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Published 2026-08-29 · updated 2026-09-06

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Id	Result	Programme	Status	Evidence	Published
MF-002 finding	An upper bound on the multiplicative complexity of a three-column interior adder tile`MC(g) ≤ 9` 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.	Adders, counters and the heap law	PROVED	RECEIPTED	2026-08-29
MF-004 finding	Functional census and degree bounds for the two-row cyclic model`\|S₂\| = 575,968, \|{Φ(S) : S ∈ S₂}\| = 32,768, and ∀S ∈ S₂, deg(Φ(S)) ≤ 3`	Symmetry, state encodings and search gauges	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-006 finding	Simultaneous and cyclic witness definitions in the Lean 4.28 kernel`Lean 4.28 accepts simultaneous and cyclic witness definitions with soundness, completeness, exact cost, computable witnesses, no sorryAx`	Formal verification and machine-checked proof	PROVED	LEAN-VERIFIED	2026-08-29
MF-008 finding	Affine equivalence classes of minimum-rank eight carry-state encodings`{e ∈ E : e passes minimum-rank filter} / affine equivalence = {natural, orbit29 = [0,1,2,3,4,7,6,5]}`	Symmetry, state encodings and search gauges	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-009 finding	Quadratic hull closure of the Z-counter transition relation`\|H₂(R_∂) ∩ W_same\| = 24; exactly three canonical quadratic rows A·B=C are necessary and sufficient`	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-010 finding	An impossibility theorem for the χ₀₁ catalyst and the full-domain p7 tile`deg(f) = 9, 80 of 140 top monomials ∉ I = ⟨c₁c₂⟩ ⟹ χ₀₁ catalyst cannot realize full-domain p7 tile`	Cipher S-boxes, χ, and quantum gate counts	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-011 finding	On the zero sets of products of affine forms on F₂³`For affine forms ℓ_1,...,ℓ_m: 𝔽₂³ → 𝔽₂ and P = ∏_{j=1}^m ℓ_j, \|Z(P)\| ∈ {0,4,6,7,8}`	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-012 finding	An impossibility theorem for pinning the three-input AND under quadratic systems`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}`	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-014 finding	Direct-sum additivity barrier for extension-field batching over F₂ᵐ`Conjecture MF-014: batching over F₂ᵐ via Karatsuba or CRT cannot beat direct-sum additivity under the register's multiplicative cost model`	Direct sums, wedges and the p14 frontier	CONJECTURE	OPEN QUESTION	2026-08-29
MF-016 finding	Formal verification of large cryptographic circuits via cone-local proofsPer-cone proofs composed via existing semantic theorems compile no-sorry in seconds to minutes when monolithic `bv_decide` is intractable	Formal verification and machine-checked proof	METHOD	RECEIPTED	2026-08-29
MF-017 finding	Sparse exact synthesis of a 69,862-variable allocator`Sparse exact CEGIS with 120× row-order symmetry reduction solved a 69,862-variable instance in 12 rounds (26 s); monolithic timed out`	Symmetry, state encodings and search gauges	METHOD	RECEIPTED	2026-08-29
MF-018 finding	A soundness condition for subset-UNSAT certificates under symmetry breaking`A subset-UNSAT certificate is sound only when ∀g ∈ G, g(R) = R`	Symmetry, state encodings and search gauges	METHOD	RECEIPTED	2026-08-29
MF-020 finding	Full-domain replay for candidate XAG validity`∀x ∈ {0,1}^n, C(x) = f(x)`	The SHA-256 record and exact synthesis	METHOD	EXHAUSTIVE CHECK	2026-08-29
MF-022 finding	A profile for quadratic pinning and defect structure`delta2^flip ≤ delta2^vert`	The quadratic hull and its defects	METHOD	RECEIPTED	2026-08-29
MF-023 finding	Exact multiplicative complexity of two exposed-sum ripple chains`MC(I_L) = 2L`	Adders, counters and the heap law	PROVED	CERTIFIED PROOF	2026-08-29
MF-024 finding	Rank-one-target deficiency of stationary full covers on GF(2)⁵`The condition of using all ten spare codewords holds if and only if delta ∈ {4, 20}`	The quadratic hull and its defects	PROVED	CERTIFIED PROOF	2026-08-29
MF-025 finding	The false path 8 -> 0 -> 0 in the natural five-code Ghost-P5 quadratic hull`No subset of H achieves soundness via repetition alone (two-column sequential composition admits exact false path 8 -> 0 -> 0)`	The quadratic hull and its defects	PROVED	CERTIFIED PROOF	2026-08-29
MF-027 finding	Verifier-level audit of finite phase-trellis relations`Subset construction gives exact completeness/soundness checks for finite phase relations; natural sample & ITER19 K0 adapter fail soundness`	Formal verification and machine-checked proof	METHOD	EXHAUSTIVE CHECK	2026-08-29
MF-028 finding	A twelve-wedge prefix obstruction for the period-two carry law in p14 circuits`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`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-031 finding	An impossibility theorem for deterministic state-only lifts of the c0c2 quadratic hull`No deterministic state-only feature set can repair the relation, as maximal lift replay retains 23,808 of 24,576 old wrong hull points`	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-032 finding	Copy-locality of target-bearing products at the rank-tight boundary over GF(2)`For affine combinations L and R of separated signals, if LR is separated and target-bearing, then LR is copy-local`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-033 finding	A structural characterization of the c0c2 quadratic defectOn 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`.	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-035 finding	A conjectured rank-two bilinear custom gate for the p14 bridge`MF-035 is a conjecture.`	Direct sums, wedges and the p14 frontier	CONJECTURE	OPEN QUESTION	2026-08-29
MF-036 finding	A census of one-block affine data-split pins for c0c2 hull repair`The corrected census spans all 8,184 pointwise-distinct cases.`	The quadratic hull and its defects	EXHAUSTED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-037 finding	A wedge-span obstruction for the exact Cartesian period-two carry law`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`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-038 finding	On gate savings under repeated period doubling`Conditional on a verifier-legal p14 fusion, C(2r) ≤ 2 C(r) - Delta_r`	Direct sums, wedges and the p14 frontier	CONJECTURE	OPEN QUESTION	2026-08-29
MF-039 finding	A conjectural lower bound for the multiplicative complexity of T2`MC(T2) ≥ 15`	Direct sums, wedges and the p14 frontier	CONJECTURE	OPEN QUESTION	2026-08-29
MF-040 finding	An upper bound on the multiplicative complexity of the natural (S,S+R) component`MC(component) ≤ 8 for the 13-input, 7-output natural (S,S+R) component`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-042 finding	Exact GL(2,2) symmetry quotient for catalyst parameterization`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`	Symmetry, state encodings and search gauges	METHOD	RECEIPTED	2026-08-29
