{ "snapshot": "2026-09-06", "source": "Acashic Research Institute mathematics division registers", "entries": [ { "id": "MF-002", "kind": "finding", "title": "An upper bound on the multiplicative complexity of a three-column interior adder tile", "title_plain": "The verified three-column interior adder tile uses nine nonlinear products for its stated interface, compared with twelve nonlinear gates reported for the cited MDFA generator.", "formal_line": "MC(g) ≤ 9", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "construction", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-002/", "receipt": "HANDOFF 2.6; verified nine-product circuit comparison via cirbo.", "external_value": "MPC, FHE, ZK, and masking circuit designers.", "statement": "The verified three-column interior adder tile uses nine nonlinear products for its stated interface, compared with twelve nonlinear gates reported for the cited MDFA generator.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "While the 2026 STACS MDFA/cirbo generator is used as the comparison point at twelve reported nonlinear gates, entry MF-077 clarifies that it does not represent the state-of-the-art baseline for AND count.", "condition": "" }, { "id": "MF-004", "kind": "finding", "title": "Functional census and degree bounds for the two-row cyclic model", "title_plain": "An exhaustive census of the two-row cyclic model finds 575,968 systems but only 32,768 distinct Boolean functions, and every realized function has degree at most 3.", "formal_line": "|S₂| = 575,968, |{Φ(S) : S ∈ S₂}| = 32,768, and ∀S ∈ S₂, deg(Φ(S)) ≤ 3", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "symmetry-and-encodings", "school": "boolean function analysis", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-004/", "receipt": "Exhaustive function census plus independent replay; DL cyclic_2row_function_census_results.json and CONT cyclic_2row_census_recheck.json.", "external_value": "Boolean-function analysis and expressiveness of low-row quadratic systems", "statement": "An exhaustive census of the two-row cyclic model finds 575,968 systems but only 32,768 distinct Boolean functions, and every realized function has degree at most 3.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-006", "kind": "finding", "title": "Simultaneous and cyclic witness definitions in the Lean 4.28 kernel", "title_plain": "The pinned Lean 4.28 kernel accepts simultaneous and cyclic witness definitions with soundness, completeness, exact cost, and computable witnesses, with no sorryAx.", "formal_line": "Lean 4.28 accepts simultaneous and cyclic witness definitions with soundness, completeness, exact cost, computable witnesses, no sorryAx", "status": "PROVED", "tier": "lean", "tier_label": "LEAN-VERIFIED", "program": "formal-verification", "school": "formal verification", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-006/", "receipt": "Pinned Lean 4.28 replay of six theorem receipts with no sorryAx; HANDOFF section 2.7 and phase_trellis_p5_verifier_legality_audit.json.", "external_value": "Formal verification and proof-system verifier designers", "statement": "The pinned Lean 4.28 kernel accepts simultaneous and cyclic witness definitions with soundness, completeness, exact cost, and computable witnesses, with no sorryAx.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-008", "kind": "finding", "title": "Affine equivalence classes of minimum-rank eight carry-state encodings", "title_plain": "Under the stated minimum-rank filter, the 40,320 encodings of the eight carry states collapse into two affine-equivalence orbits represented by the natural encoding and orbit29.", "formal_line": "{e ∈ E : e passes minimum-rank filter} / affine equivalence = {natural, orbit29 = [0,1,2,3,4,7,6,5]}", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "symmetry-and-encodings", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "CAVEAT", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-008/", "receipt": "Exact orbit enumeration cited at HANDOFF section 5.1.", "external_value": "Boolean and cryptographic circuit synthesis using affine symmetry reduction of finite-state encodings", "statement": "Under the stated minimum-rank filter, the 40,320 encodings of the eight carry states collapse into two affine-equivalence orbits represented by the natural encoding and orbit29.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-009", "kind": "finding", "title": "Quadratic hull closure of the Z-counter transition relation", "title_plain": "For the Z-counter relation, the boundary-only transition leaves 24 of 192 wrong same-input points in the quadratic hull; retaining garbage g0 or g1 closes the hull but g2 does not, and three canonical quadratic rows are necessary and sufficient.", "formal_line": "|H₂(R_∂) ∩ W_same| = 24; exactly three canonical quadratic rows A·B=C are necessary and sufficient", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-009/", "receipt": "Exhaustive canonical A*B=C enumeration of 524,800 factor pairs per case with 21 graph-vanishing equations, CONT certificates, and replay transcripts.", "external_value": "Boolean relation and quadratic-constraint researchers", "statement": "For the Z-counter relation, the boundary-only transition leaves 24 of 192 wrong same-input points in the quadratic hull; retaining garbage g0 or g1 closes the hull but g2 does not, and three canonical quadratic rows are necessary and sufficient.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-010", "kind": "finding", "title": "An impossibility theorem for the χ₀₁ catalyst and the full-domain p7 tile", "title_plain": "The chi_01 catalyst cannot realize the full-domain p7 tile: the outgoing function has degree 9 and 80 of its 140 top monomials lie outside the ideal generated by incoming c1 and c2.", "formal_line": "deg(f) = 9, 80 of 140 top monomials ∉ I = ⟨c₁c₂⟩ ⟹ χ₀₁ catalyst cannot realize full-domain p7 tile", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-010/", "receipt": "state_feature_transport_census_results.json; two independent implementations.", "external_value": "NONE", "statement": "The chi_01 catalyst cannot realize the full-domain p7 tile: the outgoing function has degree 9 and 80 of its 140 top monomials lie outside the ideal generated by incoming c1 and c2.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-011", "kind": "finding", "title": "On the zero sets of products of affine forms on F₂³", "title_plain": "A product of affine forms on F2^3 cannot vanish on exactly the two states {0,1}, so the pure state-affine-product mechanism behind the zero-slack state-{0,1} wall is proved.", "formal_line": "For affine forms ℓ_1,...,ℓ_m: 𝔽₂³ → 𝔽₂ and P = ∏_{j=1}^m ℓ_j, |Z(P)| ∈ {0,4,6,7,8}", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-011/", "receipt": "Independent exact zero-set proofs from batches A and B; BRIEF section B.1 and two replay receipts.", "external_value": "Boolean-function analysis and algebraic circuit lower-bound research", "statement": "A product of affine forms on F2^3 cannot vanish on exactly the two states {0,1}, so the pure state-affine-product mechanism behind the zero-slack state-{0,1} wall is proved.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-012", "kind": "finding", "title": "An impossibility theorem for pinning the three-input AND under quadratic systems", "title_plain": "For the fixed graph w = x1x2x3, no quadratic system can pin w at the positive input (1,1,1), while extending this obstruction to the carry systems remains conjectural.", "formal_line": "No quadratic system in (x1,x2,x3,w) can pin w=1 at x=(1,1,1) for R=Graph(w=x1x2x3)={(x1,x2,x3,w)∈F2^4:w=x1x2x3}", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-012/", "receipt": "Independent batch proofs and replay receipts; batch B checks all 256 three-input Boolean functions and the AND3 certificate.", "external_value": "R1CS and arithmetization researchers", "statement": "For the fixed graph w = x1x2x3, no quadratic system can pin w at the positive input (1,1,1), while extending this obstruction to the carry systems remains conjectural.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-014", "kind": "finding", "title": "Direct-sum additivity barrier for extension-field batching over F₂ᵐ", "title_plain": "The conjecture is that batching over an extension field F_2^m using Karatsuba or CRT cannot reduce multiplicative cost below direct-sum additivity in this cost model.", "formal_line": "Conjecture MF-014: batching over F₂ᵐ via Karatsuba or CRT cannot beat direct-sum additivity under the register's multiplicative cost model", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "algebraic complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-014/", "receipt": "BRIEF section B.5, probe 04; claimed proof is unverified.", "external_value": "Algebraic-complexity and cryptographic circuit researchers.", "statement": "The conjecture is that batching over an extension field F_2^m using Karatsuba or CRT cannot reduce multiplicative cost below direct-sum additivity in this cost model.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-016", "kind": "finding", "title": "Formal verification of large cryptographic circuits via cone-local proofs", "title_plain": "For large cryptographic circuits, cone-local proofs transported through existing semantic theorems compile without sorryAx in seconds to minutes when monolithic bv_decide is intractable.", "formal_line": "Per-cone proofs composed via existing semantic theorems compile no-sorry in seconds to minutes when monolithic `bv_decide` is intractable", "status": "METHOD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "formal-verification", "school": "formal verification", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-016/", "receipt": "HANDOFF section 4 compilation receipt for cone-local and block-local no-sorry proofs.", "external_value": "Formal-verification practitioners optimizing cryptographic circuits", "statement": "For large cryptographic circuits, cone-local proofs transported through existing semantic theorems compile without sorryAx in seconds to minutes when monolithic bv_decide is intractable.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-017", "kind": "finding", "title": "Sparse exact synthesis of a 69,862-variable allocator", "title_plain": "A sparse exact CEGIS loop that adds only replay-discovered failures and reduces row-order symmetry makes a 69,862-variable allocator tractable in 12 SAT/replay/refine rounds taking 26 seconds.", "formal_line": "Sparse exact CEGIS with 120× row-order symmetry reduction solved a 69,862-variable instance in 12 rounds (26 s); monolithic timed out", "status": "METHOD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "symmetry-and-encodings", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-017/", "receipt": "BRIEF section A.2; 12 SAT/replay/refine rounds with 120x row-order symmetry reduction.", "external_value": "SAT-based exact synthesis of Boolean and XAG circuits", "statement": "A sparse exact CEGIS loop that adds only replay-discovered failures and reduces row-order symmetry makes a 69,862-variable allocator tractable in 12 SAT/replay/refine rounds taking 26 seconds.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-018", "kind": "finding", "title": "A soundness condition for subset-UNSAT certificates under symmetry breaking", "title_plain": "A subset-UNSAT certificate is sound only when the cared-row subset is setwise invariant under the broken symmetry group used by the encoding.", "formal_line": "A subset-UNSAT certificate is sound only when ∀g ∈ G, g(R) = R", "status": "METHOD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "symmetry-and-encodings", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "CAVEAT", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-018/", "receipt": "BRIEF section B.4.5; audit rule applied to four registry certificates.", "external_value": "Formal verification and auditing of symmetry-reduced SAT certificates in exact circuit synthesis", "statement": "A subset-UNSAT certificate is sound only when the cared-row subset is setwise invariant under the broken symmetry group used by the encoding.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-020", "kind": "finding", "title": "Full-domain replay for candidate XAG validity", "title_plain": "ABC overlap counts and partial-domain CEGIS fits can suggest false XAG improvements, so a circuit claim must be checked by replay over the full domain.", "formal_line": "∀x ∈ {0,1}^n, C(x) = f(x)", "status": "METHOD", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "sha-256-record", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-020/", "receipt": "HANDOFF section 3, including three mirage wins and full-domain replay checks.", "external_value": "Boolean circuit synthesis and formal verification practitioners.", "statement": "ABC overlap counts and partial-domain CEGIS fits can suggest false XAG improvements, so a circuit claim must be checked by replay over the full domain.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-022", "kind": "finding", "title": "A profile for quadratic pinning and defect structure", "title_plain": "This diagnostic profiles selected false points using quadratic zero-column counts, outside-hull rejection rank, fibre-sensitive defect projections, and minimum graph-vanishing-quadratic support covers.", "formal_line": "delta2^flip ≤ delta2^vert", "status": "METHOD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-022/", "receipt": "CONT quadratic_pinning_profile_live_results.json, validated on c0c2, ITER19, AND3, and affine Ghost suites.", "external_value": "Researchers auditing quadratic closures of Boolean relations", "statement": "This diagnostic profiles selected false points using quadratic zero-column counts, outside-hull rejection rank, fibre-sensitive defect projections, and minimum graph-vanishing-quadratic support covers.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-023", "kind": "finding", "title": "Exact multiplicative complexity of two exposed-sum ripple chains", "title_plain": "For two length-L ripple chains sharing one operand, exposing every sum and both final carries has exact multiplicative complexity 2L, because the outputs recover all carries and the ordinary construction attains the bound.", "formal_line": "MC(I_L) = 2L", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-023/", "receipt": "Package-3 joint-carry certificate; exact-rank replay for L=1 through 10 and independent Package-3 replay.", "external_value": "Circuit-complexity researchers studying shared-operand adders.", "statement": "For two length-L ripple chains sharing one operand, exposing every sum and both final carries has exact multiplicative complexity 2L, because the outputs recover all carries and the ordinary construction attains the bound.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-024", "kind": "finding", "title": "Rank-one-target deficiency of stationary full covers on GF(2)⁵", "title_plain": "Among five-bit translations that preserve both required semantic parities affinely, only shifts 4 and 20 use all ten spare codewords, and all four corresponding stationary full-cover cases are rank-one-target deficient and dead.", "formal_line": "The condition of using all ten spare codewords holds if and only if delta ∈ {4, 20}", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-024/", "receipt": "Package-3 phase_translation_cover_certificate and CONT affine_translation_covers_delta2_vert_flip scan.", "external_value": "Finite-state Boolean circuit and relation designers", "statement": "Among five-bit translations that preserve both required semantic parities affinely, only shifts 4 and 20 use all ten spare codewords, and all four corresponding stationary full-cover cases are rank-one-target deficient and dead.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-025", "kind": "finding", "title": "The false path 8 -> 0 -> 0 in the natural five-code Ghost-P5 quadratic hull", "title_plain": "The complete 66-equation natural five-code Ghost-P5 quadratic hull has a two-column false path 8 -> 0 -> 0, so repeating that local equation family cannot make it sound.", "formal_line": "No subset of H achieves soundness via repetition alone (two-column sequential composition admits exact false path 8 -> 0 -> 0)", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-025/", "receipt": "Package-3 global quadratic hull composition certificate, package3 replay receipt, and reconciliation receipt.", "external_value": "Quadratic relaxations of Boolean constraints and arithmetization or proof-system research", "statement": "The complete 66-equation natural five-code Ghost-P5 quadratic hull has a two-column false path 8 -> 0 -> 0, so repeating that local equation family cannot make it sound.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-027", "kind": "finding", "title": "Verifier-level audit of finite phase-trellis relations", "title_plain": "It provides a verifier-level audit for finite phase-trellis relations, checking completeness, soundness, and Court-Zero legality, and the tested natural and ITER19 K0 samples fail soundness.", "formal_line": "Subset construction gives exact completeness/soundness checks for finite phase relations; natural sample & ITER19 K0 adapter fail soundness", "status": "METHOD", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "formal-verification", "school": "formal verification", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-027/", "receipt": "A phase-trellis verifier-legality audit and an exact ITER19 K0 adapter replay.", "external_value": "Formal verification and proof-system verifier engineering.", "statement": "It provides a verifier-level audit for finite phase-trellis relations, checking completeness, soundness, and Court-Zero legality, and the tested natural and ITER19 K0 samples fail soundness.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-028", "kind": "finding", "title": "A twelve-wedge prefix obstruction for the period-two carry law in p14 circuits", "title_plain": "For the exact 26-input Cartesian period-two carry law, no circuit whose first twelve gates are the verified input-only wedges can finish with only two further gates.", "formal_line": "Fixing gates 0–11 of a p14 circuit to 12 input-only quadratic wedge generators cannot realize the 26-input Cartesian period-two carry law", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-028/", "receipt": "Exact period-two prefix rank screen with full-table digest and an independent Return C recovery.", "external_value": "XOR-AND circuit-synthesis and multi-operand arithmetic research", "statement": "For the exact 26-input Cartesian period-two carry law, no circuit whose first twelve gates are the verified input-only wedges can finish with only two further gates.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-031", "kind": "finding", "title": "An impossibility theorem for deterministic state-only lifts of the c0c2 quadratic hull", "title_plain": "The complete deterministic state-only lift of the c0c2 relation already spans all 256 Boolean functions of the three state bits, so no further deterministic state-only coordinate can enlarge its degree-two evaluation space and 23,808 old wrong hull points remain.", "formal_line": "No deterministic state-only feature set can repair the relation, as maximal lift replay retains 23,808 of 24,576 old wrong hull points", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "impossibility theorem", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-031/", "receipt": "Exact maximal-lift replay plus c0c2 residue/defect and independent task replay receipts.", "external_value": "Boolean relation and circuit-synthesis researchers", "statement": "The complete deterministic state-only lift of the c0c2 relation already spans all 256 Boolean functions of the three state bits, so no further deterministic state-only coordinate can enlarge its degree-two evaluation space and 23,808 old wrong hull points remain.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-032", "kind": "finding", "title": "Copy-locality of target-bearing products at the rank-tight boundary over GF(2)", "title_plain": "In a rank-tight Boolean direct-sum completion, every genuinely target-bearing product of separated signals must be copy-local, but after one mixed slack product a later cross-copy product can expose a new separated direction.", "formal_line": "For affine combinations L and R of separated signals, if L*R is separated and target-bearing, then L*R is copy-local", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "algebraic complexity", "function_type": "structure theorem", "page_verdict": "CAVEAT", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-032/", "receipt": "Independent proof, exhaustive 65,536-case core check, and multiple component-shell replay receipts.", "external_value": "Algebraic complexity and direct-sum circuit theory", "statement": "In a rank-tight Boolean direct-sum completion, every genuinely target-bearing product of separated signals must be copy-local, but after one mixed slack product a later cross-copy product can expose a new separated direction.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-033", "kind": "finding", "title": "A structural characterization of the c0c2 quadratic defect", "title_plain": "For every public-input fibre, the nonzero in-hull output flips form the two-dimensional set generated by next_c1 and 16 XOR next_c2 times 128, while the global semantic span of those defects has rank three.", "formal_line": "On every one of the 8,192 public-input fibres, the nonzero in-hull output flips are `e1`, `u(x)`, and `e1+u(x)`, where `e1=32` flips `next_c1` and `u(x)=16 XOR next_c2(x)*128`.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-033/", "receipt": "Per-fibre exhaustive replay and independent replay of the twisted-defect certificate.", "external_value": "Boolean relation and algebraic circuit researchers", "statement": "For every public-input fibre, the nonzero in-hull output flips form the two-dimensional set generated by next_c1 and 16 XOR next_c2 times 128, while the global semantic span of those defects has rank three.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-035", "kind": "finding", "title": "A conjectured rank-two bilinear custom gate for the p14 bridge", "title_plain": "A native rank-two bilinear gate could represent the mixed component of the proposed p14 bridge in one row, but this remains an untested conjecture outside the verifier.", "formal_line": "MF-035 is a conjecture.", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "algebraic complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-035/", "receipt": "Pro Request 3 theory return section 9 and p14 one-slack mixed-tensor manifest.", "external_value": "Bilinear circuit and tensor-rank research", "statement": "A native rank-two bilinear gate could represent the mixed component of the proposed p14 bridge in one row, but this remains an untested conjecture outside the verifier.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-036", "kind": "finding", "title": "A census of one-block affine data-split pins for c0c2 hull repair", "title_plain": "An exhaustive census of all 8,184 one-block full-affine data-split/state-addition cases finds that none clears the c0c2 relation's 24,576 old wrong hull points, so this repair route is closed.", "formal_line": "The corrected census spans all 8,184 pointwise-distinct cases.", "status": "EXHAUSTED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-036/", "receipt": "Corrected exhaustive 8,184-case census with state-addition audit, full-affine census, c2 gauge-extension, and verification receipts.", "external_value": "Researchers in Boolean constraint repair and circuit synthesis", "statement": "An exhaustive census of all 8,184 one-block full-affine data-split/state-addition cases finds that none clears the c0c2 relation's 24,576 old wrong hull points, so this repair route is closed.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-037", "kind": "finding", "title": "A wedge-span obstruction for the exact Cartesian period-two carry law", "title_plain": "For the exact period-two carry law, the twelve certified standalone wedges intersect the ten-dimensional nonlinear target quotient in dimension four, forcing at least six gate functions outside their span and ruling out any topology with nine or more wedge-span gates.", "formal_line": "dim(T ∩ W)=4, dim(T/(T ∩ W))=6; every p14 realization requires ≥6 gate functions outside W, ruling out k≥9 gate functions in W", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-037/", "receipt": "Exact wedge-span intersection computation with Return B results and independent continuation replays.", "external_value": "Lower-bound research for XOR-AND and multi-operand arithmetic circuits", "statement": "For the exact period-two carry law, the twelve certified standalone wedges intersect the ten-dimensional nonlinear target quotient in dimension four, forcing at least six gate functions outside their span and ruling out any topology with nine or more wedge-span gates.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-038", "kind": "finding", "title": "On gate savings under repeated period doubling", "title_plain": "If a verifier-legal period-two fusion exists, the conjecture is that its saving Delta_r in C(2r) <= 2C(r) - Delta_r persists or grows under further period doubling.", "formal_line": "Conditional on a verifier-legal p14 fusion, C(2r) ≤ 2 C(r) - Delta_r", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-038/", "receipt": "ZKGOLF status Return 2 report with the proposed period-four test.", "external_value": "Recursive composition of multi-operand arithmetic circuits", "statement": "If a verifier-legal period-two fusion exists, the conjecture is that its saving Delta_r in C(2r) <= 2C(r) - Delta_r persists or grows under further period doubling.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "a verifier-legal p14 fusion, write C(2r) <= 2 C(r) - Delta_r and conjecture that the saving Delta_r persists or grows under 2-/4-/8-period doubling rather than vanishing" }, { "id": "MF-039", "kind": "finding", "title": "A conjectural lower bound for the multiplicative complexity of T2", "title_plain": "The two-copy period-two carry tile may require at least fifteen products, but no proof yet rules out chained products that jointly discharge the proposed obstructions.", "formal_line": "MC(T2) ≥ 15", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-039/", "receipt": "ZKGOLF status Return 2 report; the stated falsifier is a p14 circuit passing all 67,108,864 inputs.", "external_value": "Direct-sum lower bounds in Boolean circuit complexity", "statement": "The two-copy period-two carry tile may require at least fifteen products, but no proof yet rules out chained products that jointly discharge the proposed obstructions.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-040", "kind": "finding", "title": "An upper bound on the multiplicative complexity of the natural (S,S+R) component", "title_plain": "The exact 13-input seven-output natural (S,S+R) carry component has a fully replayed eight-product circuit, establishing only the upper bound MC(component) <= 8.", "formal_line": "MC(component) ≤ 8 for the 13-input, 7-output natural (S,S+R) component", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "construction", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-040/", "receipt": "Return D report and NumPy recheck plus independent Return 6 replay over all 8,192 rows.", "external_value": "Multi-operand arithmetic circuit designers and cryptographic circuit implementers", "statement": "The exact 13-input seven-output natural (S,S+R) carry component has a fully replayed eight-product circuit, establishing only the upper bound MC(component) <= 8.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-042", "kind": "finding", "title": "Exact GL(2,2) symmetry quotient for catalyst parameterization", "title_plain": "Sorting the three nonzero vectors of a mixed two-plane gives a sound GL(2,2) quotient for catalyst parameterization, provided concrete residues are still deduplicated modulo the full signal space.", "formal_line": "Sorting u, v, and u+v spanning an independent two-plane in S/<1> yields the exact GL(2,2) symmetry quotient for catalyst parameterization", "status": "METHOD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "symmetry-and-encodings", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-042/", "receipt": "Return E Pluecker parameterization certificate with Return 6 soundness replay and verification report.", "external_value": "SAT-based exact-synthesis and symmetry-reduction research", "statement": "Sorting the three nonzero vectors of a mixed two-plane gives a sound GL(2,2) quotient for catalyst parameterization, provided concrete residues are still deduplicated modulo the full signal space.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-044", "kind": "finding", "title": "Exact scans of published p8 factors for three maximal mixed p14 skeletons", "title_plain": "For the three named maximal mixed p14 skeletons, exhaustive direct fusions and relaxed published-p8 factor universes contain no seven-gate span of all fourteen targets.", "formal_line": "Zero 14-target spans across 1,920, 1,920, and 38,400 direct-fusion assignments over GF(2) on (7,15), (7,23), and (7,39) skeletons", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-044/", "receipt": "Exact GF(2) scans of 1,920, 1,920, and 38,400 assignments plus relaxed 49, 49, and 60 candidate universes and an independent NumPy recheck.", "external_value": "Exact-synthesis search strategy for multi-operand arithmetic circuits", "statement": "For the three named maximal mixed p14 skeletons, exhaustive direct fusions and relaxed published-p8 factor universes contain no seven-gate span of all fourteen targets.