof. The untruncated assertion follows from Majority Is Stablest [16, Theorem 4.4]. To match\nits [0, 1] convention, put g = (h + 1)/2: then Eg = 1/2, each nonconstant Fourier coefficient is\nhalved, and Stabt(h) = 4 Stabt(g)−1. The Gaussian halfspace value for g is 1/4+arcsin(t)/(2π),\nwhich becomes B(t) in our normalization; see [16, Theorem 4.1 and Corollary 4.3] and [5]. Here\nis the degree-cutoff reduction to the untruncated assertion. For a cube-noise parameter u < 1,\nput hu = Tuh, using ordinary bit noise in this paragraph. Range and mean are preserved.\nUniformly over h of norm at most one," }, { "blockId": "p27-ee4fdccd3a", "page": 27, "quote": "where the same formula defines B(t) = 2 arcsin(t)/π for negative t. For −1 < t ≤0 the ratio is\nat least one, whereas for 0 < t < 1 it is less than one. At t = 1 it equals one, and as t ↓−1 it\ndiverges. Thus its minimum occurs in (0, 1).\nFix αGW < α ≤1. By continuity and the density of the rationals, choose a rational t ∈(0, 1)\nsuch that" } ], "occurrenceCount": 5 }, { "symbol": "ρ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p15-3a6aba3253", "page": 15, "quote": "Summing over z proves (6.1). If M = 0, every vector c(q, z) is zero and (6.2) is immediate.\nHenceforth assume M > 0.\nSet h = 2s and ρ = maxq wq/M. We will prove the quantitative bound below for every\nC, D > 0, and then choose its parameters. The main estimate bounds the maximum squared\nnorm by the square root of the sum of fourth powers. After a Fourier change of variables,\nmost terms in that fourth moment force the same character q in all four factors; their total is\ncontrolled by the dispersion of the character masses. Two truncations will control the remaining\nterms and the cost of this change to the array:" }, "occurrences": [ { "blockId": "p1-66e58fef7e", "page": 1, "quote": "αGW =\nmin\n−1≤ρ<1\n2 arccos ρ" }, { "blockId": "p1-1700164c77", "page": 1, "quote": "π(1 −ρ) = 0.878567 . . . .\n(1.1)" }, { "blockId": "p1-4155ac1062", "page": 1, "quote": "The expression compares two contributions of an edge whose endpoint vectors have inner product\nρ: its semidefinite objective value is (1 −ρ)/2, whereas a random hyperplane separates the\nvectors with probability arccos(ρ)/π. The minimum ratio over ρ is the uniform guarantee of\nthis rounding rule. Determining whether this ratio is optimal under ordinary NP-hardness has\nbeen a central question in approximation algorithms. Håstad’s hardness theorem [10], combined\nwith the gadgets of Trevisan, Sorkin, Sudan, and Williamson [19], rules out ratios greater than\n16/17. Khot, Kindler, Mossel, and O’Donnell [13] established optimality of αGW assuming the\nUnique Games Conjecture [12]; the analytic input is the Majority Is Stablest theorem of Mossel,\nO’Donnell, and Oleszkiewicz [16].\nWe resolve the unconditional optimality question for Max-Cut." }, { "blockId": "p15-3a6aba3253", "page": 15, "quote": "Summing over z proves (6.1). If M = 0, every vector c(q, z) is zero and (6.2) is immediate.\nHenceforth assume M > 0.\nSet h = 2s and ρ = maxq wq/M. We will prove the quantitative bound below for every\nC, D > 0, and then choose its parameters. The main estimate bounds the maximum squared\nnorm by the square root of the sum of fourth powers. After a Fourier change of variables,\nmost terms in that fourth moment force the same character q in all four factors; their total is\ncontrolled by the dispersion of the character masses. Two truncations will control the remaining\nterms and the cost of this change to the array:" }, { "blockId": "p15-7ebae8fadb", "page": 15, "quote": "2ρ + 2C2" }, { "blockId": "p17-4743fa2f17", "page": 17, "quote": "2ρ + 2C2" }, { "blockId": "p17-6b32b3b65b", "page": 17, "quote": "2ρ + 2C2" }, { "blockId": "p17-9e2b210abb", "page": 17, "quote": "If s ≥s0 and ρ ≤γ, the square-root term in (6.3) is at most ε/4. Also D ≥\n√" }, { "blockId": "p24-769ceaa222", "page": 24, "quote": "The next elementary observation accounts for the loss in decoding. Let ρ : e\nA →A be a surjective\naffine map with two-element fibers, and write" }, { "blockId": "p24-43eee9121f", "page": 24, "quote": "eH = Aff( e\nA, F2),\nι = ρ∗: H ,→eH." }, { "blockId": "p24-a466781a1e", "page": 24, "quote": "Pullback is linear on the vector spaces of affine forms, even when ρ has a nonzero translation\npart. The image of ι has codimension one, and ι(1) = 1. Its dual restriction map ι∗: eH∗→H∗" }, { "blockId": "p24-62aafa6fe1", "page": 24, "quote": "has two-element fibers and, on evaluation functionals, agrees with ρ." }, { "blockId": "p24-17f2024d92", "page": 24, "quote": "eq∈e\nA\nρ(eq)=q" }, { "blockId": "p24-e0bbb65da1", "page": 24, "quote": "so eq is an evaluation functional, hence a label in e\nA, and ρ(eq) = q. Each lifted index is counted\nonly once. Averaging over the slice tables proves Equation (8.8)." }, { "blockId": "p25-b92708bed6", "page": 25, "quote": "B\nκ\n−−→e\nA\nρ\n−−→A." }, { "blockId": "p25-c16f205107", "page": 25, "quote": "Here κ is the public edge projection at every unmarked slot and the identity at the marked slot;\nρ is the identity at every unmarked slot and the unknown map πe at the marked slot. Thus κ is\nknown to the X-labeler, while ρ is used only in the analysis.