| { |
| "source": "claude/01_MAP.md §2, librarian pass of 2026-08-25. Evidence: P proved · C computed · H heuristic · M measured law · X cited · R replicated by a later independent implementation.", |
| "results": [ |
| {"id": "A2.L2.2", "lane": "A2", "line": "degenerate ⟺ constant on an affine hyperplane ⟺ AI(f) ≤ 1", "evidence": "P"}, |
| {"id": "A2.A", "lane": "A2", "line": "stagewise degeneracy on the current affine subspace ⇒ hole in poly time", "evidence": "P"}, |
| {"id": "A2.B", "lane": "A2", "line": "AND-of-parities ⇒ Avoid ∈ P at m=n+1, no locality/degree/size bound; ⊋ NC⁰₂", "evidence": "P,C,R"}, |
| {"id": "A2.C", "lane": "A2", "line": "hereditarily-degenerate gates never stuck; classification 4/16/200/10584 degenerate, the 56 excluded 3-gates = one AGL(3,2)+complement orbit MAJ₃ ≅ z₁+z₂z₃", "evidence": "P,C,R"}, |
| {"id": "A2.6.1", "lane": "A2", "line": "quadratic degenerate ⟺ rank 0, or rank 2 with Ĝ ≠ 0 (67,728 quadratics checked)", "evidence": "P,C,R"}, |
| {"id": "A2.E2", "lane": "A2", "line": "degree-2 ∈ P at m ≥ εn², threshold tight (= CERT.C = L4.T1 = ALD.T4 = C.T1 = Ünal 2023)", "evidence": "P,C,R"}, |
| {"id": "A2.D", "lane": "A2", "line": "bilinear: range = union of 2^b subspaces; hole ⟺ kernels cover; ⟺ rank-weight balances", "evidence": "P,C"}, |
| {"id": "A2.7", "lane": "A2", "line": "hard class x_a+x_bx_c: 84–97% cracked to n=60", "evidence": "C"}, |
| {"id": "A2.11", "lane": "A2", "line": "2-piece affine fibre = all 256 gates", "evidence": "C,R"}, |
| {"id": "CERT.A", "lane": "CERT", "line": "min-degree monomial of a certificate is a hole, no search", "evidence": "P,C"}, |
| {"id": "CERT.B", "lane": "CERT", "line": "ann ≤ λ ≤ n+1", "evidence": "P,C"}, |
| {"id": "CERT.E", "lane": "CERT", "line": "exact flat count; λ > ρ*/2; Gold map ρ* = n−1", "evidence": "P,C"}, |
| {"id": "CERT.C", "lane": "CERT", "line": "dense degree-k algorithm exp(O((n^k/m)^{1/(k−1)} log n)); degree-2 at n^{1+δ} in exp(Õ(n^{1−δ}))", "evidence": "P,C,R"}, |
| {"id": "CERT.D", "lane": "CERT", "line": "ann(F_φ) ≥ AI(φ); MAJ_n ⇒ certificates cost 2^{Ω(m)}, (0ⁿ,1) a hole in linear time. Certificate cost bounds Avoid from above, never below.", "evidence": "P,C,R"}, |
| {"id": "CERT.6.1", "lane": "CERT", "line": "block cubic ann=3 vs cover (4/3)^k; pentagon ann=λ=1 holds 32 holes", "evidence": "C,R"}, |
| {"id": "CERT.6.2", "lane": "CERT", "line": "every archive family is graph-form", "evidence": "P"}, |
| {"id": "C.def", "lane": "C", "line": "ad(F) = least degree of a nonzero vanishing polynomial = least single-equation region degree", "evidence": "P"}, |
| {"id": "C.T1", "lane": "C", "line": "small-ad ⇒ deterministic Avoid (= CERT.C)", "evidence": "P"}, |
| {"id": "C.T2", "lane": "C", "line": "existence: ad ≤ e when C(M,≤e) > C(N,≤ed) or > |range|; e=2,3 rungs measured sharp", "evidence": "P,C"}, |
| {"id": "C.T3", "lane": "C", "line": "relabeling: ad(σ∘F) ≤ deg(σ⁻¹)·ad(F); the two degrees are unrelated; exactly invariant under affine relabelings", "evidence": "P,C"}, |
| {"id": "C.T5", "lane": "C", "line": "graph maps: RD(g) ≤ ad(F) ≤ RD(g)+1 (Applebaum–Lovett rational degree), to k=5", "evidence": "P,C"}, |
| {"id": "C.T6", "lane": "C", "line": "(x, MAJ_N): ad = ⌈N/2⌉ (maximal) while Avoid is O(N) (= CERT.D)", "evidence": "P,C,R"}, |
| {"id": "C.T7", "lane": "C", "line": "c-robust predicate + peelable hypergraph ⇒ f_T surjective; small expander instances all non-expanding", "evidence": "P,C"}, |