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-045", "kind": "finding", "title": "The kernel of the Boolean product-residue map on separable signal cuts", "title_plain": "For a two-copy separable Boolean signal space, the kernel of the product-residue map is exactly the direct sum of the two local product-map kernels, so cross-tensor-rank-two catalyst planes have unique residues and rank-one collisions come only from decomposable local-kernel planes.", "formal_line": "ker(μ) = ker(μ_0) ⊕ ker(μ_1)", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "algebraic complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-045/", "receipt": "Full 8,192-row component censuses for masks 7, 15, 23, and 39 plus independent kernel audit receipts.", "external_value": "Algebraic-complexity and exact-synthesis researchers", "statement": "For a two-copy separable Boolean signal space, the kernel of the product-residue map is exactly the direct sum of the two local product-map kernels, so cross-tensor-rank-two catalyst planes have unique residues and rank-one collisions come only from decomposable local-kernel planes.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-047", "kind": "finding", "title": "A lower bound on chained products for the period-two p14 target", "title_plain": "Modulo affine functions, the period-two p14 target has rank 10 while standalone input-only products occupy only a rank-4 part, so every p14 needs at least six non-input-only product functions and any topology with at least nine input-only products is impossible.", "formal_line": "rank(T) = 10, rank(T≤2) = 4, dim(T ∩ W) = 4 ⇒ every p14 construction requires ≥ 6 chained or non-input-only product functions", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-047/", "receipt": "Exact target-capture and rank-tight prefix censuses with an independent audit.", "external_value": "Multiplicative-complexity researchers studying direct-sum circuit lower bounds", "statement": "Modulo affine functions, the period-two p14 target has rank 10 while standalone input-only products occupy only a rank-4 part, so every p14 needs at least six non-input-only product functions and any topology with at least nine input-only products is impossible.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-048", "kind": "finding", "title": "Quadratic-hull screens of two-column SHA-256 seed allocations", "title_plain": "A genuinely joint two-column SHA-256 relation might let four mandatory Ch/Maj products also reject carry defects, but every supplied JSC seed allocation fails the quadratic-hull screen and the remaining constructions are untested.", "formal_line": "All four K-pairs: 92,274,688 honest rows, degree-two rank 837 of 862, ideal dimension 25; JSC-13, JSC-12, JSC-11 fail instrument gate", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-048/", "receipt": "CONT priority-7 replay JSON certificates for the two-column screen, factorized pinning, and exact delta2 profile.", "external_value": "Cryptographic circuit optimization and SHA-256 implementation research.", "statement": "A genuinely joint two-column SHA-256 relation might let four mandatory Ch/Maj products also reject carry defects, but every supplied JSC seed allocation fails the quadratic-hull screen and the remaining constructions are untested.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-049", "kind": "finding", "title": "Linear fresh-defect growth in the strongest natural five-code relaxed SHA relation", "title_plain": "For the strongest natural five-code relaxed SHA relation, composing one through six columns produces one through six independent missing-C directions and linearly growing flip and interface ranks, so this relation cannot support an o(1) terminal repair tax.", "formal_line": "For k=1..6, missing-C quotient ranks are k, raw state-flip ranks are 5k, interface ranks are 2k+3, and terminal repair tax cannot be o(1)", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-049/", "receipt": "CONT boundeddefect_sha_carry_k1_k6.priority7_replay.json.", "external_value": "Automata and cryptographic constraint-system researchers", "statement": "For the strongest natural five-code relaxed SHA relation, composing one through six columns produces one through six independent missing-C directions and linearly growing flip and interface ranks, so this relation cannot support an o(1) terminal repair tax.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-050", "kind": "finding", "title": "Exact finite-horizon minimisation of natural SHA carry states", "title_plain": "Finite-horizon minimization of the 22 natural SHA carry states gives 22 Nerode classes through bit 29, then 16, 4, and 1 terminal class, so ordinary full-alphabet recoding can affect at most 128 terminal occurrences.", "formal_line": "For the 22 natural SHA carry states, exact finite-horizon minimisation produces the identical partition across all 64 round constants: 22 classes through bit 29, 16 at bit 30, four", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "symmetry-and-encodings", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-050/", "receipt": "Fleet C SHA-carry Nerode screen and an independent verification receipt.", "external_value": "Cryptographic-circuit designers and automata/state-minimization researchers.", "statement": "Finite-horizon minimization of the 22 natural SHA carry states gives 22 Nerode classes through bit 29, then 16, 4, and 1 terminal class, so ordinary full-alphabet recoding can affect at most 128 terminal occurrences.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-051", "kind": "finding", "title": "Impossibility of realizing AND4 with three witnesses and three product rows", "title_plain": "The identity-C affine-decoder model of AND4 with four public inputs, three witnesses, three rank-one product rows, and nonempty even fibres is unsatisfiable.", "formal_line": "The finite search space for an identity-C, affine-decoder, nonempty-even-fibre realization of AND4 with four public inputs, three witnesses, and three rank-one product rows contain", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-051/", "receipt": "CaDiCaL UNSAT with an independently checked DRAT proof and Glucose and CryptoMiniSat agreement.", "external_value": "R1CS expressiveness and proof-system gadget research.", "statement": "The identity-C affine-decoder model of AND4 with four public inputs, three witnesses, three rank-one product rows, and nonempty even fibres is unsatisfiable.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-052", "kind": "finding", "title": "Closure of the ordered acyclic A*B=C catalyst class at p6", "title_plain": "The acyclic p6 state-only catalyst class is closed: across 67 deterministic state-feature subspaces, all 1,005 one-output-feature models are exact-UNSAT and all 7,035 two-output-feature models are rank-dead.", "formal_line": "Ordered acyclic A*B=C catalyst class at p6 = ∅ (1,005 exact-UNSAT, 7,035 rank-dead across 67 deterministic state-feature subspaces)", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-052/", "receipt": "An 8,040-model exact census with an independent recheck receipt.", "external_value": "Boolean-circuit exact-synthesis and circuit-complexity researchers", "statement": "The acyclic p6 state-only catalyst class is closed: across 67 deterministic state-feature subspaces, all 1,005 one-output-feature models are exact-UNSAT and all 7,035 two-output-feature models are rank-dead.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-053", "kind": "finding", "title": "Refutation of stationary additive raw/state phases by a three-column divergent carry path", "title_plain": "Among the natural raw-input-twisted edge phases, the stationary additive raw-state family cannot repair the relation when its only terminal test is the accumulated syndrome, because an exact three-column divergent path returns to the honest carry with zero relative moment for every candidate phase.", "formal_line": "Natural edge quotient has 491,040 candidates across 245,520 classes with span rank 74; 40 elementary q_i(c)*u_j are linearly independent", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "boolean function analysis", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-053/", "receipt": "Fleet A edge-phase census, independent rank-and-holonomy audit, and full phase-span holonomy screen.", "external_value": "Cryptographic-circuit and finite-state synthesis researchers.", "statement": "Among the natural raw-input-twisted edge phases, the stationary additive raw-state family cannot repair the relation when its only terminal test is the accumulated syndrome, because an exact three-column divergent path returns to the honest carry with zero relative moment for every candidate phase.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-057", "kind": "finding", "title": "On the full-word gauge of the consecutive-Maj identity", "title_plain": "The exact consecutive-Maj identity holds on all 16 local inputs, but the later receipt shows the hidden intermediate is a full 32-bit gauge, so the edge row is not a function graph and its 1,024-row figure is only conditional arithmetic, with no saving from retaining both witnesses.", "formal_line": "m_t xor m_(t+1) = (a_t xor b_t) * (a_t xor b_t xor c_t xor a_(t+1)) across all 16 local inputs", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "sha-256-record", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-057/", "receipt": "CONT consecutive-Maj verification receipts plus the later maj_gauge_maximal_defect.json word-level receipt.", "external_value": "Cryptographic circuit synthesis and hidden-state relation design.", "statement": "The exact consecutive-Maj identity holds on all 16 local inputs, but the later receipt shows the hidden intermediate is a full 32-bit gauge, so the edge row is not a function graph and its 1,024-row figure is only conditional arithmetic, with no saving from retaining both witnesses.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-058", "kind": "finding", "title": "Nonexistence of small low-witness AND4 cells", "title_plain": "Exact censuses in the arbitrary-affine-C, affine-factor, arbitrary-nonempty-fibre model rule out every three-witness/two-row cell and the surveyed two-witness cells through four rows, while selector-closure later closes the two-witness higher-row remainder.", "formal_line": "In the affine-factor, affine-decoder, arbitrary-nonempty-fibre model with arbitrary affine-C: 1.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-058/", "receipt": "Exact 3w2, 2w3, and 2w4 rank and orbit censuses, an AND2 positive control, and the verification report.", "external_value": "R1CS expressiveness and relational proof-system design.", "statement": "Exact censuses in the arbitrary-affine-C, affine-factor, arbitrary-nonempty-fibre model rule out every three-witness/two-row cell and the surveyed two-witness cells through four rows, while selector-closure later closes the two-witness higher-row remainder.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-059", "kind": "finding", "title": "Complete witness-width classification of Boolean functions on F₂⁴", "title_plain": "For every Boolean function on four input bits, the arbitrary-affine-C rank-one model has a classified minimum witness width, with AND4 requiring width 3 and the width classes containing 32, 1,120, 63,872, and 512 functions at widths 0, 1, 2, and 3.", "formal_line": "Every Boolean function `f: F₂⁴ → F₂` has a well-defined minimum witness width `ω(f)` in the finite arbitrary-affine-C rank-one model with an affine decoder and arbitrary nonempty f", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "boolean function analysis", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-059/", "receipt": "A Python generator BFS and an independent GL(4,2) enumeration with 20,160 matrices and 16 translations.", "external_value": "Boolean-function analysis and R1CS expressiveness.", "statement": "For every Boolean function on four input bits, the arbitrary-affine-C rank-one model has a classified minimum witness width, with AND4 requiring width 3 and the width classes containing 32, 1,120, 63,872, and 512 functions at widths 0, 1, 2, and 3.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-060", "kind": "finding", "title": "An impossibility theorem for the normalized two-witness selector for AND₄", "title_plain": "In the normalized two-witness selector model, every selector separates the 30 wrong witnesses away from the all-ones input, but no selector separates both wrong witnesses at the all-ones input, with the failures exactly characterized by 896 bent functions and their XORs with AND4.", "formal_line": "Every normalized 2-witness selector separates 30 wrong witnesses over x ≠ 1111, and no selector separates both wrong witnesses over x = 1111", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "boolean function analysis", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-060/", "receipt": "An independent Walsh census and the AND4 selector-geometry certificate.", "external_value": "Boolean-function analysis and R1CS selector design.", "statement": "In the normalized two-witness selector model, every selector separates the 30 wrong witnesses away from the all-ones input, but no selector separates both wrong witnesses at the all-ones input, with the failures exactly characterized by 896 bent functions and their XORs with AND4.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-061", "kind": "finding", "title": "Classification of the identity-C fixed-point consumer on toy Boolean graphs", "title_plain": "An identity-C creator/consumer mechanism rejects a named wrong point on a toy AND3 graph, and its sentinel construction compiles the stated AND3 and AND4 parity graphs in three and four rows, but the SHA instance remains UNKNOWN.", "formal_line": "1*(D⊕r)=r forces D=0; with m multiplication pins, affine parity compiles in m+1 rows (AND3 in 3 rows, AND4 in 4 rows)", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "construction", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-061/", "receipt": "All-16 base-case classification and toy and canonical AND3/AND4 compiler certificates; no SHA replay.", "external_value": "R1CS relation and gadget design.", "statement": "An identity-C creator/consumer mechanism rejects a named wrong point on a toy AND3 graph, and its sentinel construction compiles the stated AND3 and AND4 parity graphs in three and four rows, but the SHA instance remains UNKNOWN.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-062", "kind": "finding", "title": "Exact fibre cardinality of the two-round SHA-256 edge-relaxed relation", "title_plain": "For every fixed two-round SHA-256 input and every chosen next intermediate word U, the edge equations admit exactly 2^32 compatible hidden assignments, so the projected relation has a full 32-bit free intermediate.", "formal_line": "p0 = b xor (U-P), p1 = p0 xor ((a xor b) and (a xor b xor c xor U)), P = T1 + Sigma0(a) mod 2^32 yields b_(t+2)=U and fibre cardinality 2^32", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "sha-256-record", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-062/", "receipt": "Explicit algebraic parametrization, 20,000-sample replay, and independent 100,000-sample C++ crosscheck.", "external_value": "Cryptographic circuit and constraint-system researchers", "statement": "For every fixed two-round SHA-256 input and every chosen next intermediate word U, the edge equations admit exactly 2^32 compatible hidden assignments, so the projected relation has a full 32-bit free intermediate.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-063", "kind": "finding", "title": "Invertibility of odd rotation sums in F2[x]/((x+1)^{2^r})", "title_plain": "For binary word lengths that are powers of two, every XOR of an odd number of cyclic rotations is invertible, which gives a full-rank hidden gauge in the stated edge-relaxed SHA-256 two-round relation.", "formal_line": "In F2[x]/((x+1)^{2^r}), every XOR sum of an odd number of cyclic rotations is invertible (s(1)=1), so SHA-256 Sigma0 has matrix rank 32", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "sha-256-record", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-063/", "receipt": "Algebraic odd-rotation unit proof, SHA-256 Sigma0 rank-32 basis digest, and the shared MF-062 replay receipts.", "external_value": "NONE", "statement": "For binary word lengths that are powers of two, every XOR of an odd number of cyclic rotations is invertible, which gives a full-rank hidden gauge in the stated edge-relaxed SHA-256 two-round relation.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-064", "kind": "finding", "title": "An exact conservation law for gauges in GF(2) forest systems", "title_plain": "In honest-right-hand-side edge systems, the GF(2) incidence-kernel gauge space has one dimension per connected component, so a forest's N-E apparent vertex-row savings exactly equal its N-E created gauges.", "formal_line": "For a forest with N vertices and E edges over GF(2), gauge space dimension = apparent vertex-row savings = created gauges = N-E", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-064/", "receipt": "Independent audit JSON and maj_edge_forest_rank certificate; labelled-graph checks through n=6.", "external_value": "Researchers in GF(2) constraint systems and graph-based circuit composition", "statement": "In honest-right-hand-side edge systems, the GF(2) incidence-kernel gauge space has one dimension per connected component, so a forest's N-E apparent vertex-row savings exactly equal its N-E created gauges.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-065", "kind": "finding", "title": "Exact number of quadratic equations for a unique common zero in GF(2)^32", "title_plain": "When 32 gauge residuals are affine coordinates and no auxiliaries are allowed, exactly 16 quadratic equations are necessary and sufficient to force them all to zero; the resulting 512-row count for all round pairs is abstract and not a SHA construction.", "formal_line": "Forcing g₀=...=g₃₁=0 over GF(2) with degree ≤ 2 equations without auxiliary variables requires and is satisfied by exactly 16 equations", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-065/", "receipt": "Chevalley-Warning lower-bound argument, explicit 16-equation construction, and exhaustive unique-zero checks through 16 bits.", "external_value": "Researchers in Boolean equation systems and cryptographic constraint design", "statement": "When 32 gauge residuals are affine coordinates and no auxiliaries are allowed, exactly 16 quadratic equations are necessary and sufficient to force them all to zero; the resulting 512-row count for all round pairs is abstract and not a SHA construction.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "The lower bound is based on the classical Chevalley-Warning theorem, and no separate prior-art position is claimed.", "condition": "" }, { "id": "MF-069", "kind": "finding", "title": "A strict flag characterization of rank-tight acyclic full-domain XAGs", "title_plain": "An acyclic XAG with exactly the target quotient rank exists if and only if the target classes admit a strict flag whose successive directions are represented by products of affine combinations of earlier directions.", "formal_line": "dim W = r admits acyclic full-domain XAG with r AND gates iff ∃ 0 = U0 < ... < Ur = W: Ui / U_{i-1} contains class of Li Ri, Li, Ri in A + U_{i-1}", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "exact synthesis methods", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-069/", "receipt": "CONT frontier verification report, independent audit JSON, and the frontier_exact.py reachability implementation.", "external_value": "Exact Boolean-circuit synthesis and algebraic complexity researchers.", "statement": "An acyclic XAG with exactly the target quotient rank exists if and only if the target classes admit a strict flag whose successive directions are represented by products of affine combinations of earlier directions.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-070", "kind": "finding", "title": "Impossibility of p7 realizations for rank-tight q-rank-7 edges under G = W", "title_plain": "In the rank-tight G = W model, none of the 32 q-rank-7 p7-eligible directed affine-code edges admits a p7 realization.", "formal_line": "In the rank-tight model G = W, every e ∈ E₇ is p7-impossible", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-070/", "receipt": "Complete replay of all 32 screens, independent engine agreement, and replay/frontier_exact_full_replay.json.", "external_value": "NONE", "statement": "In the rank-tight G = W model, none of the 32 q-rank-7 p7-eligible directed affine-code edges admits a p7 realization.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-071", "kind": "finding", "title": "A multiplicative complexity lower bound for 28 q-rank-6 edges", "title_plain": "All 28 q-rank-6 p7-eligible affine-code edges are p6-impossible: 16 are killed by exact flag profiles and the other 12 already require at least seven products, while p7 one-catalyst existence remains unknown.", "formal_line": "Every edge in E₆ is p6-impossible", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-071/", "receipt": "Replayed frontier screens, an independent engine match, and the edge_29_0 upload.", "external_value": "NONE", "statement": "All 28 q-rank-6 p7-eligible affine-code edges are p6-impossible: 16 are killed by exact flag profiles and the other 12 already require at least seven products, while p7 one-catalyst existence remains unknown.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-072", "kind": "finding", "title": "Impossibility of rank-tight p5 realizations for q-rank-5 edges", "title_plain": "All eight q-rank-5 p7-eligible affine-code edges are rank-tight p5-impossible, leaving the p7 case as a two-catalyst extension whose existence remains unknown.", "formal_line": "In the rank-tight model G = W, every e ∈ E₅ is p5-impossible", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-072/", "receipt": "rank5_p5_flag_screens.json and an independent frontier audit.", "external_value": "NONE", "statement": "All eight q-rank-5 p7-eligible affine-code edges are rank-tight p5-impossible, leaving the p7 case as a two-catalyst extension whose existence remains unknown.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-073", "kind": "finding", "title": "Exact catalyst dimension in a rank-tight XAG", "title_plain": "Under the stated rank-tight hypotheses, a p-gate acyclic XAG computing a target quotient of rank r has catalyst dimension exactly p-r.", "formal_line": "For a p-gate acyclic XAG with W ⊆ G, dim G = p, dim W = r: catalyst dimension = p − dim W = p − r", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-073/", "receipt": "Frontier verification report containing the linear-algebra derivation.", "external_value": "NONE", "statement": "Under the stated rank-tight hypotheses, a p-gate acyclic XAG computing a target quotient of rank r has catalyst dimension exactly p-r.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-074", "kind": "finding", "title": "A counterexample to the laminar multiplication-tree normal form", "title_plain": "A four-product mixed-cancellation circuit in the tested six-input model succeeds in the unrestricted rank-tight search while the laminar-only search fails, disproving the proposed laminar multiplication-tree normal form.", "formal_line": "Laminar multiplication-tree normal form does not hold on n=6: p4 circuit has q-rank 4, unrestricted [1,2,1,1,1] vs laminar [1,2,1,1,0]", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-074/", "receipt": "CONT lens_r2c_independent_audit.json with the unrestricted and laminar flag results.", "external_value": "Exact Boolean-circuit synthesis and normal-form research.", "statement": "A four-product mixed-cancellation circuit in the tested six-input model succeeds in the unrestricted rank-tight search while the laminar-only search fails, disproving the proposed laminar multiplication-tree normal form.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-077", "kind": "finding", "title": "Multiplicative complexity of four-operand addition", "title_plain": "The audited three-product carry cell computes the full-precision sum of four n-bit integers in 3n ANDs, exactly tying the standard carry-save baseline at every tested width rather than improving multiplicative complexity.", "formal_line": "x+y+z+w+c₀+2c₁ = s+2c₀′+4c₁′ using 3 AND gates; full-precision sum of four n-bit integers uses 3n ANDs", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "audit finding", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-077/", "receipt": "The MF-076 audit certificate; exhaustive 64-row cell replay and n<=3 cascade replay.", "external_value": "Circuit-benchmark authors and designers comparing nonlinear-gate counts.", "statement": "The audited three-product carry cell computes the full-precision sum of four n-bit integers in 3n ANDs, exactly tying the standard carry-save baseline at every tested width rather than improving multiplicative complexity.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "The claimed 25% improvement is revised: standard carry-save baselines also achieve 3n AND gates, as the Cirbo/STACS generator’s 4n figure optimizes total gates instead. Only the nonlinear gate comparison specific to the Cirbo/STACS generator remains valid.", "condition": "" }, { "id": "MF-080", "kind": "finding", "title": "A lower bound on the Toffoli count of exact NCT networks implementing χ₅", "title_plain": "Every exact NOT/CNOT/Toffoli network for the 5-bit chi/Ascon permutation requires at least six Toffoli gates, including networks with clean ancillas.", "formal_line": "For every exact NOT/CNOT/Toffoli network N implementing χ₅ with clean ancillas, #Toffoli(N) ≥ MC(χ₅⁻¹) = 6", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "ciphers-and-quantum", "school": "quantum circuit synthesis", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "Boolean ANF Degree / Symmetry", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-080/", "receipt": "03-mc-lower-bounds-tcount/REPORT.md sections 1 and 3.3; bound derived from MC(chi_5 inverse)=6.", "external_value": "Reversible and quantum circuit synthesis for the Ascon/chi S-box; Toffoli-cost lower bounds.", "statement": "Every exact NOT/CNOT/Toffoli network for the 5-bit chi/Ascon permutation requires at least six Toffoli gates, including networks with clean ancillas.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "No separate prior-art claims are made. This bound is derived from the exact six-AND inverse complexity and applies strictly to NCT networks.", "condition": "" }, { "id": "MF-083", "kind": "finding", "title": "An eight-product circuit for the low four bits of four four-bit numbers", "title_plain": "An explicit acyclic eight-product circuit computes the low four bits of the sum of four four-bit operands on all 65,536 inputs, so the 4op_w4_p8 instance is satisfiable.", "formal_line": "The low four bits of x₀ + x₁ + x₂ + x₃ are computed by an acyclic XAG with 8 products", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "construction", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-083/", "receipt": "Complete 65,536-row truth-table replay with zero mismatches, plus an independent certificate replay.", "external_value": "Multiplicative-complexity and multi-operand arithmetic-circuit synthesis researchers", "statement": "An explicit acyclic eight-product circuit computes the low four bits of the sum of four four-bit operands on all 65,536 inputs, so the 4op_w4_p8 instance is satisfiable.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-084", "kind": "finding", "title": "The separated-product theorem over any field under square closure", "title_plain": "Over any field, the separated-product theorem holds when each component signal space is closed under squaring, and without that condition mixed factors can produce new separated terms.", "formal_line": "If u ∈ U ⟹ u² ∈ U and v ∈ V ⟹ v² ∈ V, the separated-product theorem holds over any field k", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "direct-sums", "school": "algebraic complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-084/", "receipt": "Transfer algebra report, checker, and transfer algebra certificate.", "external_value": "Algebraic-complexity and commutative-signal-algebra researchers", "statement": "Over any field, the separated-product theorem holds when each component signal space is closed under squaring, and without that condition mixed factors can produce new separated terms.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "This entry builds on predecessor MF-032; no position on external prior art is stated.", "condition": "" }, { "id": "MF-085", "kind": "finding", "title": "The mixed-image tax and rank-tight copy-locality", "title_plain": "When a separated target quotient has dimension r and is computed with exactly r new product gates, every independent target-bearing gate must be copy-local because products with nonzero mixed span expose at most r-1 independent new separated directions.", "formal_line": "dim(residual separated target quotient) = r with r new product gates ⟹ every independent target-bearing gate is copy-local", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "direct-sums", "school": "algebraic complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-085/", "receipt": "Transfer algebra report, Theorem 3 and Corollary 4.", "external_value": "Direct-sum circuit-complexity researchers", "statement": "When a separated target quotient has dimension r and is computed with exactly r new product gates, every independent target-bearing gate must be copy-local because products with nonzero mixed span expose at most r-1 independent new separated directions.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-086", "kind": "finding", "title": "Multiplicative complexity of independent copies of 2×2 matrix multiplication over GF(2)", "title_plain": "For s independent copies of 2x2 matrix multiplication over GF(2), CP or bilinear rank and formal-quadratic multiplicative complexity are both exactly 7s, but the unrestricted Boolean XAG model is not covered.", "formal_line": "rank_CP/bilinear(M₂^⊕s) = 7s, MC_formal-quadratic(M₂^⊕s) = 7s over GF(2)", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "direct-sums", "school": "algebraic complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-086/", "receipt": "Alder-Strassen and Strassen bounds, plus REPORT sections 7-9 and the mm2 Boolean search artifacts.", "external_value": "Algebraic-complexity and matrix-multiplication researchers", "statement": "For s independent copies of 2x2 matrix multiplication over GF(2), CP or bilinear rank and formal-quadratic multiplicative complexity are both exactly 7s, but the unrestricted Boolean XAG model is not covered.