\nThe planted zeros imply that every row in Γ and Rnew has zero linear coefficient block in\nthe marked slot. The same holds for E, EΓ, and Γ′ = Γ + EΓ, since the noise masks and context\nchange only multiply by scalars and add rows. Consequently eΓ′ = ρ∗Γ′ is known to the X-labeler:\nit is obtained by leaving every unmarked coefficient and constant coefficient unchanged and\nsetting all marked coefficients to zero. This description does not require πe, including its affine\ntranslation part.\nThe X-labeler forms the bounded raw function" }, { "blockId": "p26-3843e9cd2b", "page": 26, "quote": "h multiplies the output of gate v by (−1)h(el) at every label el. Applying this shift in two distinct\npairs leaves that output, and hence the whole word, unchanged. These operations only translate\nthe free shifts and preserve the context. Character orthogonality therefore shows that every\nshift Fourier index of eG has a common pair sum. This argument holds for each fixed choice of\nall gate tables.\nWord generation commutes with pullback of rows, pointwise in the label. Since πe = ρ ◦κ,\nrestricting each free shift of eG to ρ∗H yields exactly Ge with context Γ′ and the same tables.\nThe map ρ has two-element fibers, so Lemma 8.3 applies and gives\nX" }, { "blockId": "p26-5741bc9606", "page": 26, "quote": "ρ(eq)=q∗\nemeq ≥2−2smq∗(Ge; Γ′, Fout)." }, { "blockId": "p27-a71d41604b", "page": 27, "quote": "Proof of Theorem 1.1. Substituting ρ = −t into (1.1) gives" } ], "occurrenceCount": 19 }, { "symbol": "σ", "meaning": "No explicit definition located by the extractor.", "status": "not-located", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p10-47320fe400", "page": 10, "quote": "σeven(w) = σ(w) + σ(−w)" }, "occurrences": [ { "blockId": "p10-47320fe400", "page": 10, "quote": "σeven(w) = σ(w) + σ(−w)" }, { "blockId": "p10-67e4e14f7d", "page": 10, "quote": "2\n,\nσo(w) = σ(w) −σ(−w)" }, { "blockId": "p10-063154c490", "page": 10, "quote": "cut(σ) = 1" }, { "blockId": "p10-b7dbc8a9a7", "page": 10, "quote": "cut(σ) ≤1" }, { "blockId": "p29-e4deb03dd4", "page": 29, "quote": "Both sides have dimension ℓ+ k0; consequently RC ∩HU = {0} as well. A folded answer is\na linear function σ : RC →F2. Its unfolding on (U, L) is the restriction to L of the unique\nextension\neσU(r + h) = σ(r) + hU(h),\nr ∈RC,\nh ∈HU.\n(A.2)" }, { "blockId": "p29-70734f4f37", "page": 29, "quote": "σ 7−→σ ◦AU + hU ◦BU," } ], "occurrenceCount": 6 }, { "symbol": "τ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, "occurrences": [ { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." }, { "blockId": "p5-f046a92b79", "page": 5, "quote": "The convergence follows by first choosing a finite degree cutoff to bound the tail tj, and then\ntaking u close to one on the remaining degrees. Choose u so that this error is at most ξ/2. Let\nδ > 0 be the untruncated influence threshold for error ξ/2. Choose K with u2(K+1) ≤δ/2, and\ntake τ = δ/2. Then\nInfa(hu) ≤Inf≤K\na\n(h) + u2(K+1)∥h∥2\n2 < δ." }, { "blockId": "p18-0ab17157e8", "page": 18, "quote": "Proof. Excess stability first produces an influential slice entry. We retain that influence after the\nunsplit projection, discard the part even in its gate table, and express the remaining influence\nas an affine hash of vectors indexed by labels. The hashing lemma then forces one label to carry\nsubstantial conditional mass.\nAll constants chosen below are independent of A and of the law of ω. The degree cutoff\nK and influence threshold τ are chosen first. We leave s, m unspecified during the estimates:\nStep 4 will choose s to control the even and hashed contributions, and then m to control the\ncost of the unsplit projection." }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p18-65f0ea897b", "page": 18, "quote": "some entry z ∈Fs\n2 of some v ∈D has Inf≤K\nv,z ( ¯fω) ≥τ. Indeed, if the probability of this event is\np, the expected stability is at most B(t) + δ/2 + p." }, { "blockId": "p19-f2ddd11aec", "page": 19, "quote": "v∈D\n∥(1 −Sv) ¯fω∥2\n2,R,F ≤\ns\n2emµ ≤ητ" }, { "blockId": "p19-d0b22816e9", "page": 19, "quote": "For any fixed s, the last inequality can be ensured by increasing m. The same expectation is the\nexpectation over ω, D, R, Fout of the sum of the corresponding conditional squared norms in FD.\nMarkov’s inequality shows that this sum is at most τ/4 except on an event of probability η/2.