| {"id": "C.Cor8", "lane": "C", "line": "random d-local family with ad = n^{1−o(1)} (not explicit; does not certify hardness)", "evidence": "P,X"}, |
| {"id": "CB.1", "lane": "CB", "line": "graph form ⇒ unit propagation refutes every hole in O(n+t)", "evidence": "P,C"}, |
| {"id": "CB.2", "lane": "CB", "line": "region soundness is coNP-complete; a P-certified cover is just a P-refutation", "evidence": "P"}, |
| {"id": "CB.3", "lane": "CB", "line": "cert_P monotone under post-composition; relabeling-stable where cov_d is not", "evidence": "P,C"}, |
| {"id": "CB.4", "lane": "CB", "line": "cover number and certificate complexity incomparable both ways", "evidence": "P,X"}, |
| {"id": "CB.5", "lane": "CB", "line": "Family L (even-degree 3-uniform, m=n+1): hole in O(n^ω), cov₁=1, yet Res/PCR/SoS need 2^{Ω(n)}", "evidence": "P,X,C,R"}, |
| {"id": "CB.6", "lane": "CB", "line": "F₂-linear = coding theory; no linear map is a hard generator", "evidence": "P"}, |
| {"id": "CB.7", "lane": "CB", "line": "candidate Resolution-hard generator P_d on a c≥2 expander (predicate facts verified d=5,6,7; c≥2 not exhibited at n≤80)", "evidence": "conditional"}, |
| {"id": "L4.T1", "lane": "L4", "line": "AVOID-REL at ν_D(m) > ν_{dD}(n); degree-d ∈ P at m ≥ εn^d (= CERT.C = ALD.T4)", "evidence": "P,C"}, |
| {"id": "L4.L2.1", "lane": "L4", "line": "relation span ⊆ pullback algebra, dim |range|", "evidence": "P"}, |
| {"id": "L4.T3", "lane": "L4", "line": "AVOID-XL(e): adaptive RHS, a Nullstellensatz certificate of multiplier degree ≤ e−2; measured m ≈ ν_e/ν_{e−2}", "evidence": "P,M"}, |
| {"id": "L4.6", "lane": "L4", "line": "P_MST density-2 solved to n=38, kills it as a benchmark", "evidence": "C"}, |
| {"id": "L4.T4", "lane": "L4", "line": "NSdeg affine-invariant, additive under low-degree relabelings; not a router (CERT.D)", "evidence": "P"}, |
| {"id": "PDA.def", "lane": "PDA", "line": "avoiddeg = min_y nsdeg(y), ∃-hole analogue of generator hardness", "evidence": "definition"}, |
| {"id": "PDA.C3.1", "lane": "PDA", "line": "t=0 = the linearization algorithm; why it stalls at n²/2", "evidence": "P"}, |
| {"id": "PDA.3.3", "lane": "PDA", "line": "PC-Greedy degree = D_reg(n,m) (10/11, once below, seed-dependent); NS-Greedy D_reg or +1", "evidence": "C,R"}, |
| {"id": "PDA.4", "lane": "PDA", "line": "heuristic 2^{Θ̃(n²/m)}; m=n+1 ≈ 2^{0.9n}", "evidence": "H"}, |
| {"id": "ALD.L2", "lane": "ALD", "line": "minimal-support extraction: a hole read off any relation, no search; the F₂-distinguisher defect is irrelevant to Avoid", "evidence": "P"}, |
| {"id": "ALD.T4", "lane": "ALD", "line": "degree-d ∈ P at m ≥ εn^d, time n^{O(dD)}; = Ünal EUROCRYPT 2023 Algorithms B₁/B₂", "evidence": "P"}, |
| {"id": "ALD.T7", "lane": "ALD", "line": "monomial maps: relation degree = union-collision; graphs ⇒ degree-2 relation iff girth ≤ 4, degree-3 relation for every m ≥ n (all graphs ≤ 6 vertices)", "evidence": "P,C,R"}, |
| {"id": "ALD.P12", "lane": "ALD", "line": "NC¹-Avoid ≤ degree-3-Avoid at additive stretch (Ishai–Kushilevitz perfect encodings); degree 2 provably out of reach for perfect encodings", "evidence": "P,X"}, |
| {"id": "ALD.9", "lane": "ALD", "line": "cover number of holes ≠ hardness (= CERT.D/SHEAR, third derivation)", "evidence": "C,R"}, |
| {"id": "LA-A.Cor1.3", "lane": "LA-A", "line": "the 256 ternary predicates: 88 affine-fibre / 112 weight-3-or-5 / 56 balanced non-affine (= MAJ, SEL, XOR-AND NPN classes)", "evidence": "P,C,R"}, |