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "The 7s value is credited to Alder-Strassen and Strassen under bilinear and formal-quadratic models, but this result has not been established for unrestricted Boolean settings.", "condition": "" }, { "id": "MF-087", "kind": "finding", "title": "Matroid invariants of the quadratic hull of a Boolean relation", "title_plain": "The project's quadratic hull is the standard degree-2 Boolean closure, and its defect measures match established false-loop, syndrome-rank, nullity, and critical-number invariants; only the fibre-sensitive vertical and flip defects are described as plausibly new packaging.", "formal_line": "h₂=false-loop count, ρ₂=false-syndrome rank, δ_2,pin=nullity after deleting loops, κ_2,cover=Crapo–Rota critical exponent", "status": "IMPORTED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "method or instrument", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-087/", "receipt": "Proof-complexity report sections 2-3 and 9 plus NOTES-LIT identification.", "external_value": "Proof complexity, binary matroid theory, and SAT-encoding researchers", "statement": "The project's quadratic hull is the standard degree-2 Boolean closure, and its defect measures match established false-loop, syndrome-rank, nullity, and critical-number invariants; only the fibre-sensitive vertical and flip defects are described as plausibly new packaging.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "Most concepts and terms used here follow established literature; only delta2^vert and delta2^flip represent original formulations introduced in this work.", "condition": "" }, { "id": "MF-088", "kind": "finding", "title": "Exact multilinear separator degree of the Boolean AND graph", "title_plain": "For the graph w=x_1...x_r, the positive wrong point (1,...,1,0) has exact multilinear separator degree r, while positive-pin failure is a semantic separator obstruction rather than a proof-degree lower bound.", "formal_line": "For R_r = Graph(w = x₁...x_r) and a_r = (1^r, 0), σ_{R_r}(a_r) = r, and a_r ∈ H_d(R_r) if and only if d < r", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-088/", "receipt": "Direct multilinear-separator proof plus PC_exact_certificate.py, PC_exact_certificate.json, and the report.", "external_value": "Proof complexity and algebraic SAT-encoding researchers", "statement": "For the graph w=x_1...x_r, the positive wrong point (1,...,1,0) has exact multilinear separator degree r, while positive-pin failure is a semantic separator obstruction rather than a proof-degree lower bound.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "This result generalizes MF-012; no external prior-art comparison is stated.", "condition": "" }, { "id": "MF-089", "kind": "finding", "title": "The degree-(d) hull of Boolean relations with few forbidden points", "title_plain": "For a Boolean relation on n variables, if fewer than 2^(n-d) points are forbidden then it has no nonzero degree-at-most-d vanishing polynomial and its degree-d hull is the entire cube; in particular, width-k clauses for k>=3 are invisible to static quadratic hulls.", "formal_line": "If R ⊆ 𝔽_2^n, F = 𝔽_2^n \\ R, 0 ≤ d ≤ n, and |F| < 2^{n-d}, then I_{≤d}(R) = {0} and H_d(R) = 𝔽_2^n", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-089/", "receipt": "Reed-Muller minimum-distance proof, exhaustive width-1..8 checker, and SAT-HULL certificates.", "external_value": "SAT preprocessing and proof-complexity researchers", "statement": "For a Boolean relation on n variables, if fewer than 2^(n-d) points are forbidden then it has no nonzero degree-at-most-d vanishing polynomial and its degree-d hull is the entire cube; in particular, width-k clauses for k>=3 are invisible to static quadratic hulls.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "This step relies on standard minimum-distance properties of classical Reed-Muller codes; no original contribution or novelty is claimed.", "condition": "" }, { "id": "MF-090", "kind": "finding", "title": "Exact quadratic cover number of an OR-chain lift of a width-k clause over F2", "title_plain": "The OR-chain lift for a positive width-k clause makes the relation quadratically closed with exact quadratic cover number k-1.", "formal_line": "For the lifted relation of a positive width-k clause (k ≥ 3), κ2,cover = k − 1", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-090/", "receipt": "Chevalley-Warning proof, verification for k=3..6, and independent brute-force critical-number checks for k=3,4.", "external_value": "SAT preprocessing and proof complexity", "statement": "The OR-chain lift for a positive width-k clause makes the relation quadratically closed with exact quadratic cover number k-1.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "This result uses well-established, classical techniques, and no claim of novelty is made without a targeted literature search for this encoding theorem.", "condition": "" }, { "id": "MF-091", "kind": "finding", "title": "A minimal three-row quadratic constraint system for the inverse-or-default gadget", "title_plain": "For the fixed field interface exposing z=[x=0] and y=x^-1 when x is nonzero or d otherwise, fusion saves one allocation while retaining three rank-one rows, and three rows are minimal over fields with at least five elements.", "formal_line": "xy=1-z, xz=0, z(y-d)=0 uniquely defines z=[x=0], y=x⁻¹ (x≠0), y=d (x=0); 3 rows is minimal without auxiliary allocations over |F| ≥ 5", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "R1CS over F_p", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-091/", "receipt": "Exact algebra and fibre replay plus Circom 2.2.3 O0/O2 before-and-after compilation and a minimality certificate.", "external_value": "Prime-field R1CS gadget design and cryptographic proof systems.", "statement": "For the fixed field interface exposing z=[x=0] and y=x^-1 when x is nonzero or d otherwise, fusion saves one allocation while retaining three rank-one rows, and three rows are minimal over fields with at least five elements.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-092", "kind": "finding", "title": "Exact nonlinear cost of an eight-state controller under affine relabeling", "title_plain": "An invertible affine recoding of an FSM state preserves the exact number of AND gates and multiplicative depth; for the fully specified eight-state controller, 40,320 encodings reduce to 30 affine orbits with exact minima of 2, 3, 4, or 5 ANDs.", "formal_line": "Affine maps over F₂ preserve XOR–AND count/depth: 40,320 3-bit encodings form 30 orbits with exact minimum ANDs: 1×2, 4×3, 15×4, 10×5", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "symmetry-and-encodings", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-092/", "receipt": "REPORT.md, two complete executions, two independent verification paths, and full truth-table replay.", "external_value": "Cryptographic and Boolean circuit synthesis with exact nonlinear-cost optimization", "statement": "An invertible affine recoding of an FSM state preserves the exact number of AND gates and multiplicative depth; for the fully specified eight-state controller, 40,320 encodings reduce to 30 affine orbits with exact minima of 2, 3, 4, or 5 ANDs.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-093", "kind": "finding", "title": "Formal verification of end-to-end output equality for XAGs in Lean", "title_plain": "CourtXAG is a Lean-checked prototype that binds local equivalence evidence and an explicit port map to end-to-end output equality and exact structural and cost invariants.", "formal_line": "Lean 4.28 proves equivalence between a 30,003-gate reference hierarchy and a 20,002-gate optimized hierarchy using zero axioms", "status": "METHOD", "tier": "lean", "tier_label": "LEAN-VERIFIED", "program": "formal-verification", "school": "formal verification", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-093/", "receipt": "Lean 4.28 no-axiom prototype, separate leanchecker replays, and Court Zero canaries.", "external_value": "formal verification and compiler validation of optimized Boolean circuits", "statement": "CourtXAG is a Lean-checked prototype that binds local equivalence evidence and an explicit port map to end-to-end output equality and exact structural and cost invariants.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-097", "kind": "finding", "title": "Decomposability of the extremal top form in acyclic XOR–AND circuits", "title_plain": "For an acyclic XOR-AND circuit with r AND gates, an output of degree r+1 has a decomposable top homogeneous component, so the extremal degree-bound case used by MF-001 is sound.", "formal_line": "If C is an acyclic XOR–AND circuit with r AND gates and f is an output of degree r+1, then top_{r+1}(f) is decomposable", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "Boolean ANF Degree / Symmetry", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-097/", "receipt": "The original certificate was located and replayed as UNSAT with its degree-six support and Plucker relation checked, together with the generic 6x10 determinant identity.", "external_value": "Multiplicative-complexity lower-bound researchers", "statement": "For an acyclic XOR-AND circuit with r AND gates, an output of degree r+1 has a decomposable top homogeneous component, so the extremal degree-bound case used by MF-001 is sound.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "This result makes no separate prior-art claim. It validates the extremal lemma underlying MF-001 and distinguishes it from the non-extremal counterexample in MF-094.", "condition": "" }, { "id": "MF-099", "kind": "finding", "title": "The exact digit-sum law for full K-operand addition", "title_plain": "For all K and n, exposing every nonconstant bit of the ordinary sum of K n-bit words has exact multiplicative complexity K n minus the binary digit sum of K(2^n-1). The exact six-product FullAdd(5,2) circuit, not eight, kills the former universal (K-1)n conjecture.", "formal_line": "MC(FullAdd(K,n)) = Kn − s₂(K(2ⁿ−1)); FullAdd(5,2) = 6 ≠ 8", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-099/", "receipt": "SYNTH-I-FAMILIES.md; PROVER-AFP-CARRY.md; PROVER-AFR-CONSTRUCT.md; prover-afr-scratch/fulladd-5-2-bitheap.json; prover-afr-scratch/focal-replay-final.json; synth-n-scratch/recompute_capstone.receipt.json. Composed theorem, clean-room checked; FullAdd(5,2) replayed on all 1,024 inputs.", "external_value": "Exact product counts for complete multi-operand integer adders.", "statement": "For all K and n, exposing every nonconstant bit of the ordinary sum of K n-bit words has exact multiplicative complexity K n minus the binary digit sum of K(2^n-1). The exact six-product FullAdd(5,2) circuit, not eight, kills the former universal (K-1)n conjecture.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-100", "kind": "finding", "title": "The exact width-two law for truncated multi-operand addition", "title_plain": "For every k at least 2, the low two bits of the sum of k independent two-bit words need exactly floor(k/2) nonlinear products. This width-two law does not transfer to carry-visible or full-output addition.", "formal_line": "MC(A_{k,2}) = ⌊k/2⌋ for every k ≥ 2", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-100/", "receipt": "SYNTH-H-LOWERBOUND.md; PROVER-C-CARRYSAVE.md; prover-c-scratch/width2_rank.out; prover-g-scratch/bank-checks.json; prover-trunc-scratch/replay-all.json. Polar-rank proof plus uniform construction and full replays.", "external_value": "A closed base family for truncated multi-operand addition.", "statement": "For every k at least 2, the low two bits of the sum of k independent two-bit words need exactly floor(k/2) nonlinear products. This width-two law does not transfer to carry-visible or full-output addition.", "published": "2026-08-29", "last_reviewed": "2026-08-30", "prior_art": "", "condition": "" }, { "id": "MF-101", "kind": "finding", "title": "The universal one-gate bracket for three-operand addition", "title_plain": "For three n-bit operands, independent symbolic derivations give the universal floor 2n-4 and the standard construction gives 2n-3. The upper value is exact only for n=2,3,4,5 and for prefix-causal circuits; the unrestricted all-width equality remains open.", "formal_line": "2n−4 ≤ MC(A_{3,n}) ≤ 2n−3; the upper is exact at n = 2,3,4,5 and for prefix-causal circuits", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-101/", "receipt": "SYNTH-H-LOWERBOUND.md; PROVER-A-Q1.md; PROVER-D-ELIM.md; prover-b-scratch/extremal_game.receipt.txt; synth-n-scratch/recompute_capstone.receipt.json. General floor and bracket are symbolic; finite exact cells have complete certificates and construction replays.", "external_value": "The sharpest proved all-width lower boundary for three-operand truncated addition.", "statement": "For three n-bit operands, independent symbolic derivations give the universal floor 2n-4 and the standard construction gives 2n-3. The upper value is exact only for n=2,3,4,5 and for prefix-causal circuits; the unrestricted all-width equality remains open.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-102", "kind": "finding", "title": "A projected full-heap construction for truncated addition", "title_plain": "Projecting a complete weighted bit heap onto the low n sum bits gives a general truncated-addition construction with a transient carry-height count T(k,n). It improves the old fold families, but equality with T is not proved for unrestricted circuits.", "formal_line": "MC(A_{k,n}) ≤ T(k,n) = (k−1)(n−1) − Σ_{r=1}^{n−1}⌊(k−1)/2ʳ⌋", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "construction", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-102/", "receipt": "SYNTH-L-TRUNCADD.md; PROVER-TRUNC-SHARE.md; prover-trunc-scratch/replay-all.json; prover-trunc-scratch/restriction-summary.json. Symbolic construction theorem with complete replay at the displayed cells.", "external_value": "A smaller, structurally counted family of truncated multi-operand adders.", "statement": "Projecting a complete weighted bit heap onto the low n sum bits gives a general truncated-addition construction with a transient carry-height count T(k,n). It improves the old fold families, but equality with T is not proved for unrestricted circuits.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-103", "kind": "finding", "title": "Injected carry is exactly the flagship one-gate gap", "title_plain": "The remaining one-gate gap for three-operand addition is exactly the injected-carry gap: MC(J_m)=MC(A_{3,m+1})-1. At m=32, J_32 is the record's Add32x3Canon33 block, so 62 versus 61 is the same open mathematical fork; no row saving is claimed.", "formal_line": "MC(J_m) = MC(A_{3,m+1}) − 1; J₃₂ = Add32x3Canon33 ∈ {61,62}", "status": "PROVED", "tier": "paper", "tier_label": "PAPER PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-103/", "receipt": "PROVER-ICA-PIVOT.md; SYNTH-H-LOWERBOUND.md; RECORD-BOM.md; prover-ica-scratch/t-pivot-experiments.json; prover-ica-scratch/pivot-graded-audit.json; prover-ica-scratch/m3-k4.clean.replay.json. Exact identity and localization are symbolic; small cells are fully replayed.", "external_value": "Connects an abstract lower-bound gap directly to thirty concrete SHA-256 adder blocks.", "statement": "The remaining one-gate gap for three-operand addition is exactly the injected-carry gap: MC(J_m)=MC(A_{3,m+1})-1. At m=32, J_32 is the record's Add32x3Canon33 block, so 62 versus 61 is the same open mathematical fork; no row saving is claimed.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-104", "kind": "finding", "title": "Verified width-three constructions for nine and ten operands", "title_plain": "The best verified width-three constructions now use 10 products for nine operands and 12 for ten operands. They were replayed over all 2^27 and 2^30 inputs respectively; both are upper bounds, not exact complexity values.", "formal_line": "MC(A_{9,3}) ≤ 10 and MC(A_{10,3}) ≤ 12; both are replayed uppers, not exact values", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "construction", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-104/", "receipt": "PROVER-TRUNC-SHARE.md; SYNTH-L-TRUNCADD.md; prover-trunc-scratch/replay-all.json; prover-trunc-scratch/restriction-summary.json. Full-domain replays report zero output mismatches.", "external_value": "Concrete lower-cost circuits and benchmark targets for truncated addition.", "statement": "The best verified width-three constructions now use 10 products for nine operands and 12 for ten operands. They were replayed over all 2^27 and 2^30 inputs respectively; both are upper bounds, not exact complexity values.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-105", "kind": "finding", "title": "The live p14 frontier and its doubly conditional row saving", "title_plain": "The 14-product Cartesian two-column carry target remains live and unknown. After all filed screens, 817 named copy-swap cells survive; there is neither a replayed p14 circuit nor a general p14 impossibility proof. The advertised minus-477 rows requires both a p14 witness and rank(I+A)=3.", "formal_line": "p14 existence is LIVE/UNKNOWN; 817 named copy-swap cells survive; −477 rows requires p14 and rank(I+A)=3", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-105/", "receipt": "SYNTH-M-P14.md; staged7/pro-request3-return/sha256_p14_wedge_frontier_report.json; prover-p14a-scratch/graded_obligations_receipt.json; prover-p14a-scratch/completion_evidence_aggregate.json; synth-m-scratch/recompute_p14_intersection_receipt.json. Exact scoped survivor census; no global verdict.", "external_value": "A precise map of the surviving direct-sum search space and its conditional SHA-256 consequence.", "statement": "The 14-product Cartesian two-column carry target remains live and unknown. After all filed screens, 817 named copy-swap cells survive; there is neither a replayed p14 circuit nor a general p14 impossibility proof. The advertised minus-477 rows requires both a p14 witness and rank(I+A)=3.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-106", "kind": "finding", "title": "The 37-wall atlas of failed proof routes and scope limits", "title_plain": "The Wall Atlas records 37 scoped model failures, proof-technique ceilings, scope limits, and evidence rules. It states what each obstruction forbids and what remains open, including nonadditive floors, shared catalysts, anticipatory cancellation, p14 charging, and record-interface mismatch.", "formal_line": "W01–W37: MODEL, TECHNIQUE, SCOPE, and EVIDENCE walls with receipts R1–R46", "status": "INSTRUMENT AUDIT FINDING", "tier": "receipt", "tier_label": "RECEIPTED", "program": "further-results", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-106/", "receipt": "WALL-ATLAS.md; receipt ledger R1-R46; synth-j-scratch/replay_load_bearing_walls.receipt.json; SHA-256 038cd799ef6502abade1dea46e76cae0462e34b0495fcd070c29cf60b7a1ded6. Mixed symbolic and replayed negative knowledge, preserved with per-wall kind labels.", "external_value": "Prevents invalid lower-bound and circuit-transfer arguments from being repeated.", "statement": "The Wall Atlas records 37 scoped model failures, proof-technique ceilings, scope limits, and evidence rules. It states what each obstruction forbids and what remains open, including nonadditive floors, shared catalysts, anticipatory cancellation, p14 charging, and record-interface mismatch.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-107", "kind": "finding", "title": "Counterexamples to four former addition laws", "title_plain": "Four former addition rules are false: universal FullAdd cost (K-1)n, the old greedy truncated equality, the former U(k,n) final or tight family, and CS-base for every k at least 9. Explicit replayed circuits supply the counterexamples.", "formal_line": "FullAdd (K−1)n, greedy truncated equality, U(k,n) tightness, and CS-base for k ≥ 9 are refuted", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-107/", "receipt": "STATE-OF-PROGRAM.md section 2; SYNTH-I-FAMILIES.md; SYNTH-L-TRUNCADD.md; PROVER-ICC-DIRECTSUM.md; prover-afr-scratch/focal-replay-final.json; prover-trunc-scratch/replay-all.json; prover-icc-scratch/bitheap-k9-base-refutation.json; prover-icc-scratch/bitheap-k9-independent-replay.json.", "external_value": "A compact correction ledger for multi-operand addition claims.", "statement": "Four former addition rules are false: universal FullAdd cost (K-1)n, the old greedy truncated equality, the former U(k,n) final or tight family, and CS-base for every k at least 9. Explicit replayed circuits supply the counterexamples.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-108", "kind": "finding", "title": "The verified 22,215-row SHA-256 bill remains unchanged", "title_plain": "The verified canonical SHA-256 leader remains 22,215 ordered identity-C rows, with allocations, constraints, and atomic component costs all reconciling to 22,215. Today's mathematics claims no row improvement; 20,531 and the p14 totals 21,738 or 22,074 remain conditional only.", "formal_line": "verified rows = 22,215; authorized row change = 0; no row improvement is claimed", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "formal verification", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-108/", "receipt": "RECORD-BOM.md; SYNTH-L-TRUNCADD.md; SYNTH-M-P14.md; synth-n-scratch/recompute_capstone.receipt.json; record-22215/Main.lean; record-22215/Cost.lean; record-22215/Rounds16CanonMerged.lean. Source reconciliation and independent arithmetic recomputation.", "external_value": "An auditable bill of materials and an explicit zero row delta for the current research update.", "statement": "The verified canonical SHA-256 leader remains 22,215 ordered identity-C rows, with allocations, constraints, and atomic component costs all reconciling to 22,215. Today's mathematics claims no row improvement; 20,531 and the p14 totals 21,738 or 22,074 remain conditional only.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-109", "kind": "finding", "title": "Is the projected-heap upper bound always exact?", "title_plain": "It remains open whether every truncated k-operand n-bit sum meets the projected-heap count T(k,n). The cheapest falsifier seeks six products for A_(6,3); the sharp first stable fork asks whether A_(9,3) costs 9 or 10.", "formal_line": "CONJECTURE: MC(A_{k,n}) = T(k,n); the first sharp stable fork is A_{9,3} ∈ {9,10}", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-109/", "receipt": "SYNTH-L-TRUNCADD.md; EXACT-ATLAS.md; PROVER-TRUNC-SHARE.md; prover-trunc-scratch/seeded-core-k6-p6.result.json; prover-trunc-scratch/seeded-core-early-q-k6-p6.result.json. One fixed-seed subclass is UNSAT; the unrestricted question is not decided.", "external_value": "A precise falsification target for the remaining truncated-addition lower-bound problem.", "statement": "It remains open whether every truncated k-operand n-bit sum meets the projected-heap count T(k,n). The cheapest falsifier seeks six products for A_(6,3); the sharp first stable fork asks whether A_(9,3) costs 9 or 10.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-110", "kind": "finding", "title": "Correcting two truncated-addition cells", "title_plain": "The five-operand width-three truncated sum is exactly 5, while the four-operand width-four cell remains only between 6 and 8. Repeated exact-8 labels have no retained lower certificate and are not accepted as evidence.", "formal_line": "MC(A_{5,3}) = 5; MC(A_{4,4}) ∈ [6,8], not exact 8 on retained evidence", "status": "INSTRUMENT AUDIT FINDING", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "exact synthesis methods", "function_type": "audit finding", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-110/", "receipt": "STATE-OF-PROGRAM.md; PROVER-ICC-DIRECTSUM.md; EXACT-ATLAS.md; prover-e-scratch/flow-bank-receipt.json; prover-c-scratch/degree_product_bound.out. A_(5,3) uses the p4 decision hash 1816e37e911e3ee363e0b6eb5fc2ae60b655d4298d1c4db3d587d923c88d560f plus a five-gate replay; A_(4,4) has only a replayed p8 upper and proved floor 6.", "external_value": "Keeps finite benchmark cells aligned with the receipts actually in custody.", "statement": "The five-operand width-three truncated sum is exactly 5, while the four-operand width-four cell remains only between 6 and 8. Repeated exact-8 labels have no retained lower certificate and are not accepted as evidence.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "MF-111", "kind": "finding", "title": "Bounds on the multiplicative complexity of the injected-carry family", "title_plain": "The multiplicative complexity of the injected-carry family J_m is proven to lie between 2m-3 and 2m-2 for all m >= 2, with exact values 2, 4, 6 for m = 2, 3, 4.", "formal_line": "2m-3 ≤ MC(J_m) ≤ 2m-2 for m ≥ 2; MC(J_2)=2, MC(J_3)=4, MC(J_4)=6, J_5 ∈ [7, 8]", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-111/", "receipt": "zkgolf-decomp/reports/CERT-SYNTH.md, zkgolf-decomp/reports/PROVER-ICA-PIVOT.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/cert-synth-scratch/recompute.receipt.json, zkgolf-decomp/cert-tensor-scratch/transfer-audit.receipt.json, zkgolf-decomp/prover-ica-scratch/m3-k4.clean.replay.json", "external_value": "NONE", "statement": "The multiplicative complexity of the injected-carry family J_m is proven to lie between 2m-3 and 2m-2 for all m >= 2, with exact values 2, 4, 6 for m = 2, 3, 4.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-112", "kind": "finding", "title": "Status of the FCNS reduction chain for uniform lower bounds", "title_plain": "The FCNS reduction chain remains an open route for proving uniform lower bounds rather than a fully established general theorem.", "formal_line": "Exact exclusion is Γ_{m, 2m-3} = ∅; local-response theorem proved, but summation across sites and SBD remain unproved (OPEN).", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "algebraic complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-112/", "receipt": "module-data.receipt.txt (SHA-256 68f78bfaa2b3a1c31dc0d514fbcdf3be28f3548bd7d52c94bb47e73666e162aa), exact_xag.receipt.json (SHA-256 0702ccae456b9865d27cbc3f46a628a828219b6c5dd03035b681f866811baad1), prefix-screen.receipt.json (SHA-256 a7ef24de17806ae977182df864f71d19a0a0cef5b87f7cf2743caf7ff47ca0b6), and reports CERT-SYNTH.md, CERT-TENSOR.md, CERT-SYZ.md, CERT-COALG.md, FCNS-A-TMI.md, STATE-OF-PROGRAM-V2.md", "external_value": "NONE", "statement": "The FCNS reduction chain remains an open route for proving uniform lower bounds rather than a fully established general theorem.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-113", "kind": "finding", "title": "Shortest-path and LP-dual formulation of multiplicative complexity via semantic gate states", "title_plain": "Multiplicative complexity is shown to equal shortest-path distance on the complete semantic gate-state graph, yielding an exact linear programming dual based on Lipschitz potentials.", "formal_line": "MC equals shortest-path distance on the complete semantic gate-state graph; its LP dual is the 1-Lipschitz potential problem", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "symmetry-and-encodings", "school": "multiplicative complexity", "function_type": "method or instrument", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-113/", "receipt": "zkgolf-decomp/reports/CERT-DUAL.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/cert-dual-scratch/dual-certificate.receipt.json, zkgolf-decomp/cert-dual-scratch/j3-normal-form-dual.lp", "external_value": "Researchers designing lower bounds and certificate verification systems for multiplicative complexity via linear programming duality.", "statement": "Multiplicative complexity is shown to equal shortest-path distance on the complete semantic gate-state graph, yielding an exact linear programming dual based on Lipschitz potentials.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-114", "kind": "finding", "title": "Exact restriction loss conservation law for multiplicative complexity", "title_plain": "The drop in multiplicative complexity when restricting a Boolean function splits exactly into killed cost plus residual inefficiency, with a 1-Lipschitz lifted potential.", "formal_line": "MC(f) - MC(f|_R) = k_R + e_R with lifted potential k_R + ψ_R 1-Lipschitz; J_2 spectrum has four (1,0) and six (0,1).", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-114/", "receipt": "zkgolf-decomp/reports/CERT-DUAL.md, zkgolf-decomp/reports/CERT-COALG.md, zkgolf-decomp/reports/COUNCIL-LAWVERE.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, dual-certificate.receipt.json (SHA-256 01fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4), finite_certificate.receipt.json (SHA-256 c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6), lawvere_five_cent.receipt.json", "external_value": "Complexity theorists analyzing circuit lower bounds and inductive complexity measures via restriction methods.", "statement": "The drop in multiplicative complexity when restricting a Boolean function splits exactly into killed cost plus residual inefficiency, with a 1-Lipschitz lifted potential.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-115", "kind": "finding", "title": "Exact Kummer endpoint valuation and boundary-alias count for FullAdd", "title_plain": "Proves an exact 2-adic valuation formula for the multiplicative complexity of multi-operand addition and determines its boundary-alias count.", "formal_line": "MC(FullAdd(K,n)) = v_2((KB_n)! / (B_n!)^K) for K,n ≥ 1, B_n = 2^n - 1; eventual gap is (r-1)K - 2^r + 2 for r = ⌈log_2 K⌉", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-115/", "receipt": "zkgolf-decomp/reports/COUNCIL-HARDY.md, zkgolf-decomp/reports/COUNCIL-MINE.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, carry_calculus.py, residual_hankel.py", "external_value": "NONE", "statement": "Proves an exact 2-adic valuation formula for the multiplicative complexity of multi-operand addition and determines its boundary-alias count.