\nOn the intersection with the influential-entry event, the triangle inequality for the projection\ndefining that influence gives" }, { "blockId": "p19-d3d4a81378", "page": 19, "quote": "Inf≤K\nv,z ( ¯fω) −∥(1 −Sv) ¯fω∥2,FD ≥\n√τ" }, { "blockId": "p19-377627618d", "page": 19, "quote": "v∈D\nmax\nz\nInfv,z(Hv) ≥ητ" }, { "blockId": "p31-17b5d598ec", "page": 31, "quote": "τr = min{C/2, 1/2},\nτi = 10−9τ 12\ni+1\n(0 ≤i < r)." }, { "blockId": "p31-b1ccf446e3", "page": 31, "quote": "ζr = τ0/2,\nζi = 10−9ζ12\ni+1\n(0 ≤i < r)." }, { "blockId": "p31-d15ae732c7", "page": 31, "quote": "2q−ℓ≤τ0/4.\n(A.6)" }, { "blockId": "p31-12962946c5", "page": 31, "quote": "Assume q ≤ℓ, 3k0 −r > ℓ, (A.6), and ek0 ≤τ0/4. For Q ⊆W ⊆X of codimension i ≤r, say\nthat (g, W) occurs if g : W →F2 is linear and" }, { "blockId": "p32-32ec19b100", "page": 32, "quote": "dW + (1 −dW )2q−ℓ≤ek0 + 2q−ℓ≤τ0/2 < τi," }, { "blockId": "p32-a6e82621e3", "page": 32, "quote": "ek0 ≤τ0/4,\n2q−k0 ≤δα(ℓ)/8.\n(A.8)" }, { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, { "blockId": "p34-be24debe49", "page": 34, "quote": "Here L has dimension ℓ. Set S0 = W ∩VV and γ = f|Q. Every equality FV [L] = f|L in this\ndisplay is retained in Tγ. Since ζi ≤τ0/2 ≤τi/2, the quotient pair" }, { "blockId": "p34-2d55acb379", "page": 34, "quote": "c∗:= 2−r 0.3τ0" } ], "occurrenceCount": 18 }, { "symbol": "ψ", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p36-5f81808622", "page": 36, "quote": "There are no edges within clusters. This defines a simple graph on Np6 vertices, constructible\nby enumerating all possible pairs. Take Q = p12.\nWe first estimate its edge count on arbitrary rectangles between two clusters. Fix a residue\nset I and subsets S, T ⊆F6\np. Writing ψ(a) = exp(2πia/p), Fourier inversion gives" }, "occurrences": [ { "blockId": "p36-5f81808622", "page": 36, "quote": "There are no edges within clusters. This defines a simple graph on Np6 vertices, constructible\nby enumerating all possible pairs. Take Q = p12.\nWe first estimate its edge count on arbitrary rectangles between two clusters. Fix a residue\nset I and subsets S, T ⊆F6\np. Writing ψ(a) = exp(2πia/p), Fourier inversion gives" }, { "blockId": "p36-66fc5872b6", "page": 36, "quote": "a∈I\nψ(−ha),\n|βh| ≤1." }, { "blockId": "p36-9ce14bea1c", "page": 36, "quote": "For h ̸= 0, the matrix Mh(x, y) = ψ(hx · y) satisfies" } ], "occurrenceCount": 3 }, { "symbol": "ω", "meaning": "Candidate definition context; inspect the source.", "status": "candidate", "scope": "Unresolved; symbols can be rebound in different sections.", "source": { "blockId": "p17-a7689c913b", "page": 17, "quote": "This section contains the analytic statement used in the reduction. Its constants do not depend\non the dimension of the affine label space. The functions may depend on an additional random\nparameter ω, whose law is independent of all rows, gate tables, and the random slice. In the\napplication, ω will be a tuple of right endpoints of the game.\nLet ϕω : {−1, 1}A →[−1, 1] be odd, meaning ϕω(−w) = −ϕω(w), and set" }, "occurrences": [ { "blockId": "p17-a7689c913b", "page": 17, "quote": "This section contains the analytic statement used in the reduction. Its constants do not depend\non the dimension of the affine label space. The functions may depend on an additional random\nparameter ω, whose law is independent of all rows, gate tables, and the random slice. In the\napplication, ω will be a tuple of right endpoints of the game.\nLet ϕω : {−1, 1}A →[−1, 1] be odd, meaning ϕω(−w) = −ϕω(w), and set" }, { "blockId": "p17-c44d5d344e", "page": 17, "quote": "fω(R, F) = ϕω(W(R, F)),\n¯fω = Trfω,\nr = 1 −µ/d." }, { "blockId": "p17-01d2e48829", "page": 17, "quote": "The word, tree, and random slice D are those of Section 3. Write Fout for the tables outside D.\nOnce ω, D, R, Fout are fixed, ¯fω is a function of the independent slice-table bits." }, { "blockId": "p18-ea72a11348", "page": 18, "quote": "The dependence on v, D, and ω is suppressed. The sets of indices in this definition are disjoint\nas q varies. For a fixed q, the mass includes every way to distribute that common sum between\nthe two children of each pair. We seek such mass for the original smoothed function ¯fω. The\nunsplit projection and a table-degree cutoff will serve only to find a label carrying this mass;\nneither projection is part of the conclusion. Keeping all the indices with a given common sum\nwill let us compare masses under affine pullback in Section 8.\nIn particular, if |f| ≤1, then at every fixed context" }, { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" }, { "blockId": "p18-01097b485f", "page": 18, "quote": "Here we sample ω, the random slice D, and a uniform node v ∈D; conditional on these, we\nsample Fout and Γ independently and uniformly." }, { "blockId": "p18-0ab17157e8", "page": 18, "quote": "Proof. Excess stability first produces an influential slice entry. We retain that influence after the\nunsplit projection, discard the part even in its gate table, and express the remaining influence\nas an affine hash of vectors indexed by labels. The hashing lemma then forces one label to carry\nsubstantial conditional mass.