MF-044 finding	Exact scans of published p8 factors for three maximal mixed p14 skeletons`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`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-045 finding	The kernel of the Boolean product-residue map on separable signal cuts`ker(μ) = ker(μ_0) ⊕ ker(μ_1)`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-047 finding	A lower bound on chained products for the period-two p14 target`rank(T) = 10, rank(T≤2) = 4, dim(T ∩ W) = 4 ⇒ every p14 construction requires ≥ 6 chained or non-input-only product functions`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-048 finding	Quadratic-hull screens of two-column SHA-256 seed allocations`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`	The SHA-256 record and exact synthesis	CONJECTURE	OPEN QUESTION	2026-08-29
MF-049 finding	Linear fresh-defect growth in the strongest natural five-code relaxed SHA relation`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)`	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-050 finding	Exact finite-horizon minimisation of natural SHA carry states`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`	Symmetry, state encodings and search gauges	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-051 finding	Impossibility of realizing AND4 with three witnesses and three product rows`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`	What rank-one constraints can express	PROVED	CERTIFIED PROOF	2026-08-29
MF-052 finding	Closure of the ordered acyclic AB=C catalyst class at p6`Ordered acyclic AB=C catalyst class at p6 = ∅ (1,005 exact-UNSAT, 7,035 rank-dead across 67 deterministic state-feature subspaces)`	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-053 finding	Refutation of stationary additive raw/state phases by a three-column divergent carry path`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`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-057 finding	On the full-word gauge of the consecutive-Maj identity`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`	The SHA-256 record and exact synthesis	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-058 finding	Nonexistence of small low-witness AND4 cells`In the affine-factor, affine-decoder, arbitrary-nonempty-fibre model with arbitrary affine-C: 1.`	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-059 finding	Complete witness-width classification of Boolean functions on F₂⁴Every 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	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-060 finding	An impossibility theorem for the normalized two-witness selector for AND₄`Every normalized 2-witness selector separates 30 wrong witnesses over x ≠ 1111, and no selector separates both wrong witnesses over x = 1111`	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-061 finding	Classification of the identity-C fixed-point consumer on toy Boolean graphs`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)`	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-062 finding	Exact fibre cardinality of the two-round SHA-256 edge-relaxed relation`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`	The SHA-256 record and exact synthesis	PROVED	PAPER PROOF	2026-08-29
MF-063 finding	Invertibility of odd rotation sums in F2[x]/((x+1)^{2^r})`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`	The SHA-256 record and exact synthesis	PROVED	PAPER PROOF	2026-08-29
MF-064 finding	An exact conservation law for gauges in GF(2) forest systems`For a forest with N vertices and E edges over GF(2), gauge space dimension = apparent vertex-row savings = created gauges = N-E`	What rank-one constraints can express	PROVED	PAPER PROOF	2026-08-29
MF-065 finding	Exact number of quadratic equations for a unique common zero in GF(2)^32`Forcing g₀=...=g₃₁=0 over GF(2) with degree ≤ 2 equations without auxiliary variables requires and is satisfied by exactly 16 equations` Prior art: The lower bound is based on the classical Chevalley-Warning theorem, and no separate prior-art position is claimed.	The quadratic hull and its defects	PROVED	PAPER PROOF	2026-08-29
MF-069 finding	A strict flag characterization of rank-tight acyclic full-domain XAGs`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}`	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED	2026-08-29
MF-070 finding	Impossibility of p7 realizations for rank-tight q-rank-7 edges under G = W`In the rank-tight model G = W, every e ∈ E₇ is p7-impossible`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-071 finding	A multiplicative complexity lower bound for 28 q-rank-6 edges`Every edge in E₆ is p6-impossible`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-072 finding	Impossibility of rank-tight p5 realizations for q-rank-5 edges`In the rank-tight model G = W, every e ∈ E₅ is p5-impossible`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-073 finding	Exact catalyst dimension in a rank-tight XAG`For a p-gate acyclic XAG with W ⊆ G, dim G = p, dim W = r: catalyst dimension = p − dim W = p − r`	Direct sums, wedges and the p14 frontier	PROVED	PAPER PROOF	2026-08-29
MF-074 finding	A counterexample to the laminar multiplication-tree normal form`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]`	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED NEGATIVE RESULT	2026-08-29
MF-077 finding	Multiplicative complexity of four-operand addition`x+y+z+w+c₀+2c₁ = s+2c₀′+4c₁′ using 3 AND gates; full-precision sum of four n-bit integers uses 3n ANDs` 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.	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-080 finding	A lower bound on the Toffoli count of exact NCT networks implementing χ₅`For every exact NOT/CNOT/Toffoli network N implementing χ₅ with clean ancillas, #Toffoli(N) ≥ MC(χ₅⁻¹) = 6` Prior art: No separate prior-art claims are made.	Cipher S-boxes, χ, and quantum gate counts	PROVED	PAPER PROOF	2026-08-29
MF-083 finding	An eight-product circuit for the low four bits of four four-bit numbers`The low four bits of x₀ + x₁ + x₂ + x₃ are computed by an acyclic XAG with 8 products`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-084 finding	The separated-product theorem over any field under square closure`If u ∈ U ⟹ u² ∈ U and v ∈ V ⟹ v² ∈ V, the separated-product theorem holds over any field k` Prior art: This entry builds on predecessor MF-032; no position on external prior art is stated.	Direct sums, wedges and the p14 frontier	PROVED	CERTIFIED PROOF	2026-08-29
MF-085 finding	The mixed-image tax and rank-tight copy-locality`dim(residual separated target quotient) = r with r new product gates ⟹ every independent target-bearing gate is copy-local`	Direct sums, wedges and the p14 frontier	PROVED	PAPER PROOF	2026-08-29