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-116", "kind": "finding", "title": "Strict syntactic-equivariant inverse-χ costs at widths 3, 5, and 7", "title_plain": "The exact multiplicative costs for symmetric implementations of inverse-chi S-boxes at sizes 3, 5, and 7 are 3, 10, and 21, establishing the exact cost penalty of requiring symmetry.", "formal_line": "E_3 = 3, E_5 = 10, E_7 = 21 for strict syntactic-equivariant inverse-χ; equivariance taxes are 0, 4, 12 over unrestricted costs 3, 6, 9", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-116/", "receipt": "zkgolf-decomp/reports/EXP-EQ-TAX.md, zkgolf-decomp/reports/SB-TAX.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, equivariant-family-replay.json, eq-tax-certificate.json, n7_plain_receipt.json, n7_seeded_receipt.json", "external_value": "Symmetric cryptographers and circuit designers optimizing equivariant round functions such as Keccak and Ascon inverse-χ layers.", "statement": "The exact multiplicative costs for symmetric implementations of inverse-chi S-boxes at sizes 3, 5, and 7 are 3, 10, and 21, establishing the exact cost penalty of requiring symmetry.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-117", "kind": "finding", "title": "Bounds on the strict width-nine cost E_9", "title_plain": "The strict width-nine cost E_9 is proved to lie between 18 and 36, leaving the exact value open.", "formal_line": "18 ≤ E_9 ≤ 36", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-117/", "receipt": "zkgolf-decomp/reports/SB-TAX.md, zkgolf-decomp/reports/EXP-EQ-TAX.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/sb-tax-scratch/n9_one_full_floor_receipt.json (SHA-256 5a5237e61432a07345684ff252cb112ad2aeb2606ffda9bcfab3500ca009b25e), zkgolf-decomp/sb-tax-scratch/cleanroom_upper.py (SHA-256 c03bb6312c08c9b35633cb3817c69e41ba3da401e3ad5957a594709adef51a00)", "external_value": "NONE", "statement": "The strict width-nine cost E_9 is proved to lie between 18 and 36, leaving the exact value open.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-118", "kind": "finding", "title": "Exact shear symmetry of multiplicative complexity in constant addition", "title_plain": "Constant addition exhibits an exact shear symmetry under shift by 2^(n-2), yielding uniform multiplicative complexities of 2 at width two and 5 at width three.", "formal_line": "MC(F_{n,K}) = MC(F_{n,K+2^{n-2}}), with MC(F_{2,K}) = 2 and MC(F_{3,K}) = 5 for all constant offsets K", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-118/", "receipt": "zkgolf-decomp/reports/CONST-THEORY.md, CONST-BUILD.md, STATE-OF-PROGRAM-V2.md; AX162 receipts: heap-and-width32.receipt.json (97495ea7...), small-grid-lower.receipt.json (af5365a9...)", "external_value": "Circuit complexity theorists and zero-knowledge compiler designers optimizing modular constant addition gadgets.", "statement": "Constant addition exhibits an exact shear symmetry under shift by 2^(n-2), yielding uniform multiplicative complexities of 2 at width two and 5 at width three.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-119", "kind": "finding", "title": "Formula for canonical constant-heap complexity H_n(K)", "title_plain": "For the canonical constant-heap class, the complexity formula H_n(K) is proven to equal 4n - 6 - λ_n(K).", "formal_line": "H_n(K) = 4n - 6 - λ_n(K) for the canonical constant-heap class", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-119/", "receipt": "zkgolf-decomp/const-theory-scratch/heap-and-width32.receipt.json, zkgolf-decomp/const-build-scratch/heap-verification-receipt.json, zkgolf-decomp/const-build-scratch/independent-module-replay.json, zkgolf-decomp/reports/CONST-THEORY.md, zkgolf-decomp/reports/CONST-BUILD.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md", "external_value": "NONE", "statement": "For the canonical constant-heap class, the complexity formula H_n(K) is proven to equal 4n - 6 - λ_n(K).", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-121", "kind": "finding", "title": "Refutation of the general carry-bond tensor floor", "title_plain": "A proposed general carry-bond and tensor lower bound floor is false because literal injection of the carry sector fails at the second step.", "formal_line": "Full-carry sector injection into product-state space fails at J_2: degree-4 indicators not in two-gate span; general floor is refuted.", "status": "REFUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "algebraic complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-121/", "receipt": "zkgolf-decomp/reports/VERIFY-FEYNMAN.md, CERT-TENSOR.md, STATE-OF-PROGRAM-V2.md, bond-results.json (SHA-256 5103dbbacd613a110945546000f64ed543b1ea361e638367380470bb76219c49), verify_bond.py (SHA-256 c05660e5454507676dc420378ee6ae976b677837df8a5b38e79a2ae9e836aea7)", "external_value": "Researchers studying tensor rank and lower-bound techniques for addition circuits and product-state embeddings.", "statement": "A proposed general carry-bond and tensor lower bound floor is false because literal injection of the carry sector fails at the second step.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-122", "kind": "finding", "title": "Exact complexity of the canonical heap-prefix class", "title_plain": "Under the restriction that the initial carry state is closed and canonical, the multiplicative complexity equals T(k,n), but this bound does not transfer to the unrestricted case.", "formal_line": "Complexity equals T(k,n) for the canonical heap-prefix class; transfer counterexample shows this does not prove unrestricted MC(A_{k,n})=T(k,n)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-122/", "receipt": "zkgolf-decomp/reports/EXP-FORK-CN.md, SB-HEAP.md, FCNS-C-POTENTIAL.md, STATE-OF-PROGRAM-V2.md, carry-top-quotients.json, heap-affine-sections.json, state-exposure-checks.json, quotient-spans.json, bank-span-compare.json", "external_value": "Researchers studying multiplicative complexity lower bounds and prefix decomposition methods in boolean circuits.", "statement": "Under the restriction that the initial carry state is closed and canonical, the multiplicative complexity equals T(k,n), but this bound does not transfer to the unrestricted case.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-123", "kind": "finding", "title": "Necessary carrier sequence and connected-leakage bounds for hypothetical p14", "title_plain": "Establishes necessary carrier sequence structures and connected-leakage rank bounds for hypothetical 14-multiplication circuits.", "formal_line": "0→H_6→E_10→C_4→0; rank Q_2(fg) ≤ 2r+2s+rs+2; first mixed catalyst connected rank ≤ 2", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-123/", "receipt": "zkgolf-decomp/reports/SB-P14INJ.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, two_jet_and_mixed_rank_receipt.json, census_crosscheck_receipt.json", "external_value": "NONE", "statement": "Establishes necessary carrier sequence structures and connected-leakage rank bounds for hypothetical 14-multiplication circuits.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-124", "kind": "finding", "title": "Reduction of named p14 census residual to 221 cells", "title_plain": "Automated search eliminated 596 candidate decomposition cells, reducing the surviving named p14 candidate pool from 817 to 221 cells while leaving general p14 open.", "formal_line": "596 named p14 cells proved dead (327 s=0, 269 s=1); 68 s=1 timed out and 153 multi-catalyst remain, leaving 221 named surviving cells", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": true, "paper_url": "/research/papers/mf-124/", "receipt": "zkgolf-decomp/reports/EXP-CENSUS-S1.md, zkgolf-decomp/reports/SB-P14INJ.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/exp-census-scratch/aggregate-audit.json, zkgolf-decomp/sb-p14inj-scratch/census_crosscheck_receipt.json, zkgolf-decomp/EXP-CENSUS-PROGRESS.log, zkgolf-decomp/sb-p14inj-scratch/component_p7_probe_receipt.json", "external_value": "NONE", "statement": "Automated search eliminated 596 candidate decomposition cells, reducing the surviving named p14 candidate pool from 817 to 221 cells while leaving general p14 open.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result supersedes the 817-cell snapshot from MF-105; the general p14 SAT/UNSAT problem remains open.", "condition": "" }, { "id": "MF-125", "kind": "finding", "title": "Orbit and comodule anatomy of obtained J_3 witnesses", "title_plain": "The obtained J_3 circuit witnesses partition into exactly seven unpointed gate-space orbits and ten pointed restriction-comodule types, though completeness over all optima remains open.", "formal_line": "Obtained J_3 witnesses partition into 7 unpointed gate-space orbits and 10 pointed restriction-comodule types; full census remains open.", "status": "COMPUTED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "adders-and-counters", "school": "exact synthesis methods", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-125/", "receipt": "module-data.receipt.txt, finite_certificate.receipt.json, j3.analysis.json; reports CERT-SYZ.md, CERT-COALG.md, EXP-ANAT-OPTIMA.md, STATE-OF-PROGRAM-V2.md", "external_value": "Researchers studying equivalence classes, comodule representations, and orbit classification of optimal boolean circuit witnesses.", "statement": "The obtained J_3 circuit witnesses partition into exactly seven unpointed gate-space orbits and ten pointed restriction-comodule types, though completeness over all optima remains open.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-128", "kind": "finding", "title": "Exact unrestricted multiplicative complexity of binary GF(8) multiplication", "title_plain": "Multiplication in the finite field GF(8) requires exactly six AND gates over the binary field in general acyclic circuits.", "formal_line": "MC_XAG,F_2(F_8 × F_8 → F_8) = 6 in basis F_2[r]/(r^3 + r + 1)", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-128/", "receipt": "zkgolf-decomp/FREE-05.md, zkgolf-decomp/free-05-scratch/gf8-prefix-bootstrap.receipt.json, zkgolf-decomp/free-05-scratch/gf8-six-product-replay.receipt.json, zkgolf-decomp/free-verify-scratch/gf8-cleanroom.receipt.json, zkgolf-decomp/free-verify-scratch/decisive-rerun.receipt.json", "external_value": "Cryptographers and hardware circuit designers implementing Galois field arithmetic with minimal multiplicative gates.", "statement": "Multiplication in the finite field GF(8) requires exactly six AND gates over the binary field in general acyclic circuits.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-129", "kind": "finding", "title": "Exact multiplicative complexity of the forward χ mapping", "title_plain": "The forward χ transformation requires exactly n AND gates for every width n ≥ 3 and is equivalent to Wolfram rule 210 up to output rotation.", "formal_line": "MC(χ_n) = n for every n ≥ 3, with coordinate rule χ_i = Rule210_{i+1}", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-129/", "receipt": "zkgolf-decomp/surface-verify-scratch/independent-surface.receipt.json (SHA-256 44fee010c465c25c5f7c41c4fb3978a825d87e65db09479eb9c8a76d588b8391), zkgolf-decomp/surface-verify-scratch/independent-proof-audit.md (SHA-256 ff828f2a98a8c8825cec56a3a04365b5800ef2e3464640c1e2e8c364d5d8cd63), zkgolf-decomp/SURFACE-VERIFY.md (Claim 1 SOUND)", "external_value": "Cryptographers and zero-knowledge circuit designers optimizing Keccak χ layer evaluations across arbitrary register widths.", "statement": "The forward χ transformation requires exactly n AND gates for every width n ≥ 3 and is equivalent to Wolfram rule 210 up to output rotation.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-130", "kind": "finding", "title": "Exact multiplicative complexity of all powers of the width-four chi map", "title_plain": "The multiplicative complexity of the width-four chi map over GF(2) is exactly 0 at power 0, 4 at all positive odd powers, and 5 at all positive even powers.", "formal_line": "MC(χ₄⁰)=0, MC(χ₄)=4, MC(χ₄²)=5, MC(χ₄³)=4; MC(χ₄^k)=5 for even k≥2, MC(χ₄^k)=4 for odd k≥1", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-130/", "receipt": "zkgolf-decomp/free-06-scratch/chi-square-symbolic.receipt.json, zkgolf-decomp/free-06-scratch/chi4-semigroup.receipt.json", "external_value": "Circuit designers optimizing iterated Keccak-style chi transformations in arithmetic circuit and zero-knowledge environments.", "statement": "The multiplicative complexity of the width-four chi map over GF(2) is exactly 0 at power 0, 4 at all positive odd powers, and 5 at all positive even powers.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-131", "kind": "finding", "title": "Classification of One-Word XOR-Mask Transports for Four-Word Addition", "title_plain": "One-word XOR-mask transports of four-word modular addition with modified constants are completely classified in the public-output affine family.", "formal_line": "Affine transport exists iff 2^(n-2)|M and 2^(n-2)|Δ; componentwise affine correction exists iff 2^(n-2)|M and Δ ≡ M (mod 2^(n-1)).", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "exact synthesis methods", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-131/", "receipt": "zkgolf-decomp/free-04-scratch/mask-shear.receipt.json, zkgolf-decomp/free-04-scratch/output-mixing.receipt.json, zkgolf-decomp/free-04-scratch/mask-shear.verify.json", "external_value": "NONE", "statement": "One-word XOR-mask transports of four-word modular addition with modified constants are completely classified in the public-output affine family.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-132", "kind": "finding", "title": "The Boolean ramification polytope common evaluation conjecture", "title_plain": "A proposed general equality for the Boolean ramification polytope remains an open conjecture because its definition fails at the zero-variable boundary, though specific costed lower bounds survive.", "formal_line": "MC(F) ≥ r+D-2-c(Q) survives, but unqualified polytope equality is undefined at V=0 and remains a CONJECTURE (GAP)", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "quadratic-hull", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-132/", "receipt": "zkgolf-decomp/SURFACE-LANGLANDS.md, zkgolf-decomp/SURFACE-VERIFY.md, zkgolf-decomp/surface-verify-scratch/independent-surface.receipt.json (SHA-256 44fee010c465c25c5f7c41c4fb3978a825d87e65db09479eb9c8a76d588b8391)", "external_value": "NONE", "statement": "A proposed general equality for the Boolean ramification polytope remains an open conjecture because its definition fails at the zero-variable boundary, though specific costed lower bounds survive.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-133", "kind": "finding", "title": "Finite classification of the primary-affine pairing-one localization component", "title_plain": "The primary-affine pairing-one localization component is completely classified by an exact 69-polynomial Gröbner basis containing exactly eight normalized factor-plane classes.", "formal_line": "Primary-affine pairing-one kernel has 8 points (full kernel 15); reduced lex Gröbner basis has 69 polynomials; 5/3 with zero affine tail.", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "exact synthesis methods", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-133/", "receipt": "zkgolf-decomp/h-groebner-scratch/kernel-groebner-receipt.json (SHA-256 35ae2fc6b8e6b29e2d0458f85a18b7a7271989c0b5bb3e7882885f45e1faaa95), zkgolf-decomp/h-groebner-scratch/kernel-groebner-check.json (SHA-256 7e6df0dc461792d1c27485b76d1f2034ae0100410a62dbb6bf3ec1f2b2d08345), zkgolf-decomp/h-razborov-scratch/checker-receipt.json (SHA-256 bb5cde4bd47bc6d334bd0250528b4eeaa12470fa5f61b0f7f6222be04cea2cc4), and zkgolf-decomp/h-rota-scratch/selector-fibre-poset.verification.json (SHA-256 ccb3789a20009e25653a76b49ccae5f7eed045574e5fe92ce0291cd01f886d07)", "external_value": "NONE", "statement": "The primary-affine pairing-one localization component is completely classified by an exact 69-polynomial Gröbner basis containing exactly eight normalized factor-plane classes.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-134", "kind": "finding", "title": "Exact joint multiplicative complexity of chi_n and its square", "title_plain": "For all widths n >= 4, the joint multiplicative complexity of evaluating chi_n alongside its composition chi_n^2 is shown to be exactly 2n.", "formal_line": "MC(chi_n, chi_n^2) = 2n for all n >= 4", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-134/", "receipt": "zkgolf-decomp/COUNCIL3-VERIFY-EXPOSURE.md, COUNCIL3-INV-TURING.md, COUNCIL3-REF-GODEL.md, COUNCIL3-REF-ARNOLD.md", "external_value": "Cryptographic circuit designers analyzing multi-round Keccak chi composition and sharing defect lower bounds.", "statement": "For all widths n >= 4, the joint multiplicative complexity of evaluating chi_n alongside its composition chi_n^2 is shown to be exactly 2n.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-135", "kind": "finding", "title": "Three sound whole-function multiplicative complexity lower-bound floors", "title_plain": "Three new provable lower bounds on multiplicative complexity were established using graph additive energy, Walsh butterfly growth, and isotropic switching over affine flats.", "formal_line": "MC(F) >= ceil(-log2 delta(F)/log2(8/5)), MC(F) >= max(ceil Lambda, ceil Sigma), MC(F) >= ceil(E_H MC(F|H) + alpha_{N,d})", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-135/", "receipt": "zkgolf-decomp/COUNCIL3-VERIFY-FLOORS.md", "external_value": "Provides general, invariant whole-function lower-bound instruments for boolean function analysis and cryptographic circuit complexity.", "statement": "Three new provable lower bounds on multiplicative complexity were established using graph additive energy, Walsh butterfly growth, and isotropic switching over affine flats.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-136", "kind": "finding", "title": "Exact multiplicative complexity of 5-by-3 addition and six addition floors", "title_plain": "The multiplicative complexity of adding a 5-bit integer and a 3-bit integer is exactly 5, and six other addition lower bounds are improved.", "formal_line": "MC(Add(5,3)) = 5, with Add(5,4) >= 8, Add(5,5) >= 10, Add(6,3) >= 6, Add(6,4) >= 9, Add(7,3) >= 7, Add(7,4) >= 11", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-136/", "receipt": "COUNCIL3-VERIFY-FLOORS.md, MF-135 floor (ii)", "external_value": "Circuit designers and cryptographers implementing small integer additions with minimal non-linear gates.", "statement": "The multiplicative complexity of adding a 5-bit integer and a 3-bit integer is exactly 5, and six other addition lower bounds are improved.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-137", "kind": "finding", "title": "The packing lemma lower bound for multiplicative complexity", "title_plain": "A solver-free lower bound shows that the multiplicative complexity of a function is at least the dimension of its nonlinear output span modulo affine functions plus the minimum gate cost in that span minus one.", "formal_line": "MC(F) ≥ dim(V) + μ(V) - 1, where V is nonlinear output span mod affine and μ(V) is min gate cost of any nonzero scalar class in V", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "further-results", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-137/", "receipt": "zkgolf-decomp/RECORD-WINOGRAD-P5.md", "external_value": "Researchers seeking fast, solver-free lower bounds on multiplicative complexity using rank and degree tests.", "statement": "A solver-free lower bound shows that the multiplicative complexity of a function is at least the dimension of its nonlinear output span modulo affine functions plus the minimum gate cost in that span minus one.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-138", "kind": "finding", "title": "Exact multiplicative complexity of the two-column C7 carry transducer", "title_plain": "The multiplicative complexity of the two-column C7 carry transducer is exactly six AND gates.", "formal_line": "MC(T) = 6 for the 11-input, 5-output two-column C7 carry transducer", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-138/", "receipt": "RECORD-WINOGRAD-P5.md", "external_value": "Designers synthesizing multi-operand addition networks and compressor trees.", "statement": "The multiplicative complexity of the two-column C7 carry transducer is exactly six AND gates.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-139", "kind": "finding", "title": "Gate-class dimension increment per C7 column", "title_plain": "Each added column in the C7 family requires exactly three new gate dimensions beyond the previous prefix, forcing any circuit extending that prefix to cost three gates per column.", "formal_line": "For c ∈ {2,3,4}, C7 c-column target requires 3 new gate-class dimensions beyond the 3(c-1)-gate prefix, giving exact cost 3c under prefix constraint", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-139/", "receipt": "zkgolf-decomp/record-scratch/wider_seam/prefix2.py, c7_correct.py, gate_rank.py, zkgolf-decomp/RECORD-WIDER-SEAM-PROGRESS.log", "external_value": "Researchers designing prefix-constrained synthesis techniques or dimension-counting lower bounds for structured boolean circuits.", "statement": "Each added column in the C7 family requires exactly three new gate dimensions beyond the previous prefix, forcing any circuit extending that prefix to cost three gates per column.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-140", "kind": "finding", "title": "Exact multiplicative complexity of A_{4,3} and constant-added addition F_{3,K}", "title_plain": "Adding any constant modulo 8 to a four-input addition requires exactly five multiplications, matching the zero-constant case, though this freeness fails to lift to larger widths.", "formal_line": "MC(A_{4,3}) = 5 = MC(F_{3,K}) for every K mod 8; MC(F_{2,K}) = 2 = MC(A_{4,2})", "status": "COMPUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-140/", "receipt": "RECORD-CONSTFREE.md, RECORD-CONSTLIFT.md", "external_value": "NONE", "statement": "Adding any constant modulo 8 to a four-input addition requires exactly five multiplications, matching the zero-constant case, though this freeness fails to lift to larger widths.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-141", "kind": "finding", "title": "Full linear rank of multiplicative rows in the SHA-256 leader circuit", "title_plain": "All 22,215 product equations in the SHA-256 circuit achieve full rank over GF(2), ruling out row elimination via linear combinations.", "formal_line": "Rank of 22,215 product equations modulo affine forms is 22,215/22,215 over GF(2); no linear product row elimination exists.", "status": "COMPUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "arithmetization and proof systems", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-141/", "receipt": "zkgolf-decomp/UNIT4-FREE03-PROGRESS.log", "external_value": "Circuit optimizers and zero-knowledge proof designers seeking reductions in SHA-256 multiplication count.", "statement": "All 22,215 product equations in the SHA-256 circuit achieve full rank over GF(2), ruling out row elimination via linear combinations.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-142", "kind": "finding", "title": "Multiplicative complexity bounds for chi_5^2", "title_plain": "The multiplicative complexity of the squared five-bit Chi permutation is narrowed to either 6 or 7 AND gates.", "formal_line": "6 ≤ MC(chi_5^2) ≤ 7", "status": "COMPUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "construction", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-142/", "receipt": "zkgolf-decomp/CHI5SQ-SEAM.md", "external_value": "Cryptographers optimizing Keccak Chi round evaluation in arithmetic circuits.", "statement": "The multiplicative complexity of the squared five-bit Chi permutation is narrowed to either 6 or 7 AND gates.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Prior bracket was [6,10].", "condition": "" }, { "id": "MF-143", "kind": "finding", "title": "Refutation of three proposed multiplicative complexity laws", "title_plain": "Three proposed complexity formulas for odd constant multipliers, square catalyst overhead, and odd modular reciprocals are disproved by exact counterexamples.", "formal_line": "MC(11 x mod 128) = 6 ≠ 5, kappa_sq(7) ∈ {2,3} ≠ 1, and MC(I_9) = 6 ≠ 5, refuting three conjectured complexity laws.", "status": "REFUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "further-results", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-143/", "receipt": "CONST-MULT-LAW.md, SQUARE-CATALYST.md", "external_value": "Researchers analyzing multiplicative complexity lower bounds and scaling laws for modular arithmetic operations.", "statement": "Three proposed complexity formulas for odd constant multipliers, square catalyst overhead, and odd modular reciprocals are disproved by exact counterexamples.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-144", "kind": "finding", "title": "The packing filtration lower bound and master identity for multiplicative complexity", "title_plain": "Multiplicative complexity is characterized by a master identity and lower bounds over a filtration of subspace packing numbers.", "formal_line": "MC(F) = dim V + max_j (m_{j+1}(V) - j - 1) and MC(F) ≥ dim V' - j + m_{j+1}(V') - 1 for all V' ≤ V", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "quadratic-hull", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-144/", "receipt": "geometry/REPORT.md §§3–4, 6; geometry/restricted.py, sweep.py, sweep_results.json, SWEEP-TABLE.md", "external_value": "Researchers establishing circuit lower bounds or multiplicative complexity bounds for boolean functions and cryptographic S-boxes.", "statement": "Multiplicative complexity is characterized by a master identity and lower bounds over a filtration of subspace packing numbers.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "The gate-span and rank arguments are credited to Schnorr (1989), Boyar–Peralta–Pochuev (2000), and Boyar–Find (2018). However, the filtration over j was not located as an explicit identity in prior literature.", "condition": "" }, { "id": "MF-145", "kind": "finding", "title": "Packing floors on quadratic multiplicative complexity for linear spaces of quadratics", "title_plain": "A lower bound on the quadratic multiplicative complexity of a space of quadratic forms was proved via coordinate code weight counting, matching exact values on two-dimensional planes.", "formal_line": "qMC(W) ≥ ceil(sum_{w ≠ 0} mc(w) / 2^{d-1}) for d-dim space W; d=2 yields qMC(W) ≥ ceil((mc(q1)+mc(q2)+mc(q3))/2)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "quadratic-hull", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-145/", "receipt": "geometry/REPORT.md §4, geometry/exact_m2.py, exact_m2_result.json, wave1-ms3/u3_fano_floor.py, wave1-ms3/REPORT.md §2 U3", "external_value": "Researchers studying the multiplicative complexity of systems of quadratic forms and cryptographic S-boxes.", "statement": "A lower bound on the quadratic multiplicative complexity of a space of quadratic forms was proved via coordinate code weight counting, matching exact values on two-dimensional planes.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This is a partial result. Griesmer- and simplex-style counting methods apply to the decomposable-coordinate code, while Mirwald–Schnorr's normal form plausibly resolves the `d = 2` case.", "condition": "" }, { "id": "MF-146", "kind": "finding", "title": "Deficit law lower bound for all-quadratic spaces", "title_plain": "Every all-quadratic space with nonzero classes of cost μ satisfies a multiplicative complexity lower bound of dim V - 2 + ceil(3μ/2).", "formal_line": "MC(F) ≥ dim V - 2 + ⌈3μ/2⌉ for all-quadratic V with every nonzero class of cost μ", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "quadratic-hull", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-146/", "receipt": "geometry/mcgeom.py (Target.bound_corC), sweep_results.json", "external_value": "Researchers bounding multiplicative complexity of quadratic spaces in algebraic complexity and cryptography.", "statement": "Every all-quadratic space with nonzero classes of cost μ satisfies a multiplicative complexity lower bound of dim V - 2 + ceil(3μ/2).", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "PARTIAL (composite of MF-144/MF-145).", "condition": "" }, { "id": "MF-147", "kind": "finding", "title": "Basis-free lower bound on the multiplicative complexity of GF(2^k) multiplication", "title_plain": "Proves a general basis-free lower bound of ceil(5k/2) - 2 for the multiplicative complexity of finite field multiplication over GF(2^k).", "formal_line": "MC(GF(2^k) multiplication) ≥ ⌈5k/2⌉ - 2 in the unrestricted XAG model, basis-free", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-147/", "receipt": "geometry/REPORT.md §5, sweep_results.json", "external_value": "Algebraic complexity theorists and symmetric cryptographers evaluating non-linear gate complexity of Galois field arithmetic.", "statement": "Proves a general basis-free lower bound of ceil(5k/2) - 2 for the multiplicative complexity of finite field multiplication over GF(2^k).", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result appears to be new: existing bilinear and tensor-rank literature (Chudnovsky–Chudnovsky; Shparlinski–Tsfasman–Vlăduţ; Ballet et al.) bounds a different model, and NIST lists the multiplicative complexity (MC) of vectorial functions on `>= 5` bits as an open problem.", "condition": "" }, { "id": "MF-148", "kind": "finding", "title": "Exact unrestricted multiplicative complexity of 3-term binary polynomial multiplication", "title_plain": "The exact multiplicative complexity of multiplying two 3-term polynomials over GF(2) is proved to be 6 in the unrestricted model via three independent derivations.", "formal_line": "MC(polymul_3 over F2) = 6 in the unrestricted model", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-148/", "receipt": "geometry/replay_polymul3.py (SHA-256 047ea83d…0ce907), polymul3_upper_replay.json, floors/replay_clmul3.py, out_family2.json, wave1-gf16/verify_finisher.json, fullscan_floors.json", "external_value": "Circuit designers and cryptographic implementers optimizing small binary field and polynomial arithmetic over GF(2).", "statement": "The exact multiplicative complexity of multiplying two 3-term polynomials over GF(2) is proved to be 6 in the unrestricted model via three independent derivations.