\nAll constants chosen below are independent of A and of the law of ω. The degree cutoff\nK and influence threshold τ are chosen first. We leave s, m unspecified during the estimates:\nStep 4 will choose s to control the even and hashed contributions, and then m to control the\ncost of the unsplit projection." }, { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, { "blockId": "p19-d0b22816e9", "page": 19, "quote": "For any fixed s, the last inequality can be ensured by increasing m. The same expectation is the\nexpectation over ω, D, R, Fout of the sum of the corresponding conditional squared norms in FD.\nMarkov’s inequality shows that this sum is at most τ/4 except on an event of probability η/2.\nOn the intersection with the influential-entry event, the triangle inequality for the projection\ndefining that influence gives" }, { "blockId": "p20-56ef8758e5", "page": 20, "quote": "Step 3: express odd influences as affine hashes.\nFix ω, D, v, Γ, Fout. The remaining row\ncoordinates Lgb are independent uniform elements of H. On the odd part, the common pair\nsum q is a label, by (2.1), and hence is nonzero. An unsplit shift index with common sum q\nputs q in the second child on a subset I ⊆[s] and in the first child elsewhere. Let aq,I(FD) be\nits coefficient in Hodd\nv\n.\nSet" }, { "blockId": "p21-a65ea046f4", "page": 21, "quote": "For each node, condition on ω, D, v, Γ, Fout and integrate the free shifts. Partition these\ncontexts into three classes:" }, { "blockId": "p30-8b8d7dd945", "page": 30, "quote": "P(ω),\nqedge =\nX" }, { "blockId": "p30-729a768a81", "page": 30, "quote": "ω∈E\nω7→e" } ], "occurrenceCount": 13 } ], "issues": [ { "id": "issue-1", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p3-5f5b709dd7", "page": 3, "quote": "the completeness error is made arbitrarily small. Theorem 2.2 states this input. Appendix A\nderives it from the published construction, specifying both the valid-subspace conditioning\nand the auxiliary candidate lists used in the soundness proof.\nTogether with Majority Is\nStablest, that construction supplies the external foundation of the argument. The Unique Games\nConjecture enters only as historical context. The tree Fourier analysis, hashing bound, and\nhidden-coordinate decoding are proved here." }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-2", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p4-2adef6757d", "page": 4, "quote": "The resulting decoding loss is independent of the source alphabet. This is essential: the\nalphabet dimension depends on the soundness we request, so a dimension-dependent loss could\nmake the soundness choice circular. We first fix that loss and the source soundness, then the\nalphabet and the number of product factors, and finally the source completeness error. The\nquantifier order in Theorem 2.2 permits exactly this sequence." }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-3", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p6-de8818c7d6", "page": 6, "quote": "In this paper the alphabets are Fn+1\n2\nand Fn\n2, and every πe : Fn+1\n2\n→Fn\n2 is surjective affine.\nTheorem 2.2 supplies NP-hard gaps with this structure. We discard endpoints with zero marginal\nprobability whenever conditional edge distributions are used.\nThe starting theorem has the quantifier order" }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-4", "kind": "label-mismatch", "message": "Text says Theorem B.1; candidate heading is Lemma B.1. Check the mismatch.", "source": { "blockId": "p6-7b93d7ee44", "page": 6, "quote": "A weighted graph is likewise specified by a rational distribution on unordered edges, allowing\nloops. A cut assigns signs to its vertices and satisfies an edge when its signs differ. Its value is\nits satisfaction probability. Loops have zero cut weight. The conversion in Theorem B.1 keeps a\ncommon scale even when loops are deleted." }, "related": [ { "blockId": "p35-be0e319110", "page": 35, "quote": "Lemma B.1. Let G be a graph on N vertices with nonnegative rational weights wuv on unordered\ndistinct vertex pairs, and P\nu 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" } ] }, { "id": "issue-5", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p9-39b0cb63ad", "page": 9, "quote": "Fix a rational t ∈(0, 1), tree parameters s, m, a rational 0 < µ < 1/2, and r = 1 −µ/d. The\nparameters will be chosen in Section 9. Let G be an affine game as in Theorem 2.2, with edge\nlaw λ and alphabet dimensions n + 1, n. Let T ≥1 be an integer. We use the product label\nspaces\nB = (Fn+1\n2\n)T ,\nA = (Fn\n2)T ," }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-6", "kind": "label-mismatch", "message": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch.", "source": { "blockId": "p9-1809233374", "page": 9, "quote": "2. Independently of all game data, draw a random slice D. Draw F, F ′ as in Theorem 3.2:\ncorresponding entries are independent t-correlated uniform pairs on D, and identical\nuniform pairs off D." }, "related": [ { "blockId": "p7-aa2f88a795", "page": 7, "quote": "Lemma 3.2 (Complementary traversal and correlation). For every fixed collection of tables\nF, the root output as a multilinear polynomial in the k leaf signs has only degree-d monomials.