MF-086 finding	Multiplicative complexity of independent copies of 2×2 matrix multiplication over GF(2)`rank_CP/bilinear(M₂^⊕s) = 7s, MC_formal-quadratic(M₂^⊕s) = 7s over GF(2)` 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.	Direct sums, wedges and the p14 frontier	PROVED	PAPER PROOF	2026-08-29
MF-087 finding	Matroid invariants of the quadratic hull of a Boolean relation`h₂=false-loop count, ρ₂=false-syndrome rank, δ_2,pin=nullity after deleting loops, κ_2,cover=Crapo–Rota critical exponent` Prior art: Most concepts and terms used here follow established literature; only delta2^vert and delta2^flip represent original formulations introduced in this work.	The quadratic hull and its defects	IMPORTED	PAPER PROOF	2026-08-29
MF-088 finding	Exact multilinear separator degree of the Boolean AND graph`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` Prior art: This result generalizes MF-012; no external prior-art comparison is stated.	The quadratic hull and its defects	PROVED	PAPER PROOF	2026-08-29
MF-089 finding	The degree-(d) hull of Boolean relations with few forbidden points`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` Prior art: This step relies on standard minimum-distance properties of classical Reed-Muller codes; no original contribution or novelty is claimed.	The quadratic hull and its defects	PROVED	PAPER PROOF	2026-08-29
MF-090 finding	Exact quadratic cover number of an OR-chain lift of a width-k clause over F2`For the lifted relation of a positive width-k clause (k ≥ 3), κ2,cover = k − 1` 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.	The quadratic hull and its defects	PROVED	PAPER PROOF	2026-08-29
MF-091 finding	A minimal three-row quadratic constraint system for the inverse-or-default gadget`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`	What rank-one constraints can express	PROVED	PAPER PROOF	2026-08-29
MF-092 finding	Exact nonlinear cost of an eight-state controller under affine relabeling`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`	Symmetry, state encodings and search gauges	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-093 finding	Formal verification of end-to-end output equality for XAGs in Lean`Lean 4.28 proves equivalence between a 30,003-gate reference hierarchy and a 20,002-gate optimized hierarchy using zero axioms`	Formal verification and machine-checked proof	METHOD	LEAN-VERIFIED	2026-08-29
MF-097 finding	Decomposability of the extremal top form in acyclic XOR–AND circuits`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` Prior art: This result makes no separate prior-art claim.	Direct sums, wedges and the p14 frontier	PROVED	CERTIFIED PROOF	2026-08-29
MF-099 finding	The exact digit-sum law for full K-operand addition`MC(FullAdd(K,n)) = Kn − s₂(K(2ⁿ−1)); FullAdd(5,2) = 6 ≠ 8`	Adders, counters and the heap law	PROVED	PAPER PROOF	2026-08-29
MF-100 finding	The exact width-two law for truncated multi-operand addition`MC(A_{k,2}) = ⌊k/2⌋ for every k ≥ 2`	Adders, counters and the heap law	PROVED	PAPER PROOF	2026-08-29
MF-101 finding	The universal one-gate bracket for three-operand addition`2n−4 ≤ MC(A_{3,n}) ≤ 2n−3; the upper is exact at n = 2,3,4,5 and for prefix-causal circuits`	Adders, counters and the heap law	PROVED	PAPER PROOF	2026-08-29
MF-102 finding	A projected full-heap construction for truncated addition`MC(A_{k,n}) ≤ T(k,n) = (k−1)(n−1) − Σ_{r=1}^{n−1}⌊(k−1)/2ʳ⌋`	Adders, counters and the heap law	PROVED	PAPER PROOF	2026-08-29
MF-103 finding	Injected carry is exactly the flagship one-gate gap`MC(J_m) = MC(A_{3,m+1}) − 1; J₃₂ = Add32x3Canon33 ∈ {61,62}`	Adders, counters and the heap law	PROVED	PAPER PROOF	2026-08-29
MF-104 finding	Verified width-three constructions for nine and ten operands`MC(A_{9,3}) ≤ 10 and MC(A_{10,3}) ≤ 12; both are replayed uppers, not exact values`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-08-29
MF-105 finding	The live p14 frontier and its doubly conditional row saving`p14 existence is LIVE/UNKNOWN; 817 named copy-swap cells survive; −477 rows requires p14 and rank(I+A)=3`	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-08-29
MF-106 finding	The 37-wall atlas of failed proof routes and scope limits`W01–W37: MODEL, TECHNIQUE, SCOPE, and EVIDENCE walls with receipts R1–R46`	Further results	INSTRUMENT AUDIT FINDING	RECEIPTED NEGATIVE RESULT	2026-08-29
MF-107 finding	Counterexamples to four former addition laws`FullAdd (K−1)n, greedy truncated equality, U(k,n) tightness, and CS-base for k ≥ 9 are refuted`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
MF-108 finding	The verified 22,215-row SHA-256 bill remains unchanged`verified rows = 22,215; authorized row change = 0; no row improvement is claimed`	The SHA-256 record and exact synthesis	PROVED	RECEIPTED	2026-08-29
MF-109 finding	Is the projected-heap upper bound always exact?`CONJECTURE: MC(A_{k,n}) = T(k,n); the first sharp stable fork is A_{9,3} ∈ {9,10}`	Adders, counters and the heap law	OPEN	OPEN QUESTION	2026-08-29
MF-110 finding	Correcting two truncated-addition cells`MC(A_{5,3}) = 5; MC(A_{4,4}) ∈ [6,8], not exact 8 on retained evidence`	Adders, counters and the heap law	INSTRUMENT AUDIT FINDING	RECEIPTED	2026-08-29
MF-111 finding	Bounds on the multiplicative complexity of the injected-carry family`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]`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-112 finding	Status of the FCNS reduction chain for uniform lower bounds`Exact exclusion is Γ_{m, 2m-3} = ∅; local-response theorem proved, but summation across sites and SBD remain unproved (OPEN).`	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-09-04
MF-113 finding	Shortest-path and LP-dual formulation of multiplicative complexity via semantic gate states`MC equals shortest-path distance on the complete semantic gate-state graph; its LP dual is the 1-Lipschitz potential problem`	Symmetry, state encodings and search gauges	PROVED	CERTIFIED PROOF	2026-09-04
MF-114 finding	Exact restriction loss conservation law for multiplicative complexity`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).`	Adders, counters and the heap law	PROVED	CERTIFIED PROOF	2026-09-04
MF-115 finding	Exact Kummer endpoint valuation and boundary-alias count for FullAdd`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⌉`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-116 finding	Strict syntactic-equivariant inverse-χ costs at widths 3, 5, and 7`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`	Cipher S-boxes, χ, and quantum gate counts	PROVED	CERTIFIED PROOF	2026-09-04