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This is a known value obtained using classical Karatsuba multiplication for the NIST binary-polynomial category. It remains only a partial result regarding exactness in an unrestricted model.", "condition": "" }, { "id": "MF-149", "kind": "finding", "title": "A lower bound on multiplicative complexity of 2x2 matrix multiplication over F2", "title_plain": "The unrestricted multiplicative complexity of 2x2 matrix multiplication over F2 is at least 6, narrowing its range to 6 or 7.", "formal_line": "MC(M_2 over F2) >= 6, narrowing MC(M_2 over F2) to [6,7]", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-149/", "receipt": "geometry/bilinear_targets.py (matmul2_target), sweep_results.json key matmul2", "external_value": "Researchers studying matrix multiplication algorithms and non-bilinear multiplicative complexity over small finite fields.", "statement": "The unrestricted multiplicative complexity of 2x2 matrix multiplication over F2 is at least 6, narrowing its range to 6 or 7.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Winograd’s rank-7 optimality result applies specifically to bilinear algorithms; no unrestricted bound has been established.", "condition": "" }, { "id": "MF-150", "kind": "finding", "title": "Exact multiplicative complexity of 2×2 unsigned integer multiplier", "title_plain": "The exact multiplicative complexity of the 2×2 unsigned integer multiplier is four non-linear multiplications.", "formal_line": "MC(2×2 unsigned integer multiplier) = 4", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-150/", "receipt": "floors/family_sweep2.py, out_family2.json (mult_2x2), replay_mult2.py", "external_value": "Hardware arithmetic designers and cryptographic circuit implementers optimizing small integer multipliers.", "statement": "The exact multiplicative complexity of the 2×2 unsigned integer multiplier is four non-linear multiplications.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This is a partial result, as small multipliers are well-studied and likely already known. Please check the NIST Circuits repository before external use.", "condition": "" }, { "id": "MF-151", "kind": "finding", "title": "Exact characterization of additive quadruples destroyed by one AND gate", "title_plain": "An additive quadruple in a transcript graph survives appending an AND gate if and only if its inputs do not cover all four affine fibers of the gate, eliminating exactly 3/8 of uniform patterns.", "formal_line": "Quadruple survives e=uv iff {P(x1),P(x2),P(x3),P(x4)} ≠ GF(2)^2 for P=(u,v); uniform rainbow density killed is 24/64 = 3/8", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "quadratic-hull", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-151/", "receipt": "floors/energy58.py", "external_value": "Researchers studying additive energy decay and transcript graph methods for Boolean circuit lower bounds.", "statement": "An additive quadruple in a transcript graph survives appending an AND gate if and only if its inputs do not cover all four affine fibers of the gate, eliminating exactly 3/8 of uniform patterns.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "The Fourier identity (-1)^{uv} = (1 + (-1)^u + (-1)^v - (-1)^{u+v})/2 is standard, but the additive-quadruple reformulation was not found in prior literature and appears to be new.", "condition": "" }, { "id": "MF-152", "kind": "finding", "title": "Sharpness of the 5/8 law and multiplicative complexity of n-bit addition", "title_plain": "The 5/8 lower bound constant is unconditionally sharp across multiple function families and reproves the exact multiplicative complexity n - 1 for n-bit addition without SAT solvers.", "formal_line": "delta = (5/8)^p attained at MC = p for disjoint ANDs, mod 2^n addition (MC = n - 1), and full addition (MC = n); Floor A <= 1.4747 m", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-152/", "receipt": "floors/energy58.py, transfer_energy.py, out_transfer_k2.json, transfer_full.py, scaling_probe.py, out_scaling.json", "external_value": "Complexity theorists and cryptographers seeking analytical lower-bound techniques for multiplicative complexity of arithmetic circuits.", "statement": "The 5/8 lower bound constant is unconditionally sharp across multiple function families and reproves the exact multiplicative complexity n - 1 for n-bit addition without SAT solvers.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "While the value n - 1 was previously reported in BPP (2000, abstract only), the sharpness of this internal constant appears to be new.", "condition": "" }, { "id": "MF-153", "kind": "finding", "title": "Solver-Free Multiplicative Complexity Lower Bounds from Degree and Walsh Floors", "title_plain": "Degree and Walsh spectral floors establish solver-free lower bounds on the multiplicative complexity of integer multiplication, carryless multiplication, and addition circuits.", "formal_line": "MC(3×3) ≥ 6, MC(4×4) ≥ 9, MC(clmul_4) ≥ 8, MC(clmul_5) ≥ 11, MC(Add(4,5)) ≥ 8 via degree and Walsh floor methods", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "further-results", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-153/", "receipt": "floors/out_family.json, out_family2.json, out_mixed.json, out_grid_add.json; carry/walsh_atlas.py, degree_atlas.py, out/walsh_atlas.json", "external_value": "Circuit complexity researchers analyzing non-asymptotic lower bounds for arithmetic circuits without SAT solvers.", "statement": "Degree and Walsh spectral floors establish solver-free lower bounds on the multiplicative complexity of integer multiplication, carryless multiplication, and addition circuits.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "These bounds are previously established: the degree floor is credited to Schnorr, and the Walsh floor is documented under entry MF-135.", "condition": "" }, { "id": "MF-154", "kind": "finding", "title": "Exact multiplicative complexity of chi_5^2", "title_plain": "The two-round Keccak nonlinear layer chi_5^2 requires exactly 7 AND gates, resolving the previous bracket of 6 to 7.", "formal_line": "MC(chi_5^2) = 7, closing the [6,7] bracket at the top, with eps_5 = 10 - 7 = 3 = eps_4", "status": "COMPUTED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-154/", "receipt": "chi/out_s04_chi5pow2_k6.json, out_s04b_chi5pow2_k6_nosym.json, out_s06_controls.json, out_s07_chi5pow2_k7.json, out_s09_chi5pow2_k7_replay.json, circuit_chi5pow2_k7.txt", "external_value": "Designers and cryptanalysts of Keccak-family primitives and researchers studying multiplicative complexity of iterated cryptographic S-boxes.", "statement": "The two-round Keccak nonlinear layer chi_5^2 requires exactly 7 AND gates, resolving the previous bracket of 6 to 7.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This appears to be a new result. As re-confirmed on 2026-09-01, no exact MC of any chi iterate has been published, and NIST lists no chi row.", "condition": "" }, { "id": "MF-155", "kind": "finding", "title": "Multiplicative complexity lower bounds and brackets for chi_6^2 and chi_7^2", "title_plain": "Exhaustive gate-budget exclusions show that two rounds of the Chi permutation require at least 8 AND gates on 6 bits and at least 9 AND gates on 7 bits, giving brackets of [8, 12] and [9, 14].", "formal_line": "MC(chi_6^2) >= 8 (bracket [8, 12]) and MC(chi_7^2) >= 9 (bracket [9, 14])", "status": "COMPUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-155/", "receipt": "chi/out_s04_chi6pow2_k7.json, out_s04b_chi6pow2_k7_nosym.json, out_s04b_chi7pow2_k8.json, out_s10_controls_w6.json", "external_value": "Cryptographers analyzing multiplicative complexity floors for iterated Keccak/Xoodoo-like Chi non-linear layers.", "statement": "Exhaustive gate-budget exclusions show that two rounds of the Chi permutation require at least 8 AND gates on 6 bits and at least 9 AND gates on 7 bits, giving brackets of [8, 12] and [9, 14].", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This is an apparently new result cataloged as entry MF-154, achieved using fewer searches.", "condition": "" }, { "id": "MF-156", "kind": "finding", "title": "The k-window law for joint multiplicative complexity of Chi iterates", "title_plain": "The joint multiplicative complexity of the first k iterates of the Chi permutation equals k times n precisely when k is at most floor(n/2), beyond which degree layer independence collapses.", "formal_line": "MC(chi_n, ..., chi_n^k) = k n for 1 <= k <= floor(n/2), sharp in k; joint rank drops below k n at k = floor(n/2) + 1", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-156/", "receipt": "chi/out_s02_powers_joint.json, out_s03_degree_layers.json, out_s03c_topform.json, out_s01_basics.json, out_s12_eps.json", "external_value": "Cryptographers analyzing the algebraic complexity and unrolled S-box implementations of Keccak and Xoodoo round functions.", "statement": "The joint multiplicative complexity of the first k iterates of the Chi permutation equals k times n precisely when k is at most floor(n/2), beyond which degree layer independence collapses.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Kriepke–Kyureghyan previously bounded Hadamard products for a single iterate using degree arguments, and the Schoone–Daemen order formula already established `chi_3^2 = id`. The joint-exposure vector statement and the `k = floor(n/2)` threshold were not located in prior literature.", "condition": "" }, { "id": "MF-157", "kind": "finding", "title": "Iterate structure, degree, and attributions for the Keccak chi mapping", "title_plain": "Iterates of the Keccak chi mapping satisfy algebraic degree j+1 up to floor(n/2) across odd and even dimensions, with multiple prior-art citations corrected.", "formal_line": "deg chi_n^j = j+1 for j ≤ ⌊n/2⌋ (FC-verified n=3..13); |im(chi_n)| = 2^n - 2^{n/2} for even n; Ascon S-box is B ∘ chi_5 ∘ A", "status": "COMPUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "structure theorem", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-157/", "receipt": "chi/out_s03c_topform.json, out_s01_basics.json, chi/BANK-CANDIDATES.md BC-08, BC-11, wave1-priorart/VERDICTS.md §1, sources/kriepke-kyureghyan-2024-801.txt", "external_value": "Cryptanalysts and circuit designers studying algebraic degree growth and multiplicative complexity of Keccak and Ascon chi layers.", "statement": "Iterates of the Keccak chi mapping satisfy algebraic degree j+1 up to floor(n/2) across odd and even dimensions, with multiple prior-art citations corrected.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Lemma A for odd n is due to Kriepke-Kyureghyan (CRYPTO 2024), while even n collisions match results from Schoone-Daemen (2024). The inverse chi is credited to Liu et al. (2022) and Biryukov et al. (2014).", "condition": "" }, { "id": "MF-158", "kind": "finding", "title": "Classification of relations implemented by single-row R1CS over F_p", "title_plain": "A single R1CS constraint over a prime field with any number of witnesses implements exactly one of four geometric relation types, establishing sharp size gaps and impossibility results.", "formal_line": "A single row A·B = C over F_p with m witnesses implements one of 4 types: F^n, {q=0}, {L≠0}∪({L=0}∩{q=0}), or {x : Δ(x) is a square}.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "R1CS over F_p", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-158/", "receipt": "zk/REPORT.md §3, zk/one_row_census.py, one_row_census.json", "external_value": "Zero-knowledge proof gadget designers and arithmetization researchers optimizing or proving lower bounds on R1CS constraint systems.", "statement": "A single R1CS constraint over a prime field with any number of witnesses implements exactly one of four geometric relation types, establishing sharp size gaps and impossibility results.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This appears to be a new, partial result applying elementary algebraic geometry to a single quadratic equation. We have not found this specific case in prior literature on Rank-1 Constraint System (R1CS) expressiveness classifications.", "condition": "" }, { "id": "MF-160", "kind": "finding", "title": "Exact R1CS Row Counts and Allocation-Independent Lower Bounds for Basic Gadgets", "title_plain": "Establishes the exact minimum number of R1CS constraint rows needed for standard zero-knowledge gadgets including IsZero, n-ary AND, n-ary OR, and 4-input XOR, proving textbook implementations are optimal.", "formal_line": "R1CS row counts: IsZero = 2, AND_n = OR_n = 2 for all n ≥ 3, XOR4 = 2 over F_p (p ≥ 7), with no 1-row systems possible.", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "R1CS over F_p", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-160/", "receipt": "zk/BANK-CANDIDATES.md ZK-C-02, ZK-C-04, ZK-C-05; zk/REPORT.md", "external_value": "Circuit designers and compiler authors targeting R1CS zero-knowledge proof systems.", "statement": "Establishes the exact minimum number of R1CS constraint rows needed for standard zero-knowledge gadgets including IsZero, n-ary AND, n-ary OR, and 4-input XOR, proving textbook implementations are optimal.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result is partial: the gadgets rely on standard implementations from circomlib, but no proofs were found showing that these R1CS row counts are optimal.", "condition": "" }, { "id": "MF-161", "kind": "finding", "title": "Row complexity of n-bit range checks and u32 addition in R1CS", "title_plain": "An n-bit integer range check can be enforced in n R1CS rows, and 32-bit addition requires between 32 and 33 rows with carry-booleanity fusion ruled out by conic point counting.", "formal_line": "n-bit range check takes n rows (lower n exact for n <= 2); u32 add takes [32,33] rows with carry-booleanity fusion ruled out by conic counts", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "R1CS over F_p", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-161/", "receipt": "zk/BANK-CANDIDATES.md ZK-C-06, ZK-C-07; zk/REPORT.md", "external_value": "Zero-knowledge proof engineers optimizing arithmetization row counts and witness allocations for integer gadgets in R1CS.", "statement": "An n-bit integer range check can be enforced in n R1CS rows, and 32-bit addition requires between 32 and 33 rows with carry-booleanity fusion ruled out by conic point counting, conditional on transferring the Bézout bound from the algebraic closure to F_p (ML-080.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result is partial: the constructions follow standard practice, but the corresponding lower bounds have not yet been located.", "condition": "transferring the Bézout bound from the algebraic closure to F_p (ML-080" }, { "id": "MF-162", "kind": "finding", "title": "Witness power, degree collapse, and allocation walls over large prime fields", "title_plain": "Over large prime fields, witnesses enable non-zero testing impossible without them, degree collapse prevents parity-degree lower bounds from transferring, and tight constraint walls hold against arbitrary allocations.", "formal_line": "{x : x != 0} needs 1 witness, XOR3/MAJ3 degree collapse breaks MF-013 over F_p, and primitive walls hold against arbitrary allocations", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "R1CS over F_p", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-162/", "receipt": "zk/BANK-CANDIDATES.md ZK-C-08, zk/REPORT.md", "external_value": "SNARK and R1CS circuit designers seeking exact lower bounds and arithmetization barriers over large prime fields.", "statement": "Over large prime fields, witnesses enable non-zero testing impossible without them, degree collapse prevents parity-degree lower bounds from transferring, and tight constraint walls hold against arbitrary allocations.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "PARTIAL / INTERNAL.", "condition": "" }, { "id": "MF-163", "kind": "finding", "title": "Direct-sum theorems and lower bounds for unrestricted multiplicative complexity", "title_plain": "The number of multiplications needed to compute two independent functions together is proved to equal the sum of their individual complexities whenever at least one function meets its linear rank floor or both have complexity two.", "formal_line": "MC(F ⊕ G) ≥ dim V(F) + MC(G) with MC(F ⊕ G) = MC(F) + MC(G) if min(e(F), e(G)) = 0 or if dim V(F) = dim V(G) = 1 and MC(F) = MC(G) = 2", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-163/", "receipt": "directsum/REPORT.md §§2–4, directsum/BANK-CANDIDATES.md DS-1..DS-5, DS-10, DS-11", "external_value": "Complexity theorists and cryptographers evaluating non-linear gate counts and direct-sum behavior in Boolean circuit lower bounds.", "statement": "The number of multiplications needed to compute two independent functions together is proved to equal the sum of their individual complexities whenever at least one function meets its linear rank floor or both have complexity two.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Theorem A and the rank floor are partly known or likely folklore (BPP 2000; Mirwald–Schnorr for the quadratic model). Theorem D appears new: no direct-sum theorem was found for unrestricted Boolean complexity, with the nearest precedents being Strassen additivity, Shitov's refutation, Feig 1984, and KRW for formula depth.", "condition": "" }, { "id": "MF-164", "kind": "finding", "title": "Exact multiplicative complexity of direct sums of monomial Boolean functions", "title_plain": "Multiplicative complexity is shown to be strictly additive across several benchmark direct sums, including proving that x1x2x3 ⊕ y1y2y3y4 requires exactly 5 AND gates.", "formal_line": "MC(x1x2x3 ⊕ y1y2y3y4) = 5, MC(x1x2x3 ⊕ y1y2y3) = 4, and additivity holds across all tested direct-sum cells", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOWCASE", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-164/", "receipt": "directsum/REPORT.md §5; wave1-directsum2/out/t12_upper.json, t10_main_k5.json, t11_orbit_*_k4.json, t14_cover.json, t10_cc3_k{3,4}.json, t10_a1_k{3,4}.json, satlib.py", "external_value": "Researchers investigating the direct-sum conjecture and exact multiplicative complexity bounds for cryptographic and algebraic circuits.", "statement": "Multiplicative complexity is shown to be strictly additive across several benchmark direct sums, including proving that x1x2x3 ⊕ y1y2y3y4 requires exactly 5 AND gates.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Using standard methods from Calik–Turan–Peralta (ePrint 2015/848, 2018/002; arXiv 2005.01778), these specific direct-sum cells fall outside the n <= 6 census and appear to be new. The additivity conjecture remains open after three searches.", "condition": "" }, { "id": "MF-165", "kind": "finding", "title": "Equality of multiplicative and quadratic complexity for 3-spaces on at most 5 variables", "title_plain": "For every 3-dimensional space of quadratic forms on at most 5 variables, its general multiplicative complexity equals its quadratic multiplicative complexity.", "formal_line": "MC(W) = qMC(W) for all 3-dimensional spaces W <= Quad(n) with n <= 5 in the general XAG model.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-165/", "receipt": "wave1-ms3/REPORT.md §2 U1; n4_dim3.json, n5_k3_circuits.json, n5_k4_reachW.pkl, n5_k4_replay.json, audit_u1.json, audit_u1b.json, u1c_negctrl.json, states_n5b.py, run_k4_n5.py, PROGRESS.log 62–126 and 04:00–04:10Z block", "external_value": "Researchers studying multiplicative complexity bounds and quadratic circuit lower bounds in XAG models.", "statement": "For every 3-dimensional space of quadratic forms on at most 5 variables, its general multiplicative complexity equals its quadratic multiplicative complexity.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "The result for dim W <= 2 is known from Mirwald–Schnorr (1992), while Boyar–Find record dim >= 3 as open. Novelty regarding prior exhaustive checks for small n remains unverified in existing literature.", "condition": "" }, { "id": "MF-166", "kind": "finding", "title": "Exact multiplicative complexity of a Fano 3-space at n = 8 and floor census at n = 4", "title_plain": "A specific 3-dimensional space of quadratic forms in 8 variables has multiplicative complexity exactly 7, achieving an excess of 4 over its dimension without higher-order transfers, alongside an exhaustive gap census at 4 variables.", "formal_line": "For explicit Fano W <= Λ^2(F2^8) with dim W = 3, m_2(W) = 6 and qMC(W) = MC(W) = 7 = dim W + 4.", "status": "CLOSED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-166/", "receipt": "wave1-ms3/u3_fano_floor.py, u3_fano_floor.json, fano.py, fano_result.json, verify_fano.py, n4_full.json, n4_dim3.json, PROGRESS.log 1–61, 101–112", "external_value": "Researchers studying lower bound gaps and coordinate code embeddings in bilinear complexity over GF(2).", "statement": "A specific 3-dimensional space of quadratic forms in 8 variables has multiplicative complexity exactly 7, achieving an excess of 4 over its dimension without higher-order transfers, alongside an exhaustive gap census at 4 variables.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result is partial, and its novelty remains unverified because simplex-code packing is conceptually a standard technique in coding theory.", "condition": "" }, { "id": "MF-167", "kind": "finding", "title": "Exact quadratic multiplicative complexity of GF(2^4) multiplication and polymul_4", "title_plain": "In the quadratic model where products are formed from linear inputs, both GF(16) multiplication and 4-term polynomial multiplication require exactly 9 multiplications, while unrestricted complexity lies in [8, 9].", "formal_line": "qMC(GF(2^4) mult) = 9, qMC(polymul_4) = 9, unrestricted MC(GF(16) mult) ∈ [8,9], MC(polymul_4) ∈ [8,9]", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-167/", "receipt": "wave1-gf16/m3_sweep_result.json, m3_filteroff_result.json, m3_gf16_hp0_result.json, m3_control_s1..s7.json, symmetry.py, polymul4_close.json, fullscan_floors.json, verify_finisher.json V4/V5/V7/V8, REPORT.md §2", "external_value": "Circuit complexity theorists and hardware designers optimizing small Galois field and polynomial multiplication circuits.", "statement": "In the quadratic model where products are formed from linear inputs, both GF(16) multiplication and 4-term polynomial multiplication require exactly 9 multiplications, while unrestricted complexity lies in [8, 9].", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "The value 9 is already known from Karatsuba and Winograd's bilinear theory. However, whether this bound is new under the stronger quadratic model, which allows multiplications to mix operands, has not been verified.", "condition": "" }, { "id": "MF-168", "kind": "finding", "title": "Exact multiplicative complexity of GF(16) inversion and bounds for GF(32)", "title_plain": "Inversion in the 16-element finite field requires exactly 5 AND gates, while inversion in the 32-element field requires between 7 and 14 AND gates.", "formal_line": "MC(GF(2^4) inversion) = 5; MC(GF(2^5) inversion) ∈ [7, 14] with qMC(x^3) = qMC(x^5) = 7", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-168/", "receipt": "wave1-gf16/gf16_inv_circuit.json, verify_finisher.json V1/V2, mcbfs.py, extract_inv.py, inversion_profile.json, gf32_inv.json, gf32_circuit.py, gf32_inv_circuit.json", "external_value": "Useful for hardware and software implementers optimizing AES tower-field S-boxes and finite field cryptographic primitives.", "statement": "Inversion in the 16-element finite field requires exactly 5 AND gates, while inversion in the 32-element field requires between 7 and 14 AND gates.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "GF(16) inversion is the core of tower-field AES S-boxes (Canright 2005; Boyar–Peralta 2010), with Stoffelen’s SAT study of 4-bit S-boxes (FSE 2016) confirming the inverter cost at 5. The novelty of the GF(32) stage cost remains unverified.", "condition": "" }, { "id": "MF-169", "kind": "finding", "title": "Exact multiplicative complexity of 7-variable majority", "title_plain": "The 7-variable majority function requires exactly 4 AND gates over the basis with free XOR operations, closing the known bracket [3, 4].", "formal_line": "MC(MAJ7) = MC(T^7_4) = 4", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-169/", "receipt": "wave1-symmetric/unit_e2_maj7.py, out/unit_e2_maj7.json, PROGRESS.log", "external_value": "Researchers studying symmetric threshold functions, bit-heap counters, and non-linear circuit minimization.", "statement": "The 7-variable majority function requires exactly 4 AND gates over the basis with free XOR operations, closing the known bracket [3, 4].", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "Boyar–Peralta (2008) established the published bounds `[3,4]`, where Thm 10 gives `<= 4` and Thm 8 / Schnorr degree give `>= 3`. The exact value is verified by an internal check without claiming novelty, as it may already appear in the NIST `Circuits` dataset of symmetric functions up to 25 variables.", "condition": "" }, { "id": "MF-170", "kind": "finding", "title": "Exhaustion of the t = 5 Two-Phase Route for 22-Variable Symmetric Functions", "title_plain": "The standard two-phase compression route with five intermediate signals cannot implement every 22-variable symmetric Boolean function in 21 AND gates because the parity device yields zero gain at width five.", "formal_line": "C(22,5)=18 at state (1,1,1,2); max_{f∈S_22} min_{free} MC(g_f)=4 yielding bound 22; parity device has gain 0 at t=5 vs gain 1 at t=4", "status": "CLOSED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "Boolean ANF Degree / Symmetry", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-170/", "receipt": "wave1-symmetric/compressor.py, unit_f_family22.py, out/unit_f_family22.json, unit_g_parity.py, out/unit_g_parity.json, out/unit_j_crosscheck.json, out/unit_l_bw13.json, unit_i_end2end22.py, out/unit_i_end2end22.json", "external_value": "Circuit complexity and cryptography researchers optimizing multiplicative complexity bounds for symmetric Boolean functions.", "statement": "The standard two-phase compression route with five intermediate signals cannot implement every 22-variable symmetric Boolean function in 21 AND gates because the parity device yields zero gain at width five.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "The H_1 cost and the 21-vs-22 statement are established results credited to BCSTP (§4.1.3, §5.1, Tables 3–4). The exhaustive family decision and the t = 5 degeneration of the parity device appear to be new contributions and remain unverified.", "condition": "" }, { "id": "MF-171", "kind": "finding", "title": "Exact multiplicative complexity of carry-calculus and threshold functions", "title_plain": "Computes 59 exact multiplicative complexity values for multi-bit carry and threshold functions, proves a lower-bound formula for the Hamming-weight bridge, and confirms symmetric function bounds.", "formal_line": "59 exact MC values for small multi-bit threshold functions; D(m)=floor(log2 m)-[m!=2^b-1] proved for bridge; MCmax(S_m)=m-1 for m<=7", "status": "COMPUTED", "tier": "solver", "tier_label": "SOLVER-CONFIRMED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "Boolean ANF Degree / Symmetry", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-171/", "receipt": "carry/threshold_final.py, out/thresholds_final.json, mc_engine.py, solver.py, weight_bridge.py, out/weight_bridge.json, symmetric_exact.py, bridge_full.py, out/symmetric_exact.json", "external_value": "Circuit designers synthesizing multi-bit carry, adder threshold components, and symmetric Boolean functions with minimal AND gates.", "statement": "Computes 59 exact multiplicative complexity values for multi-bit carry and threshold functions, proves a lower-bound formula for the Hamming-weight bridge, and confirms symmetric function bounds.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "The symmetric cases and maximum MC bound are from BCSTP, correcting an earlier attribution to Boyar–Peralta (2008). The multi-bit carry cells and closed D(m) bridge formula are new to this work.", "condition": "" }, { "id": "MF-174", "kind": "finding", "title": "Refutation of five candidate multiplicative complexity lower-bound conjectures", "title_plain": "Five conjectured bounds and structural lemmas for multiplicative complexity, including a two-class cost sum, gate-local growth, and a three-power law, were disproved with explicit counterexamples.", "formal_line": "Five refutations: MC ≱ dim V + μ₁ + μ₂ - 2, plane parity bound fails, Lemma X fails at (4,3), linear catalysis fails, MC(χ,χ²,χ³) ≠ 3n", "status": "REFUTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-174/", "receipt": "geometry/sweep_results.json (plane_refuter), wave1-ms3/lemx.py, PROGRESS.log 86–87, wave1-symmetric/out/unit_c_catalysis.json, out/unit_d_obstruction.json, chi/out_s02_powers_joint.json", "external_value": "Prevents invalid lower-bound proofs by identifying explicit counterexamples across quadratic space complexity, plane parity, and power laws.", "statement": "Five conjectured bounds and structural lemmas for multiplicative complexity, including a two-class cost sum, gate-local growth, and a three-power law, were disproved with explicit counterexamples.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result is supported by prior work, including the Mirwald–Schnorr <= 2-form import boundary, Boyar–Find’s open-problem statement, and the BCSTP redundant weight encoding.", "condition": "" }, { "id": "MF-175", "kind": "finding", "title": "Degree distribution of the two-column C7 transducer nonlinear quotient", "title_plain": "A complete algebraic normal form computation confirmed that 12 of 15 nonzero cosets in the two-column C7 transducer have degree at least 3, verifying a predicted degree separation.", "formal_line": "For the two-column C7 transducer with dim V = 4, exactly 12 nonzero cosets in V have degree ≥ 3 and 3 cosets are quadratic.", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "symmetry-and-encodings", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-175/", "receipt": "wave2-zk-gadgets/scratch/check_c7_degrees.py", "external_value": "NONE", "statement": "A complete algebraic normal form computation confirmed that 12 of 15 nonzero cosets in the two-column C7 transducer have degree at least 3, verifying a predicted degree separation.