\nEach such monomial chooses one child from every pair at every node that it reaches. Every\nresulting traversal meets every possible slice D at exactly one node. Consequently, negating all\nthe tables on a slice negates W.\nLet 0 < t < 1 and 0 < r < 1. Draw F uniformly, and construct F ′ by leaving the tables\noutside D unchanged and coupling each pair of corresponding table entries on D as uniform signs\nof correlation t, independently across entries. Independently draw R ∈Hk uniformly, and form\nR1, R2 by retaining each row of R with probability r and otherwise replacing it by an independent\nuniform row, independently in the two copies. Then, for every a ∈A and every fixed slice," } ] }, { "id": "issue-7", "kind": "label-mismatch", "message": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch.", "source": { "blockId": "p10-0e07627ed1", "page": 10, "quote": "Each coordinate of either sampled edge tuple has the original edge law.\nA union bound\nshows that all 2T sampled constraints are satisfied except with probability at most 2Ta.\nWhen all these constraints are satisfied, the two queried tuples project to the same label\nl = (ℓY (y1), . . . , ℓY (yT )) ∈A. The cut signs then have product −W(R1, F)(l)W(R2, F ′)(l). The\nword-generation data are independent of the game edges, so Theorem 3.2 gives conditional cut\nprobability (1 + tr2d)/2. On the remaining event the cut probability is nonnegative. This proves\nthe first bound when 2Ta ≤1; if 2Ta > 1 that bound is negative and is automatic. Finally,\nr2d ≥1 −2µ, and the loss due to the prefactor is at most 2Ta, giving the second bound." }, "related": [ { "blockId": "p7-aa2f88a795", "page": 7, "quote": "Lemma 3.2 (Complementary traversal and correlation). For every fixed collection of tables\nF, the root output as a multilinear polynomial in the k leaf signs has only degree-d monomials.\nEach such monomial chooses one child from every pair at every node that it reaches. Every\nresulting traversal meets every possible slice D at exactly one node. Consequently, negating all\nthe tables on a slice negates W.\nLet 0 < t < 1 and 0 < r < 1. Draw F uniformly, and construct F ′ by leaving the tables\noutside D unchanged and coupling each pair of corresponding table entries on D as uniform signs\nof correlation t, independently across entries. Independently draw R ∈Hk uniformly, and form\nR1, R2 by retaining each row of R with probability r and otherwise replacing it by an independent\nuniform row, independently in the two copies. Then, for every a ∈A and every fixed slice," } ] }, { "id": "issue-8", "kind": "label-mismatch", "message": "Text says Theorem 3.2; candidate heading is Lemma 3.2. Check the mismatch.", "source": { "blockId": "p10-12c4ff12f9", "page": 10, "quote": "where ϕy is odd and depends on no slice or row data. Thus ¯Gy has mean zero in the slice entries\nfor every fixed y, D, R, Fout, by the observation following Theorem 3.2. The later extraction\nargument, Theorem 7.1, applies to precisely such bounded odd functions of the generated word." }, "related": [ { "blockId": "p7-aa2f88a795", "page": 7, "quote": "Lemma 3.2 (Complementary traversal and correlation). For every fixed collection of tables\nF, the root output as a multilinear polynomial in the k leaf signs has only degree-d monomials.\nEach such monomial chooses one child from every pair at every node that it reaches. Every\nresulting traversal meets every possible slice D at exactly one node. Consequently, negating all\nthe tables on a slice negates W.\nLet 0 < t < 1 and 0 < r < 1. Draw F uniformly, and construct F ′ by leaving the tables\noutside D unchanged and coupling each pair of corresponding table entries on D as uniform signs\nof correlation t, independently across entries. Independently draw R ∈Hk uniformly, and form\nR1, R2 by retaining each row of R with probability r and otherwise replacing it by an independent\nuniform row, independently in the two copies. Then, for every a ∈A and every fixed slice," } ] }, { "id": "issue-9", "kind": "label-mismatch", "message": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch.", "source": { "blockId": "p10-12c4ff12f9", "page": 10, "quote": "where ϕy is odd and depends on no slice or row data. Thus ¯Gy has mean zero in the