MF-117 finding	Bounds on the strict width-nine cost E_9`18 ≤ E_9 ≤ 36`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-118 finding	Exact shear symmetry of multiplicative complexity in constant addition`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`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-119 finding	Formula for canonical constant-heap complexity H_n(K)`H_n(K) = 4n - 6 - λ_n(K) for the canonical constant-heap class`	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-121 finding	Refutation of the general carry-bond tensor floor`Full-carry sector injection into product-state space fails at J_2: degree-4 indicators not in two-gate span; general floor is refuted.`	Direct sums, wedges and the p14 frontier	REFUTED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-122 finding	Exact complexity of the canonical heap-prefix class`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)`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-123 finding	Necessary carrier sequence and connected-leakage bounds for hypothetical p14`0→H_6→E_10→C_4→0; rank Q_2(fg) ≤ 2r+2s+rs+2; first mixed catalyst connected rank ≤ 2`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-124 finding	Reduction of named p14 census residual to 221 cells`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` Prior art: This result supersedes the 817-cell snapshot from MF-105; the general p14 SAT/UNSAT problem remains open.	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-09-04
MF-125 finding	Orbit and comodule anatomy of obtained J_3 witnesses`Obtained J_3 witnesses partition into 7 unpointed gate-space orbits and 10 pointed restriction-comodule types; full census remains open.`	Adders, counters and the heap law	COMPUTED	CERTIFIED PROOF	2026-09-04
MF-128 finding	Exact unrestricted multiplicative complexity of binary GF(8) multiplication`MC_XAG,F_2(F_8 × F_8 → F_8) = 6 in basis F_2[r]/(r^3 + r + 1)`	Cipher S-boxes, χ, and quantum gate counts	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-129 finding	Exact multiplicative complexity of the forward χ mapping`MC(χ_n) = n for every n ≥ 3, with coordinate rule χ_i = Rule210_{i+1}`	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-130 finding	Exact multiplicative complexity of all powers of the width-four chi map`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`	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-131 finding	Classification of One-Word XOR-Mask Transports for Four-Word Addition`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)).`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-132 finding	The Boolean ramification polytope common evaluation conjecture`MC(F) ≥ r+D-2-c(Q) survives, but unqualified polytope equality is undefined at V=0 and remains a CONJECTURE (GAP)`	The quadratic hull and its defects	CONJECTURE	OPEN QUESTION	2026-09-04
MF-133 finding	Finite classification of the primary-affine pairing-one localization component`Primary-affine pairing-one kernel has 8 points (full kernel 15); reduced lex Gröbner basis has 69 polynomials; 5/3 with zero affine tail.`	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED	2026-09-04
MF-134 finding	Exact joint multiplicative complexity of chi_n and its square`MC(chi_n, chi_n^2) = 2n for all n >= 4`	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-135 finding	Three sound whole-function multiplicative complexity lower-bound floors`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})`	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-136 finding	Exact multiplicative complexity of 5-by-3 addition and six addition floors`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`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-137 finding	The packing lemma lower bound for multiplicative complexity`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`	Further results	PROVED	RECEIPTED	2026-09-04
MF-138 finding	Exact multiplicative complexity of the two-column C7 carry transducer`MC(T) = 6 for the 11-input, 5-output two-column C7 carry transducer`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-139 finding	Gate-class dimension increment per C7 column`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`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-140 finding	Exact multiplicative complexity of A_{4,3} and constant-added addition F_{3,K}`MC(A_{4,3}) = 5 = MC(F_{3,K}) for every K mod 8; MC(F_{2,K}) = 2 = MC(A_{4,2})`	Adders, counters and the heap law	COMPUTED	RECEIPTED	2026-09-04
MF-141 finding	Full linear rank of multiplicative rows in the SHA-256 leader circuit`Rank of 22,215 product equations modulo affine forms is 22,215/22,215 over GF(2); no linear product row elimination exists.`	The SHA-256 record and exact synthesis	COMPUTED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-142 finding	Multiplicative complexity bounds for chi_5^2`6 ≤ MC(chi_5^2) ≤ 7` Prior art: Prior bracket was [6,10].	Cipher S-boxes, χ, and quantum gate counts	COMPUTED	RECEIPTED	2026-09-04
MF-143 finding	Refutation of three proposed multiplicative complexity laws`MC(11 x mod 128) = 6 ≠ 5, kappa_sq(7) ∈ {2,3} ≠ 1, and MC(I_9) = 6 ≠ 5, refuting three conjectured complexity laws.`	Further results	REFUTED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-144 finding	The packing filtration lower bound and master identity for multiplicative complexity`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` Prior art: The gate-span and rank arguments are credited to Schnorr (1989), Boyar–Peralta–Pochuev (2000), and Boyar–Find (2018).	The quadratic hull and its defects	PROVED	RECEIPTED	2026-09-04
MF-145 finding	Packing floors on quadratic multiplicative complexity for linear spaces of quadratics`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)` Prior art: This is a partial result.	The quadratic hull and its defects	PROVED	RECEIPTED	2026-09-04
MF-146 finding	Deficit law lower bound for all-quadratic spaces`MC(F) ≥ dim V - 2 + ⌈3μ/2⌉ for all-quadratic V with every nonzero class of cost μ` Prior art: PARTIAL (composite of MF-144/MF-145).	The quadratic hull and its defects	PROVED	RECEIPTED	2026-09-04
MF-147 finding	Basis-free lower bound on the multiplicative complexity of GF(2^k) multiplication`MC(GF(2^k) multiplication) ≥ ⌈5k/2⌉ - 2 in the unrestricted XAG model, basis-free` 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.	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-148 finding	Exact unrestricted multiplicative complexity of 3-term binary polynomial multiplication`MC(polymul_3 over F2) = 6 in the unrestricted model` Prior art: This is a known value obtained using classical Karatsuba multiplication for the NIST binary-polynomial category.	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-149 finding	A lower bound on multiplicative complexity of 2x2 matrix multiplication over F2`MC(M_2 over F2) >= 6, narrowing MC(M_2 over F2) to [6,7]` Prior art: Winograd’s rank-7 optimality result applies specifically to bilinear algorithms; no unrestricted bound has been established.	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED	2026-09-04