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-176", "kind": "finding", "title": "Exact multiplicative complexity of all 16 optimal 4-bit S-box classes and standard ciphers", "title_plain": "The exact multiplicative complexity was determined for all 16 Leander-Poschmann optimal 4-bit S-box classes and six standard lightweight block ciphers using packing lower bounds and verified circuit synthesis.", "formal_line": "MC=5 for G0,G1,G2,G5,G8,G9,G12,G15,G4,G10,G14; MC=4 for G3,G6,G7,G11,G13; MC(PRINCE)=5, MC(PRESENT,GIFT,RECTANGLE,Piccolo,SKINNY)=4", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-176/", "receipt": "wave2-sbox-atlas/sbox_atlas_results.json, wave2-sbox-atlas/sbox_atlas.py", "external_value": "Cryptographers and circuit designers implementing lightweight block ciphers and optimal 4-bit S-boxes in low-AND architectures.", "statement": "The exact multiplicative complexity was determined for all 16 Leander-Poschmann optimal 4-bit S-box classes and six standard lightweight block ciphers using packing lower bounds and verified circuit synthesis.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-178", "kind": "finding", "title": "Domination of tensor slice rank floors for systems of quadratic forms", "title_plain": "Tensor slice rank lower bounds for systems of quadratic forms are provably dominated by linear dimension and provide no improvements across all 89 test units.", "formal_line": "srank(T) ≤ m for m-form tensors T, so ⌈srank(T)/3⌉ ≤ m ≤ MF-137 lower bound for all non-affine targets over F_2^n.", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "algebraic complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-178/", "receipt": "wave2-slice-rank/slice_rank_audit.json, wave2-slice-rank/test_all_cells_slice_rank.py", "external_value": "Complexity theorists evaluating slice rank or partition rank methods for multiplicative complexity lower bounds on systems of forms.", "statement": "Tensor slice rank lower bounds for systems of quadratic forms are provably dominated by linear dimension and provide no improvements across all 89 test units.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-179", "kind": "finding", "title": "Exact Multiplicative Complexity of Parallel Counters and Multi-Operand Compressors", "title_plain": "The exact multiplicative complexity over F_2 was established for all parallel counters and standard carry-save compressor slices up to 8 inputs.", "formal_line": "MC(3->2)=1, MC(4->3)=3, MC(5->3)=3, MC(6->3)=4, MC(7->3)=4, MC(4:2)=2, MC(5:2)=3, MC(6:2)=4 over F_2", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-179/", "receipt": "programs/corridor-sweep-20260901/wave2-compressors/compressor_results.json, programs/corridor-sweep-20260901/wave2-compressors/compressor_verification.py", "external_value": "Hardware arithmetic designers and cryptographic engineers optimizing low-AND multi-operand additions for MPC and ZK proofs.", "statement": "The exact multiplicative complexity over F_2 was established for all parallel counters and standard carry-save compressor slices up to 8 inputs.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result establishes closed exact bounds that appear to be new, though the overall problem remains partially solved.", "condition": "" }, { "id": "MF-180", "kind": "finding", "title": "Multiplicative complexity bounds for 2x2 matrix multiplication over GF(2)", "title_plain": "The multiplicative complexity of 2x2 matrix multiplication over GF(2) in the sequential XAG model is constrained to between 6 and 7 by combining Strassen's upper bound with a subfield invariant lower bound.", "formal_line": "6 ≤ MC(M_2(F_2)) ≤ 7 in sequential XAG over F_2, with qMC(W) ≥ 6 for an invariant 2-plane W ⊂ Λ^2(F_2^8)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-180/", "receipt": "programs/corridor-sweep-20260901/wave2-m2-closure/m2_qmc.py, programs/corridor-sweep-20260901/wave2-m2-closure/multi_satlib.py, programs/corridor-sweep-20260901/wave2-m2-closure/m2_k6.cnf", "external_value": "Algebraic complexity and exact synthesis researchers studying non-bilinear circuit lower bounds for matrix multiplication over finite fields.", "statement": "The multiplicative complexity of 2x2 matrix multiplication over GF(2) in the sequential XAG model is constrained to between 6 and 7 by combining Strassen's upper bound with a subfield invariant lower bound.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This is a partial and apparently new result, establishing the first non-bilinear lower bound >= 6 for unrestricted XAG over F_2.", "condition": "" }, { "id": "MF-181", "kind": "finding", "title": "Exact multiplicative complexity of the resolved-carry family K_m", "title_plain": "The exact multiplicative complexity of computing m low sum bits along with both resolved carry bits of four-operand addition is proved to be 2m+1 for all m ≥ 1.", "formal_line": "MC(K_m) = 2m+1 for every m ≥ 1", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-181/", "receipt": "box record-scratch/council4_fivecent/K{1..5}_heap.json; RECORD-COUNCIL4-FIVECENT.log; zkgolf-sha256/K-FAMILY-NOTE-2026-09-03.md; council4/k_family_heap.py, council4/k_family_cleanroom.py", "external_value": "Circuit designers optimizing multi-operand addition carry chains and researchers using exact synthesis degree filters.", "statement": "The exact multiplicative complexity of computing m low sum bits along with both resolved carry bits of four-operand addition is proved to be 2m+1 for all m ≥ 1.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result appears to be new as a general family statement: while Boyar–Peralta counted the redundant object, the exact count for the resolved object is not found in the known literature.", "condition": "" }, { "id": "MF-182", "kind": "finding", "title": "Unification of Additive Line Multiplicative Complexity via Greedy Column-Heap Recursion", "title_plain": "A greedy column-heap carry-save recursion reproduces all known exact multiplicative-complexity values and boundary constants for multi-operand addition.", "formal_line": "Heap recursion d_0=k+c0, d_{i+1}=k+⌈(d_i-1)/2⌉ with cost Σ⌈(d_i-1)/2⌉ matches all exact MC values for multi-operand addition and H_n.", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-182/", "receipt": "SYNTHESIS-2026-09-03.md §1 (box copy zkgolf-decomp/SYNTHESIS-2026-09-03.md); ZKGOLF-EXACT-ATLAS; fivecent_dotprice.py.", "external_value": "Hardware and zero-knowledge circuit designers synthesizing minimum-AND multi-operand additions.", "statement": "A greedy column-heap carry-save recursion reproduces all known exact multiplicative-complexity values and boundary constants for multi-operand addition.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "While the recursion is based on standard carry-save and Dadda dot reduction, the observation that it matches every exact XAG value, including the boundary constants, appears to be new.", "condition": "" }, { "id": "MF-183", "kind": "finding", "title": "Discard-tax principle for heap optimality with discarded top digits", "title_plain": "Conjectures that dropping top output digits saves only the AND gates feeding those digits exclusively, unifying several multiplicative complexity bounds.", "formal_line": "MC(S⁻) = MC(S) − (heap ANDs exclusive to the dropped digits)", "status": "CONJECTURE", "tier": "open", "tier_label": "OPEN QUESTION", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": true, "is_negative": false, "paper_url": "/research/papers/mf-183/", "receipt": "council4_fivecent/discard_H*_mod8_p*.json, fivecent_discard.py, rref_discard.py", "external_value": "NONE", "statement": "Conjectures that dropping top output digits saves only the AND gates feeding those digits exclusively, unifying several multiplicative complexity bounds.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "MF-184", "kind": "finding", "title": "Non-existence of 18-vector Kochen-Specker sets in dimension 6", "title_plain": "No Kochen-Specker vector set in dimension 6 can have 18 vectors, establishing that the minimum size in dimension 6 is at least 19.", "formal_line": "No KS set in C^6 has 18 vectors, hence m_6 >= 19 and m_6 in [19, 21]", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "open-problems", "school": "quantum circuit synthesis", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-184/", "receipt": "lanes/ks-d6/REPORT.md, BANK-CANDIDATES.md (KS6-01), frontier/ks6/w2/cubes/ (A1..A4, B: .cnf, .drat, .cadical.log, .drattrim.log, CUBES-PROGRESS.log), frontier/ks6/ks6_open_d6_n18/", "external_value": "Quantum foundations and quantum information researchers studying minimal contextuality proofs and Kochen-Specker sets.", "statement": "No Kochen-Specker vector set in dimension 6 can have 18 vectors, establishing that the minimum size in dimension 6 is at least 19, conditional on one cited computational lemma.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "This result improves the universal lower bound m_d >= 18 from Xu-Chen-Guehne (2020) in dimension 6. Together with Lisonek et al. (2014), this narrows m_6 to [19, 21].", "condition": "one cited computational lemma" }, { "id": "MF-185", "kind": "finding", "title": "Two necessary conditions on Kochen-Specker orthogonality graphs", "title_plain": "Any ray outside a basis clique is orthogonal to at most d - 2 basis rays, and any two non-orthogonal external rays must share a non-adjacent basis ray.", "formal_line": "For basis K in C^d: deg_K(v) ≤ d-2 for v ∉ K; and v ∦ w (v, w ∉ K) implies ∃u ∈ K with u ∦ v and u ∦ w.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "open-problems", "school": "quantum circuit synthesis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-185/", "receipt": "lanes/ks-d6/receipts/tools-w2/ks_cnf.py, receipts/replay/", "external_value": "SAT-based searches and pruning algorithms for Kochen-Specker vector systems and orthogonality graphs in dimension d.", "statement": "Any ray outside a basis clique is orthogonal to at most d - 2 basis rays, and any two non-orthogonal external rays must share a non-adjacent basis ray.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "This result was not found in the existing d = 3 SAT literature, which typically relies on C4-free or cross-product closure methods; its novelty remains unverified.", "condition": "" }, { "id": "MF-186", "kind": "finding", "title": "Maximum row degree bound for 16×17 Zarankiewicz extremal matrices", "title_plain": "Every 16-by-17 zero-one matrix with 133 ones and no 3-by-3 all-ones submatrix must have every row contain at most 10 ones.", "formal_line": "If A ∈ {0,1}^{16×17} has 133 ones and no 3×3 all-ones submatrix, then max row degree of A is ≤ 10", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-186/", "receipt": "lanes/zarankiewicz-16-17-3/BANK-CANDIDATES.md (ZAR-01), REPORT-LONG.md, VERIFY.md; box frontier/zarankiewicz/ and frontier/zarankiewicz/long/", "external_value": "Extremal combinatorics researchers working on the Zarankiewicz problem z(m,n;s,t) and automated SAT verification of Ramsey-type bounds.", "statement": "Every 16-by-17 zero-one matrix with 133 ones and no 3-by-3 all-ones submatrix must have every row contain at most 10 ones.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Two papers from August 2026 left the exact value of z(16,17;3) in {132, 133} unresolved.", "condition": "" }, { "id": "MF-187", "kind": "finding", "title": "Minimum degree bounds for (J4, J8; N)-graphs at orders 30 and 31", "title_plain": "Every graph on 30 vertices avoiding K_4 - e with at least 2 edges in every 8-set has minimum degree at least 4, and every such graph on 31 vertices has minimum degree at least 5.", "formal_line": "δ(G) ≥ 4 for every (J4,J8;30)-graph and δ(G) ≥ 5 for every (J4,J8;31)-graph", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-187/", "receipt": "lanes/ramsey-k4e-k8/BANK-CANDIDATES.md (RAM-05), REPORT.md; box frontier/ramsey/glue/C26_n30/, glue/C26_n31/", "external_value": "Ramsey theorists working on narrowing or determining the Ramsey number R(K4-e, K8-e).", "statement": "Every graph on 30 vertices avoiding K_4 - e with at least 2 edges in every 8-set has minimum degree at least 4, and every such graph on 31 vertices has minimum degree at least 5.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Wesley (arXiv:2606.17021) identifies R(J4,J8) in [30,32] as the next open case, building on the gluing framework from Goedgebeur-Van Overberghe (arXiv:2107.04460). An earlier reference to \"R(K_4-e, K_8)\" was a misprint for R(J4,J8), which is distinct from R(J4,K8) where 36 <= R <= 39.", "condition": "" }, { "id": "MF-188", "kind": "finding", "title": "Uniqueness of (J4,J7;26)-graphs with minimum degree at least 9", "title_plain": "Any 26-vertex graph without a 4-vertex clique minus an edge, where every 7 vertices span at least 2 edges and all degrees are at least 9, is uniquely isomorphic to the vertex-deleted Schläfli complement.", "formal_line": "(J4,J7;26)-graphs with δ >= 9 form a single isomorphism class: Schläfli complement minus one vertex (degrees 9^10 10^16, 125 edges)", "status": "EXHAUSTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-188/", "receipt": "lanes/ramsey-k4e-k8/BANK-CANDIDATES.md (RAM-06, RAM-07); box frontier/ramsey/sms/", "external_value": "Ramsey theorists and extremal graph theorists classifying critical graphs for generalized Ramsey numbers.", "statement": "Any 26-vertex graph without a 4-vertex clique minus an edge, where every 7 vertices span at least 2 edges and all degrees are at least 9, is uniquely isomorphic to the vertex-deleted Schläfli complement.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "MR91 proved uniqueness at order 27, but whether the 26-vertex statement appears in that paper remains unverified.", "condition": "" }, { "id": "MF-189", "kind": "finding", "title": "Exact sizes of small single-output median networks", "title_plain": "Single-output median comparator networks have exact minimal sizes of 13 comparators on 7 channels and 10 comparators for 6-channel lower median, with lower bounds certified by DRAT proofs.", "formal_line": "n=7 median network exact size is 13 (UNSAT at 12 DRAT-verified); 6-channel lower-median exact size is 10 (UNSAT at 9 DRAT-verified).", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-189/", "receipt": "lanes/median-11/BANK-CANDIDATES.md (MED-01, MED-03), VERIFY.md; box frontier/median/", "external_value": "NONE", "statement": "Single-output median comparator networks have exact minimal sizes of 13 comparators on 7 channels and 10 comparators for 6-channel lower median, with lower bounds certified by DRAT proofs.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Dobbelaere’s table lists 13 for n = 7 (unproven) and marks n = 9 as optimal (Smith 1996); Knuth TAOCP 5.3.4 remains unverified. Furthermore, a replay check confirmed Dobbelaere’s 22-comparator n = 10 network is a two-output design, correcting the single-output bound for n = 10 to 23 comparators rather than 22.", "condition": "" }, { "id": "MF-190", "kind": "finding", "title": "Exact B2 circuit size and machine-checkable certificate for 5-input MOD3,1", "title_plain": "The exact circuit size of the 5-input MOD3,1 function over the full 2-input binary basis is 9 gates, confirmed by an explicit 9-gate circuit and a machine-verified DRAT unsatisfiability certificate ruling out 8 gates.", "formal_line": "size_B2(MOD3,1 on 5 inputs) = 9", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-190/", "receipt": "lanes/mod3-6-b2/BANK-CANDIDATES.md (MOD3-01), VERIFY.md; box frontier/mod3/runs/", "external_value": "Circuit complexity and SAT-based exact synthesis researchers seeking verified minimal Boolean circuit benchmarks.", "statement": "The exact circuit size of the 5-input MOD3,1 function over the full 2-input binary basis is 9 gates, confirmed by an explicit 9-gate circuit and a machine-verified DRAT unsatisfiability certificate ruling out 8 gates.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Knuth (TAOCP 7.1.2, exercise 480) determined the values for n ≤ 5 using SAT solving without publishing certificates.", "condition": "" }, { "id": "MF-191", "kind": "finding", "title": "Exact constant-weight code bound A(11,4,5) = 66 via verified SAT", "title_plain": "The maximum size of a binary code of length 11, constant weight 5, and minimum distance 4 is proved to be exactly 66 using a machine-checkable DRAT certificate.", "formal_line": "A(11,4,5) = 66", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "formal verification", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-191/", "receipt": "lanes/cwcode-13-4-5/BANK-CANDIDATES.md (CWC-01)", "external_value": "Researchers in coding theory and SAT-based combinatorial verification seeking machine-checkable certificates for code bounds.", "statement": "The maximum size of a binary code of length 11, constant weight 5, and minimum distance 4 is proved to be exactly 66 using a machine-checkable DRAT certificate.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "This machine-checkable certificate builds on Brouwer's table (Johnson bound from A(10,4,5) = 36, Ostergard 2010). Companion entry CWC-02 re-derives the 1990 Brouwer-Shearer-Sloane-Smith lower bound using two explicit (13,4,5) codes of size 123, confirming A(13,4,5) in [123, 129] stands.", "condition": "" }, { "id": "MF-192", "kind": "finding", "title": "Verified structural obstructions for universal tournaments", "title_plain": "Specific small sets of tournaments cannot be simultaneously embedded into 9-vertex or 11-vertex host tournaments, verified via computer-checked unsatisfiability proofs.", "formal_line": "No 9-host for TT_6 and 12 6-tournaments; no 11-host for TT_7, QR_7, and 7 rare 7-tournaments (DRAT verified)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-192/", "receipt": "lanes/tournaments-t7/BANK-CANDIDATES.md (T7-02, T7-03), VERIFY.md; box frontier/t7/, frontier/t7-verify/", "external_value": "Extremal combinatorics and SAT verification researchers studying universal tournament lower bounds and structural obstructions.", "statement": "Specific small sets of tournaments cannot be simultaneously embedded into 9-vertex or 11-vertex host tournaments, verified via computer-checked unsatisfiability proofs.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Zhang and Szeider (CP 2023) stated a lower bound of 11 and resolved the case n = 11 using four separate SAT instances.", "condition": "" }, { "id": "MF-193", "kind": "finding", "title": "Contraction censuses and lifting obstructions for CW(112,36)", "title_plain": "All integer contractions of potential circulant weighing matrices of order 112 and weight 36 were completely counted across five moduli, proving that certain orbits cannot lift.", "formal_line": "CW(112,36) contractions: m=7 has 21 (2 orbits), m=8 has 96 (6), m=14 has 126 (3), m=16 has 1152 (24), m=28 has 420 vectors (4 orbits).", "status": "EXHAUSTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": true, "paper_url": "/research/papers/mf-193/", "receipt": "lanes/cwm-112-36/BANK-CANDIDATES.md (CWM-02), VERIFY.md, invent/; box frontier/cwm/", "external_value": "Combinatorial design theorists and sequence researchers studying circulant weighing matrices and cyclic difference sets.", "statement": "All integer contractions of potential circulant weighing matrices of order 112 and weight 36 were completely counted across five moduli, proving that certain orbits cannot lift.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "The orbit count 2 for m = 7 was previously reported in Arasu-Gordon-Zhang (2021, Table 9) under a multiplier assumption. The multiplier-free counts presented here are newly verified without that assumption.", "condition": "" }, { "id": "MF-194", "kind": "finding", "title": "Vertex-deletion averaging ladder for the Turán (3,4)-problem", "title_plain": "Deleting vertices from extremal 3-graphs yields a recursive averaging upper bound on Turán numbers that narrows the open cell at n=14 to a single triple.", "formal_line": "ex(n) ≤ ⌊n·ex(n-1)/(n-3)⌋ for ex(n, K_4^(3)); given ex(13) = 174, ex(14) ≤ 221 leaving a gap of at most one triple", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "CAVEAT", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-194/", "receipt": "lanes/turan-14/BANK-CANDIDATES.md (TUR-01), REPORT.md", "external_value": "Extremal combinatorics researchers studying exact values of Turán numbers ex(n, K_4^(3)) for small n.", "statement": "Deleting vertices from extremal 3-graphs yields a recursive averaging upper bound on Turán numbers that narrows the open cell at n=14 to a single triple.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "This result applies the standard Katona-Nemetz-Simonovits monotonicity argument to exact values. While the one-triple-wide observation at n = 14 was not found in prior literature, its novelty remains unverified, and this entry is recorded for completeness rather than as a novel claim.", "condition": "" }, { "id": "MF-195", "kind": "finding", "title": "Nonexistence of 19-vector Kochen-Specker sets in C^6 and the lower bound m_6 ≥ 20", "title_plain": "There is no 19-vector Kochen-Specker set in six dimensions, proving that the minimum size m_6 is at least 20 and narrowing the true minimum to either 20 or 21.", "formal_line": "No KS set in C^6 has 19 vectors, hence m_6 >= 20 and m_6 ∈ {20, 21} (conditional on cited lemma as in MF-184)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "quantum circuit synthesis", "function_type": "lower bound", "page_verdict": "SHOWCASE", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-195/", "receipt": "lanes/ks-d6/REPORT.md Theorem 2; box frontier/ks6/w2/cubes19/; lanes/ks-d6/verify-invent/VERDICTS.md, thm2_count.json, tests/ks6-n19-cover/PREREG.md", "external_value": "Quantum foundations and quantum information researchers studying minimal state-independent contextuality sets in dimension 6.", "statement": "There is no 19-vector Kochen-Specker set in six dimensions, proving that the minimum size m_6 is at least 20 and narrowing the true minimum to either 20 or 21, conditional on the same cited lemma as MF-184.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Searches across existing literature found no prior work excluding 19 in d = 6, though the novelty of this result remains unverified.", "condition": "the same cited lemma as MF-184" }, { "id": "MF-196", "kind": "finding", "title": "Residue-class parity theorem for circulant weighing matrices", "title_plain": "Every circulant weighing matrix of order 2^e q with odd q and even weight has an even number of nonzero entries in every residue class modulo 2^(e-1).", "formal_line": "For CW(n=2^e q, k) with q odd, e ≥ 1, k even: support S in F_2[Z_n] satisfies (x+1)^(2^(e-1)) | S, so S mod (x^(2^(e-1)) - 1) = 0", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "algebraic complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-196/", "receipt": "lanes/cwm-112-36/verify-invent/VERDICTS.md, invent/conway.md, invent/hilbert.md, receipts/cwm.json", "external_value": "Combinatorial design and weighing matrix researchers studying circulant weighing matrices or pruning branch-and-bound searches.", "statement": "Every circulant weighing matrix of order 2^e q with odd q and even weight has an even number of nonzero entries in every residue class modulo 2^(e-1).", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Prior work by Arasu, Leung, Ma, and Schmidt contains parity and 2-adic results for CW(2^e q, k). However, it remains unverified whether this exact residue-class statement appears in their published literature.", "condition": "" }, { "id": "MF-197", "kind": "finding", "title": "Machine-checked lower bound t(7) > 12 via fourteen amalgam cubes", "title_plain": "No 12-vertex tournament contains all required 7-vertex sub-tournaments, proving t(7) > 12 with a fully verified DRAT proof certificate across fourteen amalgam cubes.", "formal_line": "t(7) > 12; no 12-vertex tournament is 7-universal (all 14 amalgam cubes UNSAT, drat-trim VERIFIED)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "formal verification", "function_type": "lower bound", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-197/", "receipt": "lanes/tournaments-t7/verify-invent/VERDICTS.md, tests/amalgam-n12/PREREG.md, vetamal.py, venc3.py, box frontier/t7-vet/", "external_value": "Combinatorialists studying universal tournaments and automated reasoning researchers verifying extremal graph bounds.", "statement": "No 12-vertex tournament contains all required 7-vertex sub-tournaments, proving t(7) > 12 with a fully verified DRAT proof certificate across fourteen amalgam cubes.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "This value matches Zhang-Szeider (CP 2023), though the novelty of the amalgam-cube method and pattern-side symmetry breaking remains unverified against that paper, the Dec-2025 SMS survey, and the CP 2026 cubing paper. The open cell n = 13 (15 cubes) is the next instance.", "condition": "" }, { "id": "MF-198", "kind": "finding", "title": "Automorphism group structure of 7-universal tournaments on 13 vertices", "title_plain": "Every 7-universal tournament on 13 vertices is proved to have an automorphism group of order 1 or 3.", "formal_line": "For every 7-universal tournament T on 13 vertices, |Aut(T)| ∈ {1, 3}", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "boolean function analysis", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-198/", "receipt": "`lanes/tournaments-t7/verify-invent/VERDICTS.md`, `invent/burnside.py`, `invent/hilbert.md`; box `frontier/t7-vet/logs/`", "external_value": "Extremal combinatorics and tournament theory researchers studying universal tournaments and their symmetry groups.", "statement": "Every 7-universal tournament on 13 vertices is proved to have an automorphism group of order 1 or 3.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "The novelty of this result has not been independently verified. It incorporates and supersedes six earlier Conway seat DRAT proof-verification checks.", "condition": "" }, { "id": "MF-199", "kind": "finding", "title": "Nonexistence of 20-vector Kochen-Specker sets in C^6 and minimality of m_6 = 21", "title_plain": "No Kochen-Specker set in six dimensions can have 20 vectors, proving that the minimum size in dimension six is exactly 21 vectors conditional on the Xu-Chen-Gühne lemma.", "formal_line": "No Kochen-Specker set in C^6 has 20 vectors; hence m_6 = 21 exactly, conditional on the Xu-Chen-Gühne lemma.", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "quantum circuit synthesis", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "", "is_open": false, "is_negative": false, "paper_url": "/research/papers/mf-199/", "receipt": "lanes/ks-d6/verify-invent/tests/ks6-n20-cover/PREREG.md, box frontier/ks6vet/n20/ (PROGRESS.box.log, status_20_1..4.json, status_19_4.json), lanes/ks-d6/verify-invent/tools/, tests/ks6-n20-cover/, encoder vks.py", "external_value": "Quantum foundations researchers studying contextuality, Kochen-Specker vector systems, and GHZ-type parity proofs.", "statement": "No Kochen-Specker set in six dimensions can have 20 vectors, proving that the minimum size in dimension six is exactly 21 vectors conditional on the Xu-Chen-Gühne lemma.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "While prior work established 18 <= m_d for all d (XCG 2020) and m_6 <= 21 (LBPC 2014, \"simplest KS set admitting a symmetric parity proof\"), m_6 remained in [18, 21]. Entries MF-184, MF-195, and this work close m_6 to 21, proving d = 6 is the first dimension where the bound 18 is not attained.", "condition": "the Xu-Chen-Guehne lemma (every GHZ-type proof has >= 10 events), used here as the every-vertex degree bound deg(v) <= n - 11 = 9" }, { "id": "ML-007", "kind": "limit", "title": "Multiplicative complexity of Pascal and averaging-algebra state-feature classes", "title_plain": "Exhaustive screening of 2,097,152 Pascal and averaging-algebra state-feature classes left seven mixed-tensor-rank candidates, but none was producible by one raw-input product after the action-feature term.", "formal_line": "Exhaustive search across 2,097,152 Pascal / averaging-algebra state-feature classes yields 7 survivors at mixed-tensor-rank ≤2.", "status": "EXHAUSTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-007/", "receipt": "Founding source dossiers and fleet artifacts named by the register; no per-entry receipt file is named.", "external_value": "NONE", "statement": "Exhaustive screening of 2,097,152 Pascal and averaging-algebra state-feature classes left seven mixed-tensor-rank candidates, but none was producible by one raw-input product after the action-feature term.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-010", "kind": "limit", "title": "Affine rank of product-output functions and fixed-XOR optimization in the leader", "title_plain": "Exact scans find no free-row reduction for the leader in the stated affine-deletion and fixed structural-XOR classes, while topology-changing refactors remain outside scope.", "formal_line": "OPT_fixed-XOR = 274", "status": "EXHAUSTED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-010/", "receipt": "1,905 exact runs, global affine-factor and function-rank scans, and exact MILP plus replayed fixed-AIG XOR-cover certificates.", "external_value": "SHA-256 circuit optimization and Boolean-circuit compiler research", "statement": "Exact scans find no free-row reduction for the leader in the stated affine-deletion and fixed structural-XOR classes, while topology-changing refactors remain outside scope.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-016", "kind": "limit", "title": "Wrong Fixed Points in Rank 5/5 Deterministic Quadratic-Atlas Charts", "title_plain": "The deterministic Quadratic-Atlas route is exhausted: the two charts reaching full rank 5/5 fail the exact row-0 cover, while all other charts die earlier.", "formal_line": "Two deterministic Quadratic-Atlas charts attain rank 5/5 with 14 and 21 wrong fixed points; all others terminate at rank ≤4 or qdim 6", "status": "EXHAUSTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "quadratic-hull", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-016/", "receipt": "Founding dossiers and fleet artifacts are cited by route section, but no per-entry receipt file is stated.", "external_value": "SAT-based exact circuit synthesis", "statement": "The deterministic Quadratic-Atlas route is exhausted: the two charts reaching full rank 5/5 fail the exact row-0 cover, while all other charts die earlier.