slice entries\nfor every fixed y, D, R, Fout, by the observation following Theorem 3.2. The later extraction\nargument, Theorem 7.1, applies to precisely such bounded odd functions of the generated word." }, "related": [ { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" } ] }, { "id": "issue-10", "kind": "label-mismatch", "message": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch.", "source": { "blockId": "p11-164121ec4c", "page": 11, "quote": "If a cut has value at least (1 + B(t))/2 + δ/2, then its odd average has expected slice stability\nat least B(t) + δ. The next three sections show that this excess stability yields a label carrying\nsubstantial Fourier mass, with constants independent of the game dimension and the number of\nrepetitions; the formal statement is Theorem 7.1. Section 8 then turns that mass into a labeling\nof a single copy of the original game." }, "related": [ { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" } ] }, { "id": "issue-11", "kind": "label-mismatch", "message": "Text says Theorem 5.3; candidate heading is Lemma 5.3. Check the mismatch.", "source": { "blockId": "p18-8ee1a693e1", "page": 18, "quote": "For a node v ∈D, use the free shift coordinates (Lgb)g∈[s],b∈{0,1} and the remaining context\nΓ ∈Hk−2s from Theorem 5.3. A hat marked “shifts” expands only these 2s free rows. For a\nlabel q ∈A (equivalently, q ∈H∗with q(1) = 1), define" }, "related": [ { "blockId": "p13-1062f9bea6", "page": 13, "quote": "Lemma 5.3 (Free-shift coordinates). The change of coordinates" } ] }, { "id": "issue-12", "kind": "label-mismatch", "message": "Text says Theorem 2.1; candidate heading is Lemma 2.1. Check the mismatch.", "source": { "blockId": "p18-bdd509562c", "page": 18, "quote": "Step 1: find an influential slice entry.\nSimultaneously negating the slice tables negates the\nword. Since ϕω is odd and smoothing preserves this identity, ¯fω has mean zero in the slice bits\nfor every fixed ω, D, R, Fout. It also lies in [−1, 1]. Apply Theorem 2.1 with ξ = δ/2, obtaining\nK ≥1 and τ > 0. We may decrease τ to ensure τ ≤1. Since stability is at most one, (7.3)\nimplies that, with probability at least\nη = δ/2," }, "related": [ { "blockId": "p5-0de0ba404e", "page": 5, "quote": "Lemma 2.1 (Majority Is Stablest with a degree cutoff). Fix 0 < t < 1 and ξ > 0. There are an\ninteger K ≥1 and τ > 0, depending only on t, ξ, such that every h : {−1, 1}Ω→[−1, 1] with\nEh = 0 satisfies\nInf≤K\na\n(h) < τ for all a ∈Ω\n\n=⇒\nStabt(h) ≤B(t) + ξ." } ] }, { "id": "issue-13", "kind": "label-mismatch", "message": "Text says Theorem 5.2; candidate heading is Lemma 5.2. Check the mismatch.", "source": { "blockId": "p19-52d05b19a0", "page": 19, "quote": "By Theorem 5.2, the required bound on the projection loss is" }, "related": [ { "blockId": "p12-77d0658e91", "page": 12, "quote": "Lemma 5.2 (Average cost of the unsplit projection). Let f be a function of the code with |f| ≤1,\nlet r = 1 −µ/d with 0 < µ < 1/2, and let ¯f = Trf. Choose the random slice independently of f.\nThen\nEJ0,D\nX" } ] }, { "id": "issue-14", "kind": "label-mismatch", "message": "Text says Theorem 5.1; candidate heading is Lemma 5.1. Check the mismatch.", "source": { "blockId": "p19-20ff4ebca5", "page": 19, "quote": "By the sign action in Theorem 5.1, the even part has qv(1) = 0 on row support. Since the\nindex is unsplit, all its child indices are either 0 or qv, so they too take value zero on 1. The\ndomain-translation action in that lemma shows that the even part is invariant under translations\nof the domain of Fv, with the rows and outside tables fixed. These translations act transitively\non its entries. All entry influences at v are therefore equal. By (7.7)," }, "related": [ { "blockId": "p11-9c0d842b01", "page": 11, "quote": "Lemma 5.1 (Row symmetries). Let f be a function of the code. Every nonzero row Fourier\ncoefficient of f satisfies, at each internal node v," } ] }, { "id": "issue-15", "kind": "label-mismatch", "message": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch.", "source": { "blockId": "p21-2a433dcc94", "page": 21, "quote": "The factor |B| counts the entries whose influences contain that Fourier monomial; it is at most\nthe monomial’s total degree, which is at most K. Thus Mv ≤K at every context. The total-mass\nidentity in Theorem 6.1, together with (7.11), also gives" }, "related": [ { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." } ] }, { "id": "issue-16", "kind": "label-mismatch", "message": "Text says Theorem 6.1; candidate heading is Lemma 6.1. Check the mismatch.", "source": { "blockId": "p21-5c0072601e", "page": 21, "quote": "Step 4: force a heavy label.\nChoose ζ > 0 with ζK ≤Λ/4. Apply Theorem 6.1, obtaining\nγ > 0 and a lower bound on s. Choose s ≥2 large enough for that lemma and for K2−s ≤Λ/4.