MF-150 finding	Exact multiplicative complexity of 2×2 unsigned integer multiplier`MC(2×2 unsigned integer multiplier) = 4` Prior art: This is a partial result, as small multipliers are well-studied and likely already known.	Adders, counters and the heap law	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-151 finding	Exact characterization of additive quadruples destroyed by one AND gate`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` 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.	The quadratic hull and its defects	PROVED	RECEIPTED	2026-09-04
MF-152 finding	Sharpness of the 5/8 law and multiplicative complexity of n-bit addition`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` 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.	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-153 finding	Solver-Free Multiplicative Complexity Lower Bounds from Degree and Walsh Floors`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` Prior art: These bounds are previously established: the degree floor is credited to Schnorr, and the Walsh floor is documented under entry MF-135.	Further results	PROVED	RECEIPTED	2026-09-04
MF-154 finding	Exact multiplicative complexity of chi_5^2`MC(chi_5^2) = 7, closing the [6,7] bracket at the top, with eps_5 = 10 - 7 = 3 = eps_4` Prior art: This appears to be a new result.	Cipher S-boxes, χ, and quantum gate counts	COMPUTED	EXHAUSTIVE CHECK	2026-09-04
MF-155 finding	Multiplicative complexity lower bounds and brackets for chi_6^2 and chi_7^2`MC(chi_6^2) >= 8 (bracket [8, 12]) and MC(chi_7^2) >= 9 (bracket [9, 14])` Prior art: This is an apparently new result cataloged as entry MF-154, achieved using fewer searches.	Cipher S-boxes, χ, and quantum gate counts	COMPUTED	RECEIPTED	2026-09-04
MF-156 finding	The k-window law for joint multiplicative complexity of Chi iterates`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` 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`.	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-157 finding	Iterate structure, degree, and attributions for the Keccak chi mapping`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` Prior art: Lemma A for odd n is due to Kriepke-Kyureghyan (CRYPTO 2024), while even n collisions match results from Schoone-Daemen (2024). Sources: ePrint 2024/801 · ePrint 2014/474	Cipher S-boxes, χ, and quantum gate counts	COMPUTED	RECEIPTED	2026-09-04
MF-158 finding	Classification of relations implemented by single-row R1CS over F_p`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}.` Prior art: This appears to be a new, partial result applying elementary algebraic geometry to a single quadratic equation.	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-160 finding	Exact R1CS Row Counts and Allocation-Independent Lower Bounds for Basic Gadgets`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.` 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.	What rank-one constraints can express	PROVED	RECEIPTED	2026-09-04
MF-161 finding	Row complexity of n-bit range checks and u32 addition in R1CS`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` Prior art: This result is partial: the constructions follow standard practice, but the corresponding lower bounds have not yet been located.	What rank-one constraints can express	PROVED	RECEIPTED	2026-09-04
MF-162 finding	Witness power, degree collapse, and allocation walls over large prime fields`{x : x != 0} needs 1 witness, XOR3/MAJ3 degree collapse breaks MF-013 over F_p, and primitive walls hold against arbitrary allocations` Prior art: PARTIAL / INTERNAL.	What rank-one constraints can express	PROVED	RECEIPTED	2026-09-04
MF-163 finding	Direct-sum theorems and lower bounds for unrestricted multiplicative complexity`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` Prior art: Theorem A and the rank floor are partly known or likely folklore (BPP 2000; Mirwald–Schnorr for the quadratic model).	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED	2026-09-04
MF-164 finding	Exact multiplicative complexity of direct sums of monomial Boolean functions`MC(x1x2x3 ⊕ y1y2y3y4) = 5, MC(x1x2x3 ⊕ y1y2y3) = 4, and additivity holds across all tested direct-sum cells` 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.	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-09-04
MF-165 finding	Equality of multiplicative and quadratic complexity for 3-spaces on at most 5 variables`MC(W) = qMC(W) for all 3-dimensional spaces W <= Quad(n) with n <= 5 in the general XAG model.` Prior art: The result for dim W <= 2 is known from Mirwald–Schnorr (1992), while Boyar–Find record dim >= 3 as open.	The quadratic hull and its defects	PROVED	EXHAUSTIVE CHECK	2026-09-04
MF-166 finding	Exact multiplicative complexity of a Fano 3-space at n = 8 and floor census at n = 4`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.` Prior art: This result is partial, and its novelty remains unverified because simplex-code packing is conceptually a standard technique in coding theory.	Direct sums, wedges and the p14 frontier	CLOSED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-167 finding	Exact quadratic multiplicative complexity of GF(2^4) multiplication and polymul_4`qMC(GF(2^4) mult) = 9, qMC(polymul_4) = 9, unrestricted MC(GF(16) mult) ∈ [8,9], MC(polymul_4) ∈ [8,9]` Prior art: The value 9 is already known from Karatsuba and Winograd's bilinear theory.	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-168 finding	Exact multiplicative complexity of GF(16) inversion and bounds for GF(32)`MC(GF(2^4) inversion) = 5; MC(GF(2^5) inversion) ∈ [7, 14] with qMC(x^3) = qMC(x^5) = 7` 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.	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-169 finding	Exact multiplicative complexity of 7-variable majority`MC(MAJ7) = MC(T^7_4) = 4` Prior art: Boyar–Peralta (2008) established the published bounds `[3,4]`, where Thm 10 gives `<= 4` and Thm 8 / Schnorr degree give `>= 3`.	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-170 finding	Exhaustion of the t = 5 Two-Phase Route for 22-Variable Symmetric Functions`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` 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). Sources: ePrint 2019/708	Adders, counters and the heap law	CLOSED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-171 finding	Exact multiplicative complexity of carry-calculus and threshold functions`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` Prior art: The symmetric cases and maximum MC bound are from BCSTP, correcting an earlier attribution to Boyar–Peralta (2008).	Adders, counters and the heap law	COMPUTED	SOLVER-CONFIRMED	2026-09-04
MF-174 finding	Refutation of five candidate multiplicative complexity lower-bound conjectures`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` 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.	Cipher S-boxes, χ, and quantum gate counts	REFUTED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-175 finding	Degree distribution of the two-column C7 transducer nonlinear quotient`For the two-column C7 transducer with dim V = 4, exactly 12 nonzero cosets in V have degree ≥ 3 and 3 cosets are quadratic.`	Symmetry, state encodings and search gauges	PROVED	RECEIPTED	2026-09-04