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-017", "kind": "limit", "title": "Universal wrong row-0 assignments in Toffoli-lift cyclic p6 modules", "title_plain": "All 15 qdim-6 Toffoli-lift modules reach full rank but each admits a universal wrong row-0 assignment, so the examined cyclic p6 route is exhausted.", "formal_line": "All 15 qdim-6 modules in the Toffoli-lift cyclic p6 family attain full rank and admit a universal wrong row-0 assignment", "status": "EXHAUSTED", "tier": "legacy", "tier_label": "LEGACY: NO RECEIPT", "program": "ciphers-and-quantum", "school": "quantum circuit synthesis", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-017/", "receipt": "none stated", "external_value": "NONE", "statement": "All 15 qdim-6 Toffoli-lift modules reach full rank but each admits a universal wrong row-0 assignment, so the examined cyclic p6 route is exhausted.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-018", "kind": "limit", "title": "Exhaustion of the 3-gate full-domain start on the Toffoli acyclic p7 route", "title_plain": "After 786430 factor checks, the examined Toffoli acyclic p7 route is dead, while alternate prefixes remain unresolved.", "formal_line": "Real 3-gate full-domain start on Toffoli acyclic p7 route: EXHAUSTED after 786,430 factor checks; alternate prefixes: UNKNOWN", "status": "EXHAUSTED", "tier": "legacy", "tier_label": "LEGACY: NO RECEIPT", "program": "ciphers-and-quantum", "school": "quantum circuit synthesis", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-018/", "receipt": "none stated", "external_value": "NONE", "statement": "After 786430 factor checks, the examined Toffoli acyclic p7 route is dead, while alternate prefixes remain unresolved.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-020", "kind": "limit", "title": "An impossibility result for two-witness AND4 in the affine-C model", "title_plain": "After the recorded updates, the two-witness AND4 route is unsatisfiable for every finite row count in the affine-C setting, while seven singular affine-C three-witness/three-row orbits remain unknown.", "formal_line": "In the affine-C model, the two-witness AND4 route is unsatisfiable for every finite row count r.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-020/", "receipt": "Enlarged exact affine-C and arbitrary-C censuses plus selector-closure verification replays.", "external_value": "R1CS expressiveness and relational proof-system design.", "statement": "After the recorded updates, the two-witness AND4 route is unsatisfiable for every finite row count in the affine-C setting, while seven singular affine-C three-witness/three-row orbits remain unknown.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-022", "kind": "limit", "title": "Unsoundness of the fixed Ghost-P5 ITER19 allocation and K0 trellis adapter", "title_plain": "The fixed Ghost-P5 allocation and K0 trellis adapter fail soundness, while alternative allocations, lifts, and relational presentations remain unresolved.", "formal_line": "Ghost-P5 (18,801, ITER19): deg-2 hull spans 17-bit space, δvert=66, δflip=2; exact K0 adapter unsound on all 131,072 replayed assignments", "status": "LIVE-PARKED", "tier": "open", "tier_label": "OPEN QUESTION", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-022/", "receipt": "A 4,096-fibre hull census and exhaustive 131,072-assignment K0 replay, including eight real SHA round-0 cases.", "external_value": "NONE", "statement": "The fixed Ghost-P5 allocation and K0 trellis adapter fail soundness, while alternative allocations, lifts, and relational presentations remain unresolved.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-023", "kind": "limit", "title": "On the natural stationary p7 route targeting 21,738", "title_plain": "The natural stationary p7 route remains unresolved: the corrected audit counts 2,018 dead tails, but the unexhausted prefix lacks the rank-tight export needed to establish global closure.", "formal_line": "p7 target 21,738: 2,018 dead_tail / 181,441 blockers across 768 rows, 0 unresolved tails, global closure = UNKNOWN, status = LIVE-PARKED", "status": "LIVE-PARKED", "tier": "open", "tier_label": "OPEN QUESTION", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-023/", "receipt": "Direct resume audit and verification report documenting 2,018 dead tails among 181,441 blockers over 768 rows.", "external_value": "NONE", "statement": "The natural stationary p7 route remains unresolved: the corrected audit counts 2,018 dead tails, but the unexhausted prefix lacks the rank-tight export needed to establish global closure.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-024", "kind": "limit", "title": "On the quadratic hull of carry relations under state-only catalyst lifts", "title_plain": "Deterministic state-only catalyst lifts cannot repair the natural carry relation's quadratic hull, including the complete state algebra and the scoped acyclic p6 class, while broader edge and relational lifts remain unresolved.", "formal_line": "Deterministic state-only lifts fail at every row budget: 8,040 acyclic-p6 models eliminated, 8,184 anchors yield no hull repairs", "status": "PROVED DEAD", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-024/", "receipt": "Exhaustive 8,040-model acyclic census, corrected 8,184-anchor enumeration, and full quadratic-hull replays.", "external_value": "Cryptographic circuit synthesis and constraint-system design", "statement": "Deterministic state-only catalyst lifts cannot repair the natural carry relation's quadratic hull, including the complete state algebra and the scoped acyclic p6 class, while broader edge and relational lifts remain unresolved.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-025", "kind": "limit", "title": "The p8 top-form obstruction for the exact three-column boundary tile", "title_plain": "The exact functional three-column boundary tile has no p8 circuit, while the separately specified recoded or enlarged relational interface remains undecided.", "formal_line": "Exact functional four-word tile T with output degrees [1,2,4,6,9] has no p8 circuit", "status": "CLOSED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-025/", "receipt": "Exact schedule-relation scope audit, including 3,696 failures among 4,096 schedule rows for the conflated relational premise.", "external_value": "exact Boolean and arithmetic-circuit synthesis", "statement": "The exact functional three-column boundary tile has no p8 circuit, while the separately specified recoded or enlarged relational interface remains undecided.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-028", "kind": "limit", "title": "Quadratic hull closure for the canonical Z-difference counter interface", "title_plain": "Retaining garbage coordinate g0 or g1 closes the Z-difference counter's quadratic hull, and three quadratic rows are minimal for the canonical interface; g2 alone does not close it.", "formal_line": "For the canonical Z-difference counter interface A*B=C, retaining garbage coordinate g0 or g1 closes the quadratic hull.", "status": "CLOSED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-028/", "receipt": "Exhaustive canonical A*B=C enumeration of 524,800 factor pairs per case, 21 graph-vanishing equations, and explicit three-row covers.", "external_value": "NONE", "statement": "Retaining garbage coordinate g0 or g1 closes the Z-difference counter's quadratic hull, and three quadratic rows are minimal for the canonical interface; g2 alone does not close it.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-029", "kind": "limit", "title": "Exact-UNSAT of Phase B in the natural P10 missing-plane model", "title_plain": "In the natural 30-coordinate P10 model, Phase B has no genuine rank-one equation in any nonzero missing-plane class, so the proposed rejection family is empty and Phase C is not reached.", "formal_line": "Phase B is exact-UNSAT in the natural 30-coordinate P10 model; product image rank 423, quotient rank 393, rejection family empty", "status": "PROVED", "tier": "solver", "tier_label": "SOLVER-CONFIRMED", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-029/", "receipt": "Exact-UNSAT Phase-B dual-solver certificates with rank reconciliation showing product-image rank 423 and quotient rank 393.", "external_value": "NONE", "statement": "In the natural 30-coordinate P10 model, Phase B has no genuine rank-one equation in any nonzero missing-plane class, so the proposed rejection family is empty and Phase C is not reached.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-033", "kind": "limit", "title": "Finite-horizon Nerode minimisation of the 22-state carry transducer", "title_plain": "The natural 22-state carry transducer retains 22 Nerode classes through bit 29, then 16, 4, and 1 terminal class, leaving only 128 terminal occurrences mergeable and closing ordinary full-alphabet recoding as a large-saving route.", "formal_line": "Under finite-horizon Nerode minimisation, the natural 22-state carry transducer maintains 22 classes through bit 29, followed by 16, four, and one terminal class.", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "symmetry-and-encodings", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-033/", "receipt": "Exact finite-horizon Nerode minimisation with independent replay, 2,048 raw symbols projected to 128 actions, and zero right-congruence violations.", "external_value": "Cryptographic circuit synthesis and finite-state minimization", "statement": "The natural 22-state carry transducer retains 22 Nerode classes through bit 29, then 16, 4, and 1 terminal class, leaving only 128 terminal occurrences mergeable and closing ordinary full-alphabet recoding as a large-saving route.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-034", "kind": "limit", "title": "Elimination of stationary additive carry/raw phases with terminal syndrome-only checks", "title_plain": "Every stationary additive raw/state edge-phase scheme tested is ruled out when its only terminal check is the accumulated syndrome, while semantic-sensitive, position-dependent, non-additive, and relational variants remain unresolved.", "formal_line": "491,040 candidates across 245,520 classes yield rank 74 with 40 independent directions; span indicator(c=s)*{1,u0,...,u9} is 88-dimensional", "status": "PROVED DEAD", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-034/", "receipt": "Exhaustive 491,040-candidate and 245,520-class span audit plus a three-column zero-moment holonomy replay.", "external_value": "NONE", "statement": "Every stationary additive raw/state edge-phase scheme tested is ruled out when its only terminal check is the accumulated syndrome, while semantic-sensitive, position-dependent, non-additive, and relational variants remain unresolved.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-035", "kind": "limit", "title": "Fixed-interface factor-component contraction on the 22,215-row leader", "title_plain": "Every fixed-interface factor component in the 22,215-row leader is a full-rank path, so the tested local contraction route has no candidate.", "formal_line": "Fixed-interface factor-component contraction is closed on 22,215-row leader: 20,525 full-rank paths (18,837×(1,1), 1,686×(2,2), 2×(3,3))", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-035/", "receipt": "Fixed-interface factor-component screen and independent GF(2) rank crosscheck.", "external_value": "Boolean-circuit optimizer research on local contraction rules", "statement": "Every fixed-interface factor component in the 22,215-row leader is a full-rank path, so the tested local contraction route has no candidate.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-040", "kind": "limit", "title": "Piecewise cubic separation of JSC-12 cases in the flip-support subspace", "title_plain": "For the named JSC-12 witness, no single cubic in the specified flip-support subspace separates all four K graphs, although a K-piecewise pair separates the named cases.", "formal_line": "Piecewise pair (k0=0 principal, k0=1 companion) separates K10/K11; no square-free cubic meeting {middle1, middle2, final0} rejects witness", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "boolean function analysis", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-040/", "receipt": "Exact flip-support span logs plus K-piecewise companion-cubic and Identity-C lowering receipts.", "external_value": "cryptographic circuit and Boolean-relation research", "statement": "For the named JSC-12 witness, no single cubic in the specified flip-support subspace separates all four K graphs, although a K-piecewise pair separates the named cases.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-041", "kind": "limit", "title": "An impossibility result for affine third factors of the named JSC-12 witness", "title_plain": "For the named JSC-12 witness, no affine third factor for q=current1 times middle2 works across the K cases, and the viable k0=1 companion is a non-affine 36-term cubic.", "formal_line": "For JSC-12 with q = current1 * middle2, no affine M makes qM vanish on honest {q=1} while rejecting the false point on K10, K11, K1x, all-K", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "r1cs-expressiveness", "school": "algebraic complexity", "function_type": "impossibility theorem", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-041/", "receipt": "Exact affine interpolation log and K1 companion-cubic receipt.", "external_value": "cryptographic circuit and Boolean-relation research", "statement": "For the named JSC-12 witness, no affine third factor for q=current1 times middle2 works across the K cases, and the viable k0=1 companion is a non-affine 36-term cubic.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-042", "kind": "limit", "title": "A polar rank 8 separator in the JSC-12 K1x even dual coset", "title_plain": "The polar-rank search found a rank-eight JSC-12 K1x separator, but the rank-six-or-lower space was not exhausted and remains unknown.", "formal_line": "JSC-12 K1x even dual coset separator middle2 * Q has polar rank 8, Hamming weight 37, satisfying K10=K11=0 and FALSE_CUBIC=1", "status": "EXHAUSTED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "CAVEAT", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-042/", "receipt": "Polar-rank search logs and Identity-C lowering receipts, with 277,207,495 tests but no exhaustion of the rank-six-or-lower space.", "external_value": "exact-synthesis researchers studying algebraic separator searches", "statement": "The polar-rank search found a rank-eight JSC-12 K1x separator, but the rank-six-or-lower space was not exhausted and remains unknown.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-043", "kind": "limit", "title": "Closure of all 68 p7-eligible directed edges under the rank-tight affine-code model", "title_plain": "The rank-tight affine-code model is closed for all 68 p7-eligible directed edges: q-rank-7 edges are p7-dead, q-rank-6 edges are p6-dead, and q-rank-5 edges are rank-tight p5-dead, while one-catalyst extensions remain unknown.", "formal_line": "Zero-catalyst rank-tight affine-code model: 68 p7-eligible edges are closed: 32 rank-tight p7-dead, 28 p6-dead, 8 rank-tight p5-dead", "status": "PROVED", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-043/", "receipt": "Frontier exact replay and independent verification receipts.", "external_value": "NONE", "statement": "The rank-tight affine-code model is closed for all 68 p7-eligible directed edges: q-rank-7 edges are p7-dead, q-rank-6 edges are p6-dead, and q-rank-5 edges are rank-tight p5-dead, while one-catalyst extensions remain unknown.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-049", "kind": "limit", "title": "An eight-product acyclic XAG for the low four bits of x₀+x₁+x₂+x₃", "title_plain": "The p=8 branch of the four-operand, four-bit discriminator is settled by an explicit eight-product acyclic XAG that matches all 65,536 inputs, so p=8 is SAT and only p=7 remains informative.", "formal_line": "An acyclic XAG computes the low four bits of x₀+x₁+x₂+x₃ in 8 products (target 4op_w4_p8, p = 8 is SAT)", "status": "NOT DECISIVE", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "construction", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-049/", "receipt": "MF-083 eight-product witness certificate and independent full truth-table replay; the p=8 solver attempts returned no verdict.", "external_value": "Exact circuit synthesis and multi-operand-addition benchmarking.", "statement": "The p=8 branch of the four-operand, four-bit discriminator is settled by an explicit eight-product acyclic XAG that matches all 65,536 inputs, so p=8 is SAT and only p=7 remains informative.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-050", "kind": "limit", "title": "Impossibility of helper-free IsZero over finite fields with |F| ≥ 4", "title_plain": "Over every finite field with at least four elements, no degree-at-most-two equations in the exposed pair (x,z) can realize z=[x=0] without an inverse helper, while F3 is an explicit exception.", "formal_line": "For |F| ≥ 4, no family of degree-at-most-two equations in (x,z) has solution relation Γ₀ = {(0,1)} ∪ {(x,0) : x ∈ F*}", "status": "PROVED DEAD", "tier": "paper", "tier_label": "PAPER PROOF", "program": "r1cs-expressiveness", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "SHOWCASE", "cost_model": "R1CS over F_p", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-050/", "receipt": "Exact algebra and fibre replay in the quadratic-relation certificate and report.", "external_value": "R1CS gadget design and cryptographic proof systems.", "statement": "Over every finite field with at least four elements, no degree-at-most-two equations in the exposed pair (x,z) can realize z=[x=0] without an inverse helper, while F3 is an explicit exception.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-051", "kind": "limit", "title": "On the functional direct-sum additivity of M₂ over GF(2)", "title_plain": "Functional direct-sum additivity for 2x2 matrix multiplication over GF(2) is open in unrestricted sequential Boolean XAGs, because the exact 7s result is known only in bilinear and formal-quadratic models pending a six-gate impossibility test.", "formal_line": "Functional direct-sum additivity for M₂ over GF(2) in unrestricted sequential Boolean XAGs is OPEN", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "algebraic complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-051/", "receipt": "REPORT sections 8-9, mm2_boolean_mc.py, and boolean-mm-sat search artifacts; no six-gate UNSAT result.", "external_value": "Algebraic-complexity and matrix-multiplication researchers", "statement": "Functional direct-sum additivity for 2x2 matrix multiplication over GF(2) is open in unrestricted sequential Boolean XAGs, because the exact 7s result is known only in bilinear and formal-quadratic models pending a six-gate impossibility test.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "Alder-Strassen and Strassen establish 7s only within the bilinear and formal-quadratic models. The question of unrestricted Boolean functional additivity remains open.", "condition": "" }, { "id": "ML-052", "kind": "limit", "title": "On the static quadratic hull as a general SAT preprocessor", "title_plain": "A static quadratic hull is not a general SAT preprocessor: for width-k clauses with k at least 3 it is the full ambient space, although it remains useful as a local encoding lint.", "formal_line": "If |F| < 2^(n-d), then I_≤d(R) = {0} and H_d(R) = F₂^n", "status": "PROVED DEAD", "tier": "replay", "tier_label": "EXHAUSTIVE CHECK", "program": "quadratic-hull", "school": "arithmetization and proof systems", "function_type": "impossibility theorem", "page_verdict": "SHOWCASE", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-052/", "receipt": "Reed-Muller minimum-distance proof and exhaustive checker for clause widths 1 through 8.", "external_value": "SAT solving and proof complexity", "statement": "A static quadratic hull is not a general SAT preprocessor: for width-k clauses with k at least 3 it is the full ambient space, although it remains useful as a local encoding lint.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "The Reed-Muller minimum-distance property used here is a standard, classical mathematical result, and no novelty is claimed for this component.", "condition": "" }, { "id": "ML-053", "kind": "limit", "title": "Empirical limits of single-counterexample CEGIS exact synthesis", "title_plain": "Single-counterexample CEGIS exact synthesis decides small targets around ten inputs, but targets at 12 inputs or more fail to return, and even a known six-product 10-input control times out.", "formal_line": "K=4,n=2,p=1: UNSAT; K=3,n=3,p=2: UNSAT; K=3,n=3,p=3: SAT; K=2,n≤5,p=3: UNSAT", "status": "EMPIRICAL-WALL", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "CAVEAT", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-053/", "receipt": "sweep2_results.json, mf001_reaudit_results.json, exact_mc_full.py, and mf001_fullencode_results.json.", "external_value": "Exact circuit-synthesis and solver-method researchers", "statement": "Single-counterexample CEGIS exact synthesis decides small targets around ten inputs, but targets at 12 inputs or more fail to return, and even a known six-product 10-input control times out.", "published": "2026-08-29", "last_reviewed": "2026-08-29", "prior_art": "", "condition": "" }, { "id": "ML-054", "kind": "limit", "title": "Multiplicative complexity bounds for the injected-carry family J_m", "title_plain": "The multiplicative complexity of the injected-carry family J_m satisfies 2m-3 <= MC(J_m) <= 2m-2, yielding exact values 2, 4, and 6 for m = 2, 3, and 4 while leaving J_5 open in the bracket [7, 8].", "formal_line": "2m-3 ≤ MC(J_m) ≤ 2m-2 with MC(J_2)=2, MC(J_3)=4, MC(J_4)=6, and J_5 ∈ [7,8]", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-054/", "receipt": "zkgolf-decomp/reports/CERT-SYNTH.md, zkgolf-decomp/reports/PROVER-ICA-PIVOT.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/cert-synth-scratch/recompute.receipt.json, zkgolf-decomp/cert-tensor-scratch/transfer-audit.receipt.json, zkgolf-decomp/prover-ica-scratch/m3-k4.clean.replay.json", "external_value": "NONE", "statement": "The multiplicative complexity of the injected-carry family J_m satisfies 2m-3 <= MC(J_m) <= 2m-2, yielding exact values 2, 4, and 6 for m = 2, 3, and 4 while leaving J_5 open in the bracket [7, 8].", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-055", "kind": "limit", "title": "Exact gate-state duality and 1-Lipschitz potential bounds for multiplicative complexity", "title_plain": "Multiplicative complexity equals the shortest-path distance on the complete semantic gate-state graph, whose linear programming dual is a 1-Lipschitz potential problem.", "formal_line": "Gate-state shortest path equals MC with LP dual as 1-Lipschitz potential; quotient certificates require complete edge sets.", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "symmetry-and-encodings", "school": "multiplicative complexity", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-055/", "receipt": "zkgolf-decomp/reports/CERT-DUAL.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, dual-certificate.receipt.json (SHA-256 01fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4), j3-normal-form-dual.lp (SHA-256 6c4d382908e80ca879c4b3f58b8ec7004b32cbe1f4c60b473a7093099aba15fe)", "external_value": "Researchers developing LP-based circuit lower bound certificates or exact synthesis algorithms via state-graph distance.", "statement": "Multiplicative complexity equals the shortest-path distance on the complete semantic gate-state graph, whose linear programming dual is a 1-Lipschitz potential problem.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-056", "kind": "limit", "title": "Restriction conservation law for multiplicative complexity under exact restriction", "title_plain": "The change in multiplicative complexity under an exact restriction equals the number of killed gates plus residual inefficiency, yielding a 1-Lipschitz lifted potential.", "formal_line": "MC(f) - MC(f|_R) = k_R + e_R under exact restriction, with 1-Lipschitz potential k_R + ψ_R; exhaustive J_2 spectrum is 4×(1,0), 6×(0,1)", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-056/", "receipt": "Reports CERT-DUAL.md, CERT-COALG.md, COUNCIL-LAWVERE.md, STATE-OF-PROGRAM-V2.md; AX162 dual-certificate.receipt.json, finite_certificate.receipt.json, lawvere_five_cent.receipt.json", "external_value": "Complexity theorists analyzing restriction techniques and potential functions for multiplicative complexity lower bounds.", "statement": "The change in multiplicative complexity under an exact restriction equals the number of killed gates plus residual inefficiency, yielding a 1-Lipschitz lifted potential.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-057", "kind": "limit", "title": "Kummer endpoint and boundary-alias formulas", "title_plain": "Exact 2-adic valuation and product-floor boundary-alias formulas are proved, leaving unrestricted projected-heap equality open.", "formal_line": "v_2((K(2^n-1))!/((2^n-1)!)^K) and P=∑_j min(h_j, 2^{n-j}-1) proved; gap is (r-1)K-2^r+2 with r=⌈log_2 K⌉; unrestricted MC=T is open", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "structure theorem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-057/", "receipt": "zkgolf-decomp/reports/COUNCIL-HARDY.md, zkgolf-decomp/reports/COUNCIL-MINE.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, carry_calculus.py, residual_hankel.py", "external_value": "NONE", "statement": "Exact 2-adic valuation and product-floor boundary-alias formulas are proved, leaving unrestricted projected-heap equality open.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-058", "kind": "limit", "title": "Exact costs of strict syntactic equivariance for widths 3, 5, and 7", "title_plain": "In the strict syntactic-equivariant circuit model, the exact costs for inverse-chi are proved to be 3, 10, and 21 for widths 3, 5, and 7, imposing overheads of 0, 4, and 12 over unrestricted complexity.", "formal_line": "E_3 = 3, E_5 = 10, E_7 = 21 with exact taxes 0, 4, 12 over unrestricted inverse-χ multiplicative complexity", "status": "PROVED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-058/", "receipt": "zkgolf-decomp/reports/EXP-EQ-TAX.md, zkgolf-decomp/reports/SB-TAX.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/exp-eq-scratch/equivariant-family-replay.json (SHA-256 3203a1f94016c6d0f3ab302a5c145748b5a19835dd814fac7af7b485b89a9ed4), zkgolf-decomp/exp-eq-scratch/eq-tax-certificate.json (SHA-256 7d1e0bc24abf7b71b1a937aa0301211d82c342f3c171bc9f42360aaec9e9f628), zkgolf-decomp/sb-tax-scratch/n7_plain_receipt.json (SHA-256 4b8f60c073f5d282778a0c7d8e97c34009bf52ca1195fed2c11aa5dcf9bb50df), zkgolf-decomp/sb-tax-scratch/n7_seeded_receipt.json (SHA-256 ed7b3be7554901f1fdb59b16b349f804d1e8727749d97c6e77d23b224c7d54d1)", "external_value": "S-box circuit designers evaluating the exact complexity penalty of enforcing strict syntactic rotational equivariance.", "statement": "In the strict syntactic-equivariant circuit model, the exact costs for inverse-chi are proved to be 3, 10, and 21 for widths 3, 5, and 7, imposing overheads of 0, 4, and 12 over unrestricted complexity.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-059", "kind": "limit", "title": "Bounds on strict width-nine complexity E_9", "title_plain": "The exact complexity of strict width nine remains open within the proven bracket between 18 and 36.", "formal_line": "18 ≤ E_9 ≤ 36", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-059/", "receipt": "zkgolf-decomp/reports/SB-TAX.md, zkgolf-decomp/reports/EXP-EQ-TAX.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/sb-tax-scratch/n9_one_full_floor_receipt.json, zkgolf-decomp/sb-tax-scratch/cleanroom_upper.py", "external_value": "NONE", "statement": "The exact complexity of strict width nine remains open within the proven bracket between 18 and 36.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-060", "kind": "limit", "title": "Symmetry and exact small-width complexity of constant-addition shears", "title_plain": "Proves a shift symmetry for constant-addition shear functions and determines their exact multiplicative complexity for bit-widths 2 and 3.", "formal_line": "MC(F_{n,K}) = MC(F_{n,K+2^{n-2}}), with MC(F_{2,K}) = 2 and MC(F_{3,K}) = 5 for every constant K", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-060/", "receipt": "zkgolf-decomp/const-theory-scratch/heap-and-width32.receipt.json (SHA-256 97495ea7cb13222225e45598ceef9956d61e9684d0c2001e96613a41f83a4da0), zkgolf-decomp/const-theory-scratch/small-grid-lower.receipt.json (SHA-256 af5365a96feb27fcfecab8c9d8f1fc291b4aedb280e7f999958abc52a0c28a89)", "external_value": "Circuit designers optimizing constant addition and shear components in symmetric cryptographic primitives and hash functions.", "statement": "Proves a shift symmetry for constant-addition shear functions and determines their exact multiplicative complexity for bit-widths 2 and 3.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-063", "kind": "limit", "title": "SHA-256 compilation record and status of conditional sub-22,215 alternatives", "title_plain": "The verified SHA-256 compilation record stands at 22,215 rows, while proposed sub-22,215 configurations remain unverified conditional scenarios.", "formal_line": "SHA-256 record = 22215 rows; hypothetical alternatives at 20612, 20531, and 22185 remain conditional without full witnesses", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "arithmetization and proof systems", "function_type": "audit finding", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-063/", "receipt": "zkgolf-decomp/reports/RECORD-BOM.md, zkgolf-decomp/reports/CONST-THEORY.md, zkgolf-decomp/reports/CERT-SYNTH.md, zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, zkgolf-decomp/synth-n-scratch/recompute_capstone.receipt.json, zkgolf-decomp/const-theory-scratch/heap-and-width32.receipt.json", "external_value": "Zero-knowledge proof engineers and circuit optimizers implementing minimal SHA-256 arithmetizations.", "statement": "The verified SHA-256 compilation record stands at 22,215 rows, while proposed sub-22,215 configurations remain unverified conditional scenarios.