\nThen choose m as in (7.5). Finally put" }, "related": [ { "blockId": "p14-77b34dc6fe", "page": 14, "quote": "Lemma 6.1 (Vector-valued hashing). Let H be a finite-dimensional vector space over F2, write\nχq(u) = (−1)q(u) for q ∈H∗and u ∈H, let V be a real Hilbert space, let s ≥1 be an integer,\nand let\nc(q, z) ∈V,\nq ∈H∗,\nz ∈Fs\n2." } ] }, { "id": "issue-17", "kind": "label-mismatch", "message": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch.", "source": { "blockId": "p26-afd1cf8192", "page": 26, "quote": "Proof of Theorem 1.2. Fix t and ε as in the theorem. Take δ = ε. This satisfies 0 < δ < 1−B(t)\nbecause ε < (t −B(t))/4 < (1 −B(t))/4. Choose a rational 0 < µ < 1/2 with tµ ≤ε/4. Apply\nTheorem 7.1 with t, δ, µ, fixing s, m, p∗, θ, and hence k, d, r. Choose a rational" }, "related": [ { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" } ] }, { "id": "issue-18", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p26-842530a02d", "page": 26, "quote": "By Theorem 2.2, this fixes an alphabet dimension n for which the game has arbitrarily small\ncompleteness error. Choose an integer T satisfying" }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-19", "kind": "label-mismatch", "message": "Text says Theorem 4.1; candidate heading is Lemma 4.1. Check the mismatch.", "source": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "related": [ { "blockId": "p9-a96718c718", "page": 9, "quote": "Lemma 4.1 (Completeness). If Val(G) ≥1 −a, the graph has a cut of value at least" } ] }, { "id": "issue-20", "kind": "label-mismatch", "message": "Text says Theorem 4.2; candidate heading is Lemma 4.2. Check the mismatch.", "source": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "related": [ { "blockId": "p10-b68ce20ed0", "page": 10, "quote": "Lemma 4.2 (Stability identity). For every cut of the constructed graph," } ] }, { "id": "issue-21", "kind": "label-mismatch", "message": "Text says Theorem 7.1; candidate heading is Proposition 7.1. Check the mismatch.", "source": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "related": [ { "blockId": "p18-ce4b4357f2", "page": 18, "quote": "Proposition 7.1 (Dimension-independent extraction). Fix t ∈(0, 1), 0 < δ < 1 −B(t), and\n0 < µ < 1/2. There are tree parameters s ≥2, m ≥1 and constants p∗, θ > 0, depending only on\nt, δ, µ, with the following property. For any affine label space A and any family ϕω as above, if" } ] }, { "id": "issue-22", "kind": "label-mismatch", "message": "Text says Theorem 8.4; candidate heading is Proposition 8.4. Check the mismatch.", "source": { "blockId": "p27-fbd085fe50", "page": 27, "quote": "In the YES case, Theorem 4.1 gives cut value at least (1 + t)/2 −ε/2, which is stronger than\nrequired. In the NO case, suppose a cut had value at least (1+B(t))/2+ε/2. By Theorem 4.2, its\nodd average has expected slice stability at least B(t)+ε. Theorems 7.1 and 8.4 would then imply\nVal(G) ≥2−2sp∗θ/2 > b, a contradiction. Thus every cut has value less than (1 + B(t))/2 + ε/2.\nFor completeness, the test is a deterministic polynomial-size graph construction. The vertex\nset has\n|X|T 2|B| = |X|T 22(n+1)T" }, "related": [ { "blockId": "p24-0df0c48959", "page": 24, "quote": "Proposition 8.4 (Decoding). Suppose the extraction conclusion of Proposition 7.1 holds with\nconstants p∗, θ > 0. If T satisfies Equation (8.6), then the original affine game has value at least" } ] }, { "id": "issue-23", "kind": "label-mismatch", "message": "Text says Theorem B.1; candidate heading is Lemma B.1. Check the mismatch.", "source": { "blockId": "p27-7abf2de4cc", "page": 27, "quote": "Apply Theorem 1.2. Delete loops while retaining the original weight scale, so that the remaining\nweights sum to at most one. Apply Theorem B.1 with error ϵ′, producing a simple unweighted\ngraph whose maximum cut, on the common scale Q, differs from the original maximum cut\nweight by at most ϵ′. An α-approximation algorithm on that graph returns, in the YES case, a\ncut of scaled value at least the left side of (9.2). In the NO case, no cut has scaled value greater\nthan its right side. A fixed rational threshold strictly between them therefore distinguishes the\nNP-hard promise cases. This proves the theorem." }, "related": [ { "blockId": "p35-be0e319110", "page": 35, "quote": "Lemma B.1. Let G be a graph on N vertices with nonnegative rational weights wuv on unordered\ndistinct vertex pairs, and P\nu 0, one can construct in deterministic\npolynomial time a simple unweighted graph G′ and an integer scale Q such that\n\nMaxCut(G′)" } ] }, { "id": "issue-24", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p28-0d80eef515", "page": 28, "quote": "Starting instance.\nTheorem 4.1 of [6], obtained from Håstad’s theorem [10], supplies an\nabsolute constant s∗< 1 such that for every sufficiently small ε > 0 it is NP-hard to distinguish\nregular binary 3Lin instances of value at least 1 −ε from those of value at most s∗. Regularity\nincludes bounded variable occurrence, three distinct variables per equation, and at most one\nvariable shared by two distinct equations.