MF-176 finding	Exact multiplicative complexity of all 16 optimal 4-bit S-box classes and standard ciphers`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`	Cipher S-boxes, χ, and quantum gate counts	PROVED	RECEIPTED	2026-09-04
MF-178 finding	Domination of tensor slice rank floors for systems of quadratic forms`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.`	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED NEGATIVE RESULT	2026-09-04
MF-179 finding	Exact Multiplicative Complexity of Parallel Counters and Multi-Operand Compressors`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` Prior art: This result establishes closed exact bounds that appear to be new, though the overall problem remains partially solved.	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-180 finding	Multiplicative complexity bounds for 2x2 matrix multiplication over GF(2)`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)` Prior art: This is a partial and apparently new result, establishing the first non-bilinear lower bound >= 6 for unrestricted XAG over F_2.	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED	2026-09-04
MF-181 finding	Exact multiplicative complexity of the resolved-carry family K_m`MC(K_m) = 2m+1 for every m ≥ 1` 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.	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-182 finding	Unification of Additive Line Multiplicative Complexity via Greedy Column-Heap Recursion`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.` 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.	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
MF-183 finding	Discard-tax principle for heap optimality with discarded top digits`MC(S⁻) = MC(S) − (heap ANDs exclusive to the dropped digits)`	Adders, counters and the heap law	CONJECTURE	OPEN QUESTION	2026-09-04
MF-184 finding	Non-existence of 18-vector Kochen-Specker sets in dimension 6`No KS set in C^6 has 18 vectors, hence m_6 >= 19 and m_6 in [19, 21]` Prior art: This result improves the universal lower bound m_d >= 18 from Xu-Chen-Guehne (2020) in dimension 6. Sources: Xu, Chen, Guehne 2020 (PRL 124, 230401) · Lisonek, Badziag, Portillo, Cabello 2014 (PRA 89, 042101)	Exact answers in open problems	PROVED	CERTIFIED PROOF	2026-09-06
MF-185 finding	Two necessary conditions on Kochen-Specker orthogonality graphs`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.` 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. Sources: Xu, Chen, Guehne 2020	Exact answers in open problems	PROVED	EXHAUSTIVE CHECK	2026-09-06
MF-186 finding	Maximum row degree bound for 16×17 Zarankiewicz extremal matrices`If A ∈ {0,1}^{16×17} has 133 ones and no 3×3 all-ones submatrix, then max row degree of A is ≤ 10` Prior art: Two papers from August 2026 left the exact value of z(16,17;3) in {132, 133} unresolved. Sources: Afrasyab 2026 · Hou 2026	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-187 finding	Minimum degree bounds for (J4, J8; N)-graphs at orders 30 and 31`δ(G) ≥ 4 for every (J4,J8;30)-graph and δ(G) ≥ 5 for every (J4,J8;31)-graph` 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). Sources: Radziszowski, Small Ramsey Numbers (dynamic survey)	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-188 finding	Uniqueness of (J4,J7;26)-graphs with minimum degree at least 9`(J4,J7;26)-graphs with δ >= 9 form a single isomorphism class: Schläfli complement minus one vertex (degrees 9^10 10^16, 125 edges)` Prior art: MR91 proved uniqueness at order 27, but whether the 26-vertex statement appears in that paper remains unverified. Sources: Goedgebeur, Van Overberghe 2021	Exact answers in open problems	EXHAUSTED	RECEIPTED NEGATIVE RESULT	2026-09-06
MF-189 finding	Exact sizes of small single-output median networks`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).` 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. Sources: Dobbelaere, median networks table	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-190 finding	Exact B2 circuit size and machine-checkable certificate for 5-input MOD3,1`size_B2(MOD3,1 on 5 inputs) = 9` Prior art: Knuth (TAOCP 7.1.2, exercise 480) determined the values for n ≤ 5 using SAT solving without publishing certificates. Sources: Knuth, TAOCP vol. 4A, section 7.1.2 · Kulikov, Pechenev, Slezkin 2022 (MFCS) · Kojevnikov, Kulikov, Yaroslavtsev 2009	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-191 finding	Exact constant-weight code bound A(11,4,5) = 66 via verified SAT`A(11,4,5) = 66` Prior art: This machine-checkable certificate builds on Brouwer's table (Johnson bound from A(10,4,5) = 36, Ostergard 2010). Sources: Ostergard 2010, classification of binary constant weight codes	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-192 finding	Verified structural obstructions for universal tournaments`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)` Prior art: Zhang and Szeider (CP 2023) stated a lower bound of 11 and resolved the case n = 11 using four separate SAT instances. Sources: Zhang, Szeider 2023 (CP 2023)	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-193 finding	Contraction censuses and lifting obstructions for CW(112,36)`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).` Prior art: The orbit count 2 for m = 7 was previously reported in Arasu-Gordon-Zhang (2021, Table 9) under a multiplier assumption. Sources: Arasu, Gordon, Zhang 2021 (Cryptogr. Commun.) · Tan 2026 · Gordon, circulant weighing matrices table	Exact answers in open problems	EXHAUSTED	RECEIPTED NEGATIVE RESULT	2026-09-06
MF-194 finding	Vertex-deletion averaging ladder for the Turán (3,4)-problem`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` Prior art: This result applies the standard Katona-Nemetz-Simonovits monotonicity argument to exact values. Sources: Katona, Nemetz, Simonovits 1964 · Turan 1941	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-195 finding	Nonexistence of 19-vector Kochen-Specker sets in C^6 and the lower bound m_6 ≥ 20`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)` Prior art: Searches across existing literature found no prior work excluding 19 in d = 6, though the novelty of this result remains unverified. Sources: Xu, Chen, Guehne 2020 · Lisonek, Badziag, Portillo, Cabello 2014	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-196 finding	Residue-class parity theorem for circulant weighing matrices`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` Prior art: Prior work by Arasu, Leung, Ma, and Schmidt contains parity and 2-adic results for CW(2^e q, k). Sources: Arasu, Gordon, Zhang 2021 · Gordon, circulant weighing matrices table	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-197 finding	Machine-checked lower bound t(7) > 12 via fourteen amalgam cubes`t(7) > 12; no 12-vertex tournament is 7-universal (all 14 amalgam cubes UNSAT, drat-trim VERIFIED)` 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. Sources: Zhang, Szeider 2023 (CP 2023)	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-198 finding	Automorphism group structure of 7-universal tournaments on 13 vertices`For every 7-universal tournament T on 13 vertices, \|Aut(T)\| ∈ {1, 3}` Prior art: The novelty of this result has not been independently verified. Sources: Zhang, Szeider 2023 (CP 2023)	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