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-064", "kind": "limit", "title": "Infeasibility of K12 score reduction via banked rowwise chi5 replacements", "title_plain": "Direct rowwise replacement of 5-bit chi transformations cannot reduce the product count in 12-round Keccak-p[1600,12] below 19,200 products.", "formal_line": "Rowwise chi5 in Keccak-p[1600,12] is optimal at 5 products per row (19,200 total); no banked rowwise replacement reduces the score.", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-064/", "receipt": "zkgolf-decomp/REDEPLOY-K12.md, zkgolf-decomp/SURFACE-VERIFY.md", "external_value": "NONE", "statement": "Direct rowwise replacement of 5-bit chi transformations cannot reduce the product count in 12-round Keccak-p[1600,12] below 19,200 products.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-065", "kind": "limit", "title": "BLAKE3 and ARX redeployment limits from banked addition assets", "title_plain": "Banked addition constructions cannot improve the BLAKE3 compression board product count under current arsenal constraints.", "formal_line": "Direct redeployment of banked addition assets yields ≥ 10304 products on BLAKE3, exceeding the 10298-product conditional leader", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "algebraic complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-065/", "receipt": "zkgolf-decomp/REDEPLOY-ARX-ADDITION.md", "external_value": "Implementers of ARX and BLAKE3 zero-knowledge circuits assessing modular addition optimization strategies.", "statement": "Banked addition constructions cannot improve the BLAKE3 compression board product count under current arsenal constraints.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-066", "kind": "limit", "title": "Failure of Keccak-family score reduction via inverse-χ exactness", "title_plain": "Attempting to reduce Keccak-family circuit scores using exact synthesis of inverse-χ maps fails because forward-χ is already minimal and inverse paths do not reduce circuit costs.", "formal_line": "Exact inverse-χ_5 and Ascon inverse circuits yield zero solver score reductions over forward-χ (MC(χ_n) = n).", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-066/", "receipt": "zkgolf-decomp/REDEPLOY-KECCAK-FAMILY.md", "external_value": "Reference circuits and optimality certificates for inverse-χ_5 and the affine-conjugate Ascon inverse.", "statement": "Attempting to reduce Keccak-family circuit scores using exact synthesis of inverse-χ maps fails because forward-χ is already minimal and inverse paths do not reduce circuit costs.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-067", "kind": "limit", "title": "Limits of SHA-256 row reduction from banked component certificates", "title_plain": "Banked component certificates covering 91.15% of SHA-256 leader rows cannot be directly composed to reduce total rows, yielding zero immediate row reductions.", "formal_line": "Banked certificates cover 20,248 of 22,215 SHA-256 rows but license 0 reductions due to non-additivity and a 1,967-row component deficit", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "arithmetization and proof systems", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-067/", "receipt": "zkgolf-decomp/REDEPLOY-CERTIFICATION.md", "external_value": "NONE", "statement": "Banked component certificates covering 91.15% of SHA-256 leader rows cannot be directly composed to reduce total rows, yielding zero immediate row reductions.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-068", "kind": "limit", "title": "Impossibility of Constant Absorption by Memoryless Scalar Carry Codes", "title_plain": "Once an addition constant makes carry 4 reachable, no block circuit with a memoryless scalar carry boundary can beat four AND gates per column.", "formal_line": "No block with boundary state encoding scalar carry 0..4 beats 4 ANDs/col once carry 4 is reachable.", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-068/", "receipt": "RECORD-CONSTADD92.md, RECORD-CONSTFREE.md", "external_value": "Designers of low-multiplicative-complexity adder architectures absorbing round constants.", "statement": "Once an addition constant makes carry 4 reachable, no block circuit with a memoryless scalar carry boundary can beat four AND gates per column.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-069", "kind": "limit", "title": "Exact additivity of two-copy block sharing in leader pairs", "title_plain": "Sharing blocks between consecutive rounds or adders does not reduce circuit cost because the actual instances in the leader share no operand bundles.", "formal_line": "Consecutive-round Ch and Maj pairs and schedule adder pairs are exactly additive at width 32; block sharing across real pairs yields 0 savings.", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "algebraic complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-069/", "receipt": "RECORD-BLOCKSHARE.md", "external_value": "Circuit optimizers and cryptographic implementers evaluating block-sharing optimizations in SHA-family leaders.", "statement": "Sharing blocks between consecutive rounds or adders does not reduce circuit cost because the actual instances in the leader share no operand bundles.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-071", "kind": "limit", "title": "Refutation of the Maj Edge Phase -1,024 Candidate", "title_plain": "The candidate Maj edge phase -1,024 was ruled out because its phase defect grows unboundedly by 32 bits per round pair.", "formal_line": "Maj edge phase -1,024 candidate phase defect grows linearly at 32k bits for k=1..4, refuting the candidate with zero solver time.", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-071/", "receipt": "RECORD-MAJEDGE.md", "external_value": "Prevents pursuing the Maj edge phase -1,024 route in XOR-free multiplicative complexity synthesis.", "statement": "The candidate Maj edge phase -1,024 was ruled out because its phase defect grows unboundedly by 32 bits per round pair.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-072", "kind": "limit", "title": "Universal screen refutation of the JSC one-catalyst class", "title_plain": "Every acyclic one-catalyst lift for the JSC-11 allocation is shown to accept a false point inside its degree-3 evaluation hull, completely ruling out all k <= 1 lifts.", "formal_line": "For JSC-11, degree-3 evaluation hull accepts a non-honest point, killing all acyclic one-catalyst lifts k ≤ 1.", "status": "PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "algebraic complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-072/", "receipt": "RECORD-JSC.md, RECORD-JSC-K2.md, RECORD-JSC-K2-CLOSE-PROGRESS.log", "external_value": "NONE", "statement": "Every acyclic one-catalyst lift for the JSC-11 allocation is shown to accept a false point inside its degree-3 evaluation hull, completely ruling out all k <= 1 lifts.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-073", "kind": "limit", "title": "Closure of redundant-carry seam route for natural prefixes", "title_plain": "Any circuit using the natural redundant-carry seam chain incurs a cost of 3c gates, proving it cannot beat the 2.875c threshold required to improve the leader schedule.", "formal_line": "Natural seam chains cost exactly 3c (gate-class rank equals gate count for c=2,3,4), exceeding the 2.875c leader schedule step threshold", "status": "CLOSED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "multiplicative complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-073/", "receipt": "RECORD-WINOGRAD-P5.md, RECORD-WIDER-SEAM-PROGRESS.log", "external_value": "NONE", "statement": "Any circuit using the natural redundant-carry seam chain incurs a cost of 3c gates, proving it cannot beat the 2.875c threshold required to improve the leader schedule.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-074", "kind": "limit", "title": "Method limit of cube-and-conquer without algebraic reduction", "title_plain": "Splitting seam instances into cubes times out without algebraic reduction, showing that invariant reduction is necessary before automated solving.", "formal_line": "march_cu on c=3/p=8 yielded 4096 cubes timing out at 600 s; algebraic reduction enables 0.4-6 s decisions.", "status": "METHOD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "sha-256-record", "school": "exact synthesis methods", "function_type": "method or instrument", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-074/", "receipt": "RECORD-WIDER-SEAM-PROGRESS.log", "external_value": "SAT and automated reasoning practitioners tackling structured algebraic search instances.", "statement": "Splitting seam instances into cubes times out without algebraic reduction, showing that invariant reduction is necessary before automated solving.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-075", "kind": "limit", "title": "The Mirwald–Schnorr 3-form transfer gap and n = 6 enumeration barrier", "title_plain": "Equivalence between quadratic and general multiplicative complexity for 3-forms is proven only up to 5 variables, leaving 6 variables computationally out of reach and blocking packing filtration bounds.", "formal_line": "MC(W) = qMC(W) for dim W = 3 is proved only for n ≤ 5; closing it at n = 6 is an open bottleneck for packing filtration bounds at j ≥ 2", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-075/", "receipt": "geometry/REPORT.md §6, wave1-ms3/REPORT.md §4 MS-08, PROGRESS.log 86–87", "external_value": "Researchers studying multiplicative complexity of bilinear maps, GF(16) multiplication, and 3-output S-box lower bounds.", "statement": "Equivalence between quadratic and general multiplicative complexity for 3-forms is proven only up to 5 variables, leaving 6 variables computationally out of reach and blocking packing filtration bounds.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-076", "kind": "limit", "title": "Multiplicative complexity bracket for GF(2^4) multiplication", "title_plain": "The multiplicative complexity of GF(2^4) multiplication is pinned between 8 and 9 in the unrestricted circuit model, with closing the gap reduced to the 3-form transfer or exhaustive search.", "formal_line": "MC(GF(2^4) multiplication) ∈ [8, 9] in the unrestricted model", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "ciphers-and-quantum", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-076/", "receipt": "wave1-gf16/REPORT.md §3, m3_sweep_result.json", "external_value": "Cryptographic hardware designers and complexity theorists optimizing finite field arithmetic over GF(16).", "statement": "The multiplicative complexity of GF(2^4) multiplication is pinned between 8 and 9 in the unrestricted circuit model, with closing the gap reduced to the 3-form transfer or exhaustive search.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-077", "kind": "limit", "title": "Exact Multiplicative Complexity of x1x2x3 ⊕ y1y2y3y4", "title_plain": "The multiplicative complexity of the direct-sum function x1x2x3 ⊕ y1y2y3y4 is exactly 5, verified by exhaustive DRAT proofs across all first-gate symmetry orbits.", "formal_line": "MC(x1x2x3 ⊕ y1y2y3y4) = 5", "status": "CLOSED", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "direct-sums", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "AGL(n, 2) Orbits / Encodings", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-077/", "receipt": "t02_orbits.py, out/t02_orbits.json, t14_recall.py, out/t14_cover.json, run_5.sh, /tmp/ds2_k4_cubes/*.drat.log, orbit_*.drat via drat-trim", "external_value": "Complexity theorists studying direct-sum behavior of multiplicative complexity and SAT-based circuit lower bound certification.", "statement": "The multiplicative complexity of the direct-sum function x1x2x3 ⊕ y1y2y3y4 is exactly 5, verified by exhaustive DRAT proofs across all first-gate symmetry orbits.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-079", "kind": "limit", "title": "Open Status of the NIST 21-vs-22 Question for Degree-22 Symmetric Functions", "title_plain": "The question of whether degree-22 symmetric functions require 21 or 22 multiplications remains open because only the specific five-step two-phase construction route has been ruled out.", "formal_line": "NIST 21-vs-22 question for deg-22 symmetric functions remains open; t = 5 two-phase route yields 22, while t = 6, 7 remain open.", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "symmetry-and-encodings", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "Boolean ANF Degree / Symmetry", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-079/", "receipt": "wave1-symmetric/REPORT.md §4, out/unit_g_parity.json", "external_value": "Researchers studying the multiplicative complexity of symmetric Boolean functions and the maximum complexity B_6 of 6-variable functions.", "statement": "The question of whether degree-22 symmetric functions require 21 or 22 multiplications remains open because only the specific five-step two-phase construction route has been ruled out.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-082", "kind": "limit", "title": "Multiplicative complexity bounds for chi_6^2 and chi_7^2", "title_plain": "Multiplicative complexity bounds for the squared chi permutation remain bounded within [8, 12] at width 6 and [9, 14] at width 7.", "formal_line": "chi_6^2 ∈ [8, 12], chi_7^2 ∈ [9, 14]", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "ciphers-and-quantum", "school": "cryptographic sbox complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-082/", "receipt": "chi/REPORT.md §5, out_s04b_chi6pow2_k7_nosym.json", "external_value": "Cryptographic circuit designers synthesizing nonlinear chi permutation layers for Keccak-like designs.", "statement": "Multiplicative complexity bounds for the squared chi permutation remain bounded within [8, 12] at width 6 and [9, 14] at width 7.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-086", "kind": "limit", "title": "Ineffectiveness of 3-tensor slice rank for multiplicative complexity bounds", "title_plain": "Order-3 tensor slice rank cannot produce useful multiplicative complexity lower bounds because any system of m quadratic forms trivially has slice rank at most m.", "formal_line": "srank(T) ≤ m implies MC ≥ ⌈srank(T)/3⌉ ≤ ⌈m/3⌉ < m, which fails to improve upon the trivial linear dimension floor m", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "direct-sums", "school": "algebraic complexity", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-086/", "receipt": "wave2-slice-rank/slice_rank_audit.json", "external_value": "Informs researchers in algebraic complexity and tensor methods that 3-tensor slice rank methods cannot beat trivial linear bounds over F_2.", "statement": "Order-3 tensor slice rank cannot produce useful multiplicative complexity lower bounds because any system of m quadratic forms trivially has slice rank at most m.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "This result builds on tensor slice rank and partition rank techniques introduced by Tao (2016) and Naslund (2020).", "condition": "" }, { "id": "ML-087", "kind": "limit", "title": "Multiplicative complexity of cascaded k:2 carry-save compressor slices", "title_plain": "Cascaded k:2 carry-save compressor slices require exactly k-2 AND gates because intermediate carry bits cannot catalytically reduce downstream gate counts.", "formal_line": "MC(k:2) = k - 2 for cascaded Full Adder compressor slices in the sequential XAG model (MC(4:2)=2, MC(5:2)=3, MC(6:2)=4)", "status": "PROVED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "exact determination", "page_verdict": "SHOWCASE", "cost_model": "Bilinear / Tensor Rank", "is_open": false, "is_negative": false, "paper_url": "/research/papers/ml-087/", "receipt": "programs/corridor-sweep-20260901/wave2-compressors/compressor_results.json", "external_value": "Circuit architects and arithmetic synthesis tool designers optimizing compressor trees and carry-save multipliers for AND-depth or multiplicative complexity.", "statement": "Cascaded k:2 carry-save compressor slices require exactly k-2 AND gates because intermediate carry bits cannot catalytically reduce downstream gate counts.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-088", "kind": "limit", "title": "Search scale barrier for unrestricted 6-AND XAG decision of M_2(F_2)", "title_plain": "Deciding whether 2x2 matrix multiplication over F_2 can be computed using six AND gates yields a massive CNF SAT instance that remains unsolved.", "formal_line": "Deciding 6-AND unrestricted XAG for M_2(F_2) requires refuting a CNF with 155,669 clauses and 33,345 variables; open at k=6.", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "direct-sums", "school": "exact synthesis methods", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "SAT / Automated Reasoning", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-088/", "receipt": "programs/corridor-sweep-20260901/wave2-m2-closure/multi_satlib.py, programs/corridor-sweep-20260901/wave2-m2-closure/m2_k6.cnf", "external_value": "SAT solver developers and researchers studying exact synthesis of matrix multiplication circuits over F_2.", "statement": "Deciding whether 2x2 matrix multiplication over F_2 can be computed using six AND gates yields a massive CNF SAT instance that remains unsolved.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-089", "kind": "limit", "title": "Resolution tax τ_4 in its smallest open form", "title_plain": "The resolution tax τ_4 = MC(K_4) − MC(J_5) is bounded within {1, 2}, with deciding MC(J_5) at 7 gates currently hitting a synthesis solver wall.", "formal_line": "τ_4 = MC(K_4) − MC(J_5) ∈ {1,2}; MC(K_4) = 9, MC(J_5) ∈ [7,8]", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "adders-and-counters", "school": "multiplicative complexity", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "GF(2) XAG (XOR-free)", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-089/", "receipt": "ZKGOLF-COUNCIL4-DIGEST-2026-09-02.md, RECORD-RREF-SYNTH.log, RECORD-COUNCIL4-FIVECENT.log", "external_value": "Researchers studying exact multiplicative complexity bounds and solver limits for Boolean synthesis.", "statement": "The resolution tax τ_4 = MC(K_4) − MC(J_5) is bounded within {1, 2}, with deciding MC(J_5) at 7 gates currently hitting a synthesis solver wall.", "published": "2026-09-04", "last_reviewed": "2026-09-04", "prior_art": "", "condition": "" }, { "id": "ML-090", "kind": "limit", "title": "Search wall for the smallest 7-universal tournament at order 13", "title_plain": "The existence of a 13-vertex tournament containing every 7-vertex tournament remains undecided due to an exponential SAT solver bottleneck during iterative pattern-core enforcement.", "formal_line": "13 ≤ t(7) ≤ 15; n = 13 undecided under SAT and lazy pattern-core CEGAR which exceeds 1800 s timeout at 32 enforced patterns", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "open-problems", "school": "exact synthesis methods", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-090/", "receipt": "box frontier/t7/ run logs and status.json; lane lanes/tournaments-t7/WALL.md", "external_value": "Combinatorialists studying universal tournaments and SAT practitioners working on pattern-containment and subgraph isomorphism encodings.", "statement": "The existence of a 13-vertex tournament containing every 7-vertex tournament remains undecided due to an exponential SAT solver bottleneck during iterative pattern-core enforcement.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "The bounds 13 <= t(7) <= 15 were established by Zhang and Szeider (CP 2023).", "condition": "" }, { "id": "ML-091", "kind": "limit", "title": "Partial degree-cube search wall for Zarankiewicz number z(16,17;3)", "title_plain": "The Zarankiewicz number z(16,17;3) remains narrowed to {132, 133} as monolithic SAT fails and degree-cube decomposition leaves 8 sub-cubes open at row degrees 10 and 9.", "formal_line": "z(16,17;3) ∈ {132, 133} undecided; d = 17..11 refuted with DRAT, 8 of 46 row-2 sub-cubes at d ∈ {9, 10} undecided at 7200 s", "status": "LIVE-PARKED", "tier": "open", "tier_label": "OPEN QUESTION", "program": "open-problems", "school": "exact synthesis methods", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-091/", "receipt": "box frontier/zarankiewicz/long/status.json, PROGRESS.box.log, proofs/, lane WALL.md", "external_value": "NONE", "statement": "The Zarankiewicz number z(16,17;3) remains narrowed to {132, 133} as monolithic SAT fails and degree-cube decomposition leaves 8 sub-cubes open at row degrees 10 and 9.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "", "condition": "" }, { "id": "ML-092", "kind": "limit", "title": "Solver wall in SAT-based exact median network synthesis", "title_plain": "Finding the minimum number of comparators needed to compute medians remains open for 8 or more inputs because SAT encodings hit computational limits and a prior published claim for 9 inputs was not reproduced.", "formal_line": "Optimal median network size undecided for n ≥ 8; SAT solver times out on n=8 at 15, n=9 at 18, n=10 at 21, and n=11 at 24 comparators", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "open-problems", "school": "exact synthesis methods", "function_type": "conjecture open problem", "page_verdict": "CAVEAT", "cost_model": "", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-092/", "receipt": "frontier/median/PROGRESS.box.log, runs/*.status.json, lane WALL.md", "external_value": "Researchers working on median network synthesis, sorting networks, and SAT-based circuit optimization.", "statement": "Finding the minimum number of comparators needed to compute medians remains open for 8 or more inputs because SAT encodings hit computational limits and a prior published claim for 9 inputs was not reproduced.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "A verification check was unable to reproduce the published n = 9 optimality claim from Smith (1996).", "condition": "" }, { "id": "ML-093", "kind": "limit", "title": "Proof-system mismatch for the circulant weighing matrix cell CW(112,36)", "title_plain": "Standard resolution and cutting-planes SAT encodings fail on the smallest open circulant weighing matrix cell CW(112,36) and benchmark cells due to a proof-system mismatch.", "formal_line": "DRAT resolution and RoundingSat cutting planes fail on CW(112,36) and known-nonexistent CW(n,36) for n ∈ {40, 44, 50, 56}.", "status": "OPEN / PROVED DEAD", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "arithmetization and proof systems", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-093/", "receipt": "box frontier/cwm/ (PROGRESS.box.log, status/*.json, ladder JSONs under invent_k*); lane WALL.md, invent/", "external_value": "NONE", "statement": "Standard resolution and cutting-planes SAT encodings fail on the smallest open circulant weighing matrix cell CW(112,36) and benchmark cells due to a proof-system mismatch.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "", "condition": "" }, { "id": "ML-094", "kind": "limit", "title": "Exact B2 circuit size of MOD3 on 6 inputs and solver scaling wall", "title_plain": "The exact two-input Boolean circuit size for MOD3 on 6 inputs remains undecided as verifying an 11-gate lower bound timed out under multiple solvers.", "formal_line": "C_B2(MOD3,0 on 6 inputs) conjectured 12; deciding 11 gates timed out at 5400 s with UNSAT cost growing 30-100x per gate", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "open-problems", "school": "exact synthesis methods", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-094/", "receipt": "frontier/mod3/STATUS.log, runs/*.status, WALL.md", "external_value": "NONE", "statement": "The exact two-input Boolean circuit size for MOD3 on 6 inputs remains undecided as verifying an 11-gate lower bound timed out under multiple solvers.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Knuth's conjecture predicts a circuit size of 12 for MOD3,0 on 6 inputs.", "condition": "" }, { "id": "ML-095", "kind": "limit", "title": "Open status and catalogue corrections for resolution hardness h_11", "title_plain": "The resolution hardness number h_11 remains open above 28, while previous values for m up to 9 were verified and an omission in the published candidate catalogue was corrected.", "formal_line": "h_11 ≥ 28 remains undetermined; reproduced h_8 = 19, h_9 = 22; candidate census corrected to 626,973 with missing RSMU(8,10) identified", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "open-problems", "school": "arithmetization and proof systems", "function_type": "conjecture open problem", "page_verdict": "CAVEAT", "cost_model": "", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-095/", "receipt": "lanes/hardness-h11/VERIFY.md, WALL.md; box frontier/h11/", "external_value": "Researchers studying proof complexity and resolution hardness benchmarks requiring verified catalogue counts.", "statement": "The resolution hardness number h_11 remains open above 28, while previous values for m up to 9 were verified and an omission in the published candidate catalogue was corrected.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "Prior work by Peitl-Szeider established the values of h_m for m ≤ 10 (including h_10 = 26) and proved that h_11 ≥ 28.", "condition": "" }, { "id": "ML-096", "kind": "limit", "title": "Bounds and search limits on the constant-weight code size A(13,4,5)", "title_plain": "The maximum size of a constant-weight code A(13,4,5) remains bounded between 123 and 129 because CDCL SAT refutations time out and local search does not exceed 123.", "formal_line": "A(13,4,5) ∈ [123, 129]", "status": "OPEN", "tier": "open", "tier_label": "OPEN QUESTION", "program": "open-problems", "school": "exact synthesis methods", "function_type": "conjecture open problem", "page_verdict": "SHOW", "cost_model": "", "is_open": true, "is_negative": false, "paper_url": "/research/papers/ml-096/", "receipt": "lane lanes/cwcode-13-4-5/WALL.md, receipts/", "external_value": "Coding theorists tracking exact bounds and SAT limits on binary constant-weight codes.", "statement": "The maximum size of a constant-weight code A(13,4,5) remains bounded between 123 and 129 because CDCL SAT refutations time out and local search does not exceed 123.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "", "condition": "" }, { "id": "ML-097", "kind": "limit", "title": "Limits of plain CDCL with cardinality totalizers for Turán numbers ex(n, K_4^(3))", "title_plain": "Plain CDCL solvers with cardinality totalizers fail on hypergraph Turán problems ex(n, K₄^(3)) for n ≥ 10, whereas exact degree-sequence cubing resolves sub-cases efficiently.", "formal_line": "ex(14, K_4^(3)) is undecided; CDCL with totalizers hits an empirical wall above n = 9, while degree-sequence cubing refutes cubes at n = 10", "status": "EMPIRICAL-WALL", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "exact synthesis methods", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-097/", "receipt": "box frontier/turan/runs/*.json, PROGRESS.log; lane WALL.md", "external_value": "Researchers applying SAT solving, symmetry breaking, and cube-and-conquer methods to extremal hypergraph problems.", "statement": "Plain CDCL solvers with cardinality totalizers fail on hypergraph Turán problems ex(n, K₄^(3)) for n ≥ 10, whereas exact degree-sequence cubing resolves sub-cases efficiently.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "", "condition": "" }, { "id": "ML-098", "kind": "limit", "title": "Failure of SMS standalone LRAT certificate verification for Kochen-Specker n = 18", "title_plain": "A standalone LRAT certificate produced by SMS for the Kochen-Specker n = 18 problem cannot be verified by standard checkers because symmetry-breaking clauses are emitted outside the LRAT derivation chain.", "formal_line": "Standalone lrat-check of smsg -v 18 --lrat-output fails due to unintegrated --sym-break-clauses outside the LRAT chain.", "status": "PROVED DEAD", "tier": "cert", "tier_label": "CERTIFIED PROOF", "program": "open-problems", "school": "formal verification", "function_type": "negative result closed route", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-098/", "receipt": "frontier/ks6/cert/d6n18_sms.lratcheck.log, d6n18_sms.log, d6n18_symclauses.json", "external_value": "SAT and SMS proof-verification practitioners seeking valid certificate chains for symmetry-breaking solvers.", "statement": "A standalone LRAT certificate produced by SMS for the Kochen-Specker n = 18 problem cannot be verified by standard checkers because symmetry-breaking clauses are emitted outside the LRAT derivation chain.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "", "condition": "" }, { "id": "ML-099", "kind": "limit", "title": "Exclusion of 19- and 20-vector Kochen-Specker sets in C^6", "title_plain": "Kochen-Specker sets of 19 or 20 vectors in six-dimensional complex space do not exist, establishing that any such set requires at least 21 vectors.", "formal_line": "No 19- or 20-vector Kochen-Specker set exists in C^6 (both excluded via MF-195, MF-199)", "status": "CLOSED", "tier": "receipt", "tier_label": "RECEIPTED", "program": "open-problems", "school": "quantum circuit synthesis", "function_type": "impossibility theorem", "page_verdict": "SHOW", "cost_model": "", "is_open": false, "is_negative": true, "paper_url": "/research/papers/ml-099/", "receipt": "lanes/ks-d6/REPORT.md Theorem 2; BANK-CANDIDATES.md (KS6-03); frontier/ks6/w2/cubes19/, cubes20/", "external_value": "Quantum contextuality and quantum foundations researchers studying minimal Kochen-Specker vector systems.", "statement": "Kochen-Specker sets of 19 or 20 vectors in six-dimensional complex space do not exist, establishing that any such set requires at least 21 vectors.", "published": "2026-09-06", "last_reviewed": "2026-09-06", "prior_art": "", "condition": "" } ] }