| {"id": "LA-A.ThmA", "lane": "LA-A", "line": "n+1 outputs with an affine fibre ⇒ poly-time hole at minimal stretch (all 88, no locality)", "evidence": "P,C"}, |
| {"id": "LA-A.ThmB", "lane": "LA-A", "line": "private-vertex family ⇒ range membership is 2-SAT on link variables", "evidence": "P,C"}, |
| {"id": "LA-A.CorB.2", "lane": "LA-A", "line": "balanced predicate + pseudoforest link graph ⇒ surjective (Tarsi)", "evidence": "P,C"}, |
| {"id": "LA-A.CorB.3", "lane": "LA-A", "line": "pure XOR-AND private families immune (surjective) at every stretch", "evidence": "P,C,R"}, |
| {"id": "LA-A.XA", "lane": "LA-A", "line": "the joint blind spot is the XA fragment = exactly the GGNS rigid-matrix encoding; Scheme B solves everything else to n=480", "evidence": "P,M"}, |
| {"id": "LAB.T1", "lane": "LA-B", "line": "NC⁰ₖ-Avoid[n,n+1] in O*(2^{γ_k n}), γ_k = c_k/(1+c_k), c_k = C(k,2)/5.769; γ₃ < 0.3422, beats (k−2)/(k−1) for 3 ≤ k ≤ 10", "evidence": "P,C,R"}, |
| {"id": "LAB.C3", "lane": "LA-B", "line": "k=2 poly (Hall block connected ⇒ tw ≤ 2)", "evidence": "P,C"}, |
| {"id": "LAB.P4", "lane": "LA-B", "line": "expander Hall blocks have tw Ω(t): no sparsity route to poly time", "evidence": "P,C"}, |
| {"id": "SHIFT.L1", "lane": "SHIFT", "line": "v ∉ D(F) ⇒ F(x)+v a hole for every x, one evaluation", "evidence": "P,C"}, |
| {"id": "SHIFT.T2", "lane": "SHIFT", "line": "d(F) = m − ls(F); systematic and degenerate are the two ends of one parameter", "evidence": "P"}, |
| {"id": "SHIFT.T3", "lane": "SHIFT", "line": "m*(n) = 2n−2 exactly; every F with m ≥ 2n−1 has a shift, >½ of directions at m=2n", "evidence": "P,C,R"}, |
| {"id": "SHIFT.4", "lane": "SHIFT", "line": "MAJ₃ maps at m=n+1 have a shift with probability Θ(1/n) (P·n ≈ 3.7, n=8..18); size-matched control 0", "evidence": "C,R"}, |
| {"id": "SHEAR.T2", "lane": "SHEAR", "line": "shear group acts transitively on graph-form hole sets; Avoid there is one evaluation", "evidence": "P,C"}, |
| {"id": "SHEAR.3b", "lane": "SHEAR", "line": "a measure fails as a router iff some poly-size easy family makes it grow", "evidence": "P"}, |
| {"id": "SHEAR.T7", "lane": "SHEAR", "line": "block-sum family: cover_d = ∞ (all d), rect = (d+2)^{N/(d+1)}, aff = 2^{N/2} (d=1); shear → 1; Avoid O(N)", "evidence": "P,C,R"}, |
| {"id": "SHEAR.T9", "lane": "SHEAR", "line": "O(M) static check always exists", "evidence": "P"}, |
| {"id": "SHEAR.P10", "lane": "SHEAR", "line": "shift degree δ is the one surviving (computational) invariant", "evidence": "P"}, |
| {"id": "L2.FactB", "lane": "L2", "line": "variety density inside {G=0} = average correlation ≤ ‖(−1)^G‖_{U^{d+1}}", "evidence": "P"}, |
| {"id": "L2.L2", "lane": "L2", "line": "max_{deg≤2}|corr(f, g_k)| = (3/4)^k", "evidence": "P,C,R"}, |
| {"id": "L2.T1", "lane": "L2", "line": "½(4/3)^k ≤ c₂(k) ≤ 2.08k(4/3)^k, lim c₂^{1/k} = 4/3; c₂(1,2,3) = 2,3,{3 or 4}", "evidence": "P,SAT"}, |
| {"id": "L2.3", "lane": "L2", "line": "corner bottleneck an artefact; maximal regions > 2·10⁶", "evidence": "C"}, |
| {"id": "L3.T1", "lane": "L3", "line": "under Sym, every hole-set invariant is a function of |H|; φ_flat(47,6) = 4", "evidence": "P,C"}, |
| {"id": "L3.T2", "lane": "L3", "line": "degree-≤2 bijections generate A_{2^m} ⇒ no degree ladder", "evidence": "P,C,R"}, |
| {"id": "L3.P3", "lane": "L3", "line": "P*_{G_s} ≥ 2 ⟺ an exact sampling lower bound", "evidence": "P"} |
| ] |
| } |
|
|