MF-199 finding	Nonexistence of 20-vector Kochen-Specker sets in C^6 and minimality of m_6 = 21`No Kochen-Specker set in C^6 has 20 vectors; hence m_6 = 21 exactly, conditional on the Xu-Chen-Gühne lemma.` 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]. Sources: Xu, Chen, Guehne 2020 (PRL 124, 230401) · Lisonek, Badziag, Portillo, Cabello 2014 (PRA 89, 042101)	Exact answers in open problems	PROVED	RECEIPTED	2026-09-06
ML-007 limit	Multiplicative complexity of Pascal and averaging-algebra state-feature classes`Exhaustive search across 2,097,152 Pascal / averaging-algebra state-feature classes yields 7 survivors at mixed-tensor-rank ≤2.`	Direct sums, wedges and the p14 frontier	EXHAUSTED	RECEIPTED NEGATIVE RESULT	2026-08-29
ML-010 limit	Affine rank of product-output functions and fixed-XOR optimization in the leader`OPT_fixed-XOR = 274`	The SHA-256 record and exact synthesis	EXHAUSTED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-016 limit	Wrong Fixed Points in Rank 5/5 Deterministic Quadratic-Atlas Charts`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`	The quadratic hull and its defects	EXHAUSTED	RECEIPTED NEGATIVE RESULT	2026-08-29
ML-017 limit	Universal wrong row-0 assignments in Toffoli-lift cyclic p6 modules`All 15 qdim-6 modules in the Toffoli-lift cyclic p6 family attain full rank and admit a universal wrong row-0 assignment`	Cipher S-boxes, χ, and quantum gate counts	EXHAUSTED	LEGACY: NO RECEIPT NEGATIVE RESULT	2026-08-29
ML-018 limit	Exhaustion of the 3-gate full-domain start on the Toffoli acyclic p7 route`Real 3-gate full-domain start on Toffoli acyclic p7 route: EXHAUSTED after 786,430 factor checks; alternate prefixes: UNKNOWN`	Cipher S-boxes, χ, and quantum gate counts	EXHAUSTED	LEGACY: NO RECEIPT NEGATIVE RESULT	2026-08-29
ML-020 limit	An impossibility result for two-witness AND4 in the affine-C model`In the affine-C model, the two-witness AND4 route is unsatisfiable for every finite row count r.`	What rank-one constraints can express	PROVED	EXHAUSTIVE CHECK	2026-08-29
ML-022 limit	Unsoundness of the fixed Ghost-P5 ITER19 allocation and K0 trellis adapter`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`	The quadratic hull and its defects	LIVE-PARKED	OPEN QUESTION	2026-08-29
ML-023 limit	On the natural stationary p7 route targeting 21,738`p7 target 21,738: 2,018 dead_tail / 181,441 blockers across 768 rows, 0 unresolved tails, global closure = UNKNOWN, status = LIVE-PARKED`	The SHA-256 record and exact synthesis	LIVE-PARKED	OPEN QUESTION	2026-08-29
ML-024 limit	On the quadratic hull of carry relations under state-only catalyst lifts`Deterministic state-only lifts fail at every row budget: 8,040 acyclic-p6 models eliminated, 8,184 anchors yield no hull repairs`	The quadratic hull and its defects	PROVED DEAD	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-025 limit	The p8 top-form obstruction for the exact three-column boundary tile`Exact functional four-word tile T with output degrees [1,2,4,6,9] has no p8 circuit`	The SHA-256 record and exact synthesis	CLOSED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-028 limit	Quadratic hull closure for the canonical Z-difference counter interface`For the canonical Z-difference counter interface A*B=C, retaining garbage coordinate g0 or g1 closes the quadratic hull.`	The quadratic hull and its defects	CLOSED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-029 limit	Exact-UNSAT of Phase B in the natural P10 missing-plane model`Phase B is exact-UNSAT in the natural 30-coordinate P10 model; product image rank 423, quotient rank 393, rejection family empty`	What rank-one constraints can express	PROVED	SOLVER-CONFIRMED NEGATIVE RESULT	2026-08-29
ML-033 limit	Finite-horizon Nerode minimisation of the 22-state carry transducer`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.`	Symmetry, state encodings and search gauges	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-034 limit	Elimination of stationary additive carry/raw phases with terminal syndrome-only checks`491,040 candidates across 245,520 classes yield rank 74 with 40 independent directions; span indicator(c=s)*{1,u0,...,u9} is 88-dimensional`	The quadratic hull and its defects	PROVED DEAD	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-035 limit	Fixed-interface factor-component contraction on the 22,215-row leader`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))`	The SHA-256 record and exact synthesis	PROVED	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-040 limit	Piecewise cubic separation of JSC-12 cases in the flip-support subspace`Piecewise pair (k0=0 principal, k0=1 companion) separates K10/K11; no square-free cubic meeting {middle1, middle2, final0} rejects witness`	The SHA-256 record and exact synthesis	PROVED	RECEIPTED	2026-08-29
ML-041 limit	An impossibility result for affine third factors of the named JSC-12 witness`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`	What rank-one constraints can express	PROVED	RECEIPTED	2026-08-29
ML-042 limit	A polar rank 8 separator in the JSC-12 K1x even dual coset`JSC-12 K1x even dual coset separator middle2 * Q has polar rank 8, Hamming weight 37, satisfying K10=K11=0 and FALSE_CUBIC=1`	The SHA-256 record and exact synthesis	EXHAUSTED	RECEIPTED NEGATIVE RESULT	2026-08-29
ML-043 limit	Closure of all 68 p7-eligible directed edges under the rank-tight affine-code model`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`	Direct sums, wedges and the p14 frontier	PROVED	EXHAUSTIVE CHECK	2026-08-29
ML-049 limit	An eight-product acyclic XAG for the low four bits of x₀+x₁+x₂+x₃`An acyclic XAG computes the low four bits of x₀+x₁+x₂+x₃ in 8 products (target 4op_w4_p8, p = 8 is SAT)`	Adders, counters and the heap law	NOT DECISIVE	EXHAUSTIVE CHECK	2026-08-29
ML-050 limit	Impossibility of helper-free IsZero over finite fields with \|F\| ≥ 4`For \|F\| ≥ 4, no family of degree-at-most-two equations in (x,z) has solution relation Γ₀ = {(0,1)} ∪ {(x,0) : x ∈ F*}`	What rank-one constraints can express	PROVED DEAD	PAPER PROOF NEGATIVE RESULT	2026-08-29