\nFor now let ℓ≥2, k0 ≥ℓ, and β ∈(0, 1) be fixed parameters, with β rational. The two\nalphabet dimensions will be ℓand ℓ−1, so the dimension in Theorem 2.2 is n = ℓ−1. The\nsoundness argument below chooses ℓ, k0, β using only b and the absolute constant s∗." }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-25", "kind": "external-source-unlocated", "message": "Lemma 3.7 is explicitly external; identify and inspect the cited source before using it.", "source": { "blockId": "p33-c92a3e0cf5", "page": 33, "quote": "and let E3 be nonemptiness of the Q-list of FU at thresholds (τi). Event E2 is precisely the\nsmoothness condition in the source’s Equation (4.3). For E3, Lemma 3.7 of the source supplies\nan occurring pair for an α(ℓ) fraction of uniform Q whenever E1 holds; any such pair has a\nmaximal occurring extension. This uses only existence of the first list, not a bound on its\nsize. The cited arguments give these events joint probability at least δα(ℓ)/4. Intersecting\nwith E4 = {Q ∩HU = {0}} leaves probability at least δα(ℓ)/8, by (A.7)–(A.8). The event E4\ndepends only on U, Q. Consequently it does not alter the subsequent conditional law of V after\nU, Q have been fixed." }, "related": [] }, { "id": "issue-26", "kind": "label-mismatch", "message": "Text says Theorem A.1; candidate heading is Lemma A.1. Check the mismatch.", "source": { "blockId": "p34-f21c3e202c", "page": 34, "quote": "Success of the two strategies.\nThe first player chooses a uniform maximal pair (f, W) from\nthe τ-list of FU and extends f uniformly to VU, using an arbitrary assignment if its list is empty.\nThis finite list need not have a size bound independent of k0: the success estimate will hold for\nevery one of its elements. The second player uses the quotient lists just constructed.\nFix U, Q in E1 ∩E2 ∩E3 ∩E4, and any first-player choice (f, W), with i = codimVU W ≤r.\nBy Theorem A.1, HU ⊆W and f|HU = hU. Every extension chosen by the first player therefore\nsatisfies all equations of U. This is the only use of the invariant-list assertion in the source’s\ndecoding argument.\nThe covering step of [6, Lemmas 4.11 and 5.5 and the proof of Lemma 5.3], using (A.5),\ngives probability at least 0.3τi ≥0.3τ0 over V that" }, "related": [ { "blockId": "p31-89e490beb0", "page": 31, "quote": "Lemma A.1 (The invariant-list assertion on admissible advice spaces). Let X have dimension\n3k0, let H ≤X have dimension k0, and let h : H →F2 be linear. Suppose a table F assigns\na linear function to every ℓ-space, and that for every (ℓ+ k0)-space K ⊇H there is a linear\nfunction fK : K →F2 such that fK|H = h and F[L] = fK|L whenever L + H = K. Fix a\nq-space Q with Q ∩H = {0}, and set" } ] }, { "id": "issue-27", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p34-80a9431745", "page": 34, "quote": "Fixing soundness before completeness.\nFor the game in Theorem 2.2, take δ = b/4 and\nretain the valid raw weights. Their total mass qedge is at least 1/2 by the preceding rejection\nestimates. Normalization therefore gives" }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] }, { "id": "issue-28", "kind": "label-mismatch", "message": "Text says Theorem 2.2; candidate heading is Proposition 2.2. Check the mismatch.", "source": { "blockId": "p35-5bbe01a21e", "page": 35, "quote": "Effectivity and size.\nAll spaces in an individual question have bounded dimension. There\nare polynomially many k0-tuples of input equations, and each tuple has only constantly many\nsubspaces, bases, and local sampling outcomes. The folding equivalence relation is explicitly\ntestable by linear algebra. To find RC effectively, start from any (U, L) ∈C and enumerate\nthe ℓ-spaces inside L + HU, whose dimension is the fixed constant ℓ+ k0; test each candidate\nagainst (A.1) for every class member. Lemma 4.2 guarantees a successful candidate. Gaussian\nelimination then computes bases and the affine edge maps. The finite sampling distributions,\ntheir conditioning, and the aggregation of parallel edges have rational weights of polynomial\nbit length. These observations give the deterministic polynomial reduction claimed in the\nproposition. This completes the proof of Theorem 2.2." }, "related": [ { "blockId": "p6-b920c0ecc8", "page": 6, "quote": "Proposition 2.2 (Affine 2-to-1 Games Theorem). For every constant b ∈(0, 1) there is an\ninteger n ≥1 such that, for every constant a ∈(0, 1 −b), the following promise problem is\nNP-hard. Its input is a weighted bipartite game with vertex sets X, Y , respective label sets\nFn+1\n2\n, Fn\n2, and a surjective affine map" } ] } ], "coverage": { "pages": 38, "blocks": 691, "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 } }