ML-051 limit	On the functional direct-sum additivity of M₂ over GF(2)`Functional direct-sum additivity for M₂ over GF(2) in unrestricted sequential Boolean XAGs is OPEN` Prior art: Alder-Strassen and Strassen establish 7s only within the bilinear and formal-quadratic models.	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-08-29
ML-052 limit	On the static quadratic hull as a general SAT preprocessor`If \|F\| < 2^(n-d), then I_≤d(R) = {0} and H_d(R) = F₂^n` Prior art: The Reed-Muller minimum-distance property used here is a standard, classical mathematical result, and no novelty is claimed for this component.	The quadratic hull and its defects	PROVED DEAD	EXHAUSTIVE CHECK NEGATIVE RESULT	2026-08-29
ML-053 limit	Empirical limits of single-counterexample CEGIS exact synthesis`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`	The SHA-256 record and exact synthesis	EMPIRICAL-WALL	RECEIPTED	2026-08-29
ML-054 limit	Multiplicative complexity bounds for the injected-carry family J_m`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]`	Adders, counters and the heap law	OPEN	OPEN QUESTION	2026-09-04
ML-055 limit	Exact gate-state duality and 1-Lipschitz potential bounds for multiplicative complexity`Gate-state shortest path equals MC with LP dual as 1-Lipschitz potential; quotient certificates require complete edge sets.`	Symmetry, state encodings and search gauges	PROVED	CERTIFIED PROOF	2026-09-04
ML-056 limit	Restriction conservation law for multiplicative complexity under exact restriction`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)`	Adders, counters and the heap law	PROVED	CERTIFIED PROOF	2026-09-04
ML-057 limit	Kummer endpoint and boundary-alias formulas`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`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
ML-058 limit	Exact costs of strict syntactic equivariance for widths 3, 5, and 7`E_3 = 3, E_5 = 10, E_7 = 21 with exact taxes 0, 4, 12 over unrestricted inverse-χ multiplicative complexity`	Cipher S-boxes, χ, and quantum gate counts	PROVED	CERTIFIED PROOF	2026-09-04
ML-059 limit	Bounds on strict width-nine complexity E_9`18 ≤ E_9 ≤ 36`	Adders, counters and the heap law	OPEN	OPEN QUESTION	2026-09-04
ML-060 limit	Symmetry and exact small-width complexity of constant-addition shears`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`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
ML-063 limit	SHA-256 compilation record and status of conditional sub-22,215 alternatives`SHA-256 record = 22215 rows; hypothetical alternatives at 20612, 20531, and 22185 remain conditional without full witnesses`	The SHA-256 record and exact synthesis	PROVED	RECEIPTED	2026-09-04
ML-064 limit	Infeasibility of K12 score reduction via banked rowwise chi5 replacements`Rowwise chi5 in Keccak-p[1600,12] is optimal at 5 products per row (19,200 total); no banked rowwise replacement reduces the score.`	Cipher S-boxes, χ, and quantum gate counts	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-065 limit	BLAKE3 and ARX redeployment limits from banked addition assets`Direct redeployment of banked addition assets yields ≥ 10304 products on BLAKE3, exceeding the 10298-product conditional leader`	Adders, counters and the heap law	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-066 limit	Failure of Keccak-family score reduction via inverse-χ exactness`Exact inverse-χ_5 and Ascon inverse circuits yield zero solver score reductions over forward-χ (MC(χ_n) = n).`	Cipher S-boxes, χ, and quantum gate counts	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-067 limit	Limits of SHA-256 row reduction from banked component certificates`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`	The SHA-256 record and exact synthesis	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-068 limit	Impossibility of Constant Absorption by Memoryless Scalar Carry Codes`No block with boundary state encoding scalar carry 0..4 beats 4 ANDs/col once carry 4 is reachable.`	Adders, counters and the heap law	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-069 limit	Exact additivity of two-copy block sharing in leader pairs`Consecutive-round Ch and Maj pairs and schedule adder pairs are exactly additive at width 32; block sharing across real pairs yields 0 savings.`	The SHA-256 record and exact synthesis	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-071 limit	Refutation of the Maj Edge Phase -1,024 Candidate`Maj edge phase -1,024 candidate phase defect grows linearly at 32k bits for k=1..4, refuting the candidate with zero solver time.`	The SHA-256 record and exact synthesis	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-072 limit	Universal screen refutation of the JSC one-catalyst class`For JSC-11, degree-3 evaluation hull accepts a non-honest point, killing all acyclic one-catalyst lifts k ≤ 1.`	The SHA-256 record and exact synthesis	PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-073 limit	Closure of redundant-carry seam route for natural prefixes`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`	The SHA-256 record and exact synthesis	CLOSED	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-074 limit	Method limit of cube-and-conquer without algebraic reduction`march_cu on c=3/p=8 yielded 4096 cubes timing out at 600 s; algebraic reduction enables 0.4-6 s decisions.`	The SHA-256 record and exact synthesis	METHOD	RECEIPTED	2026-09-04
ML-075 limit	The Mirwald–Schnorr 3-form transfer gap and n = 6 enumeration barrier`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`	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-09-04
ML-076 limit	Multiplicative complexity bracket for GF(2^4) multiplication`MC(GF(2^4) multiplication) ∈ [8, 9] in the unrestricted model`	Cipher S-boxes, χ, and quantum gate counts	OPEN	OPEN QUESTION	2026-09-04
ML-077 limit	Exact Multiplicative Complexity of x1x2x3 ⊕ y1y2y3y4`MC(x1x2x3 ⊕ y1y2y3y4) = 5`	Direct sums, wedges and the p14 frontier	CLOSED	CERTIFIED PROOF NEGATIVE RESULT	2026-09-04
ML-079 limit	Open Status of the NIST 21-vs-22 Question for Degree-22 Symmetric Functions`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.`	Symmetry, state encodings and search gauges	OPEN	OPEN QUESTION	2026-09-04
ML-082 limit	Multiplicative complexity bounds for chi_6^2 and chi_7^2`chi_6^2 ∈ [8, 12], chi_7^2 ∈ [9, 14]`	Cipher S-boxes, χ, and quantum gate counts	OPEN	OPEN QUESTION	2026-09-04
ML-086 limit	Ineffectiveness of 3-tensor slice rank for multiplicative complexity bounds`srank(T) ≤ m implies MC ≥ ⌈srank(T)/3⌉ ≤ ⌈m/3⌉ < m, which fails to improve upon the trivial linear dimension floor m` Prior art: This result builds on tensor slice rank and partition rank techniques introduced by Tao (2016) and Naslund (2020).	Direct sums, wedges and the p14 frontier	PROVED	RECEIPTED NEGATIVE RESULT	2026-09-04
ML-087 limit	Multiplicative complexity of cascaded k:2 carry-save compressor slices`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)`	Adders, counters and the heap law	PROVED	RECEIPTED	2026-09-04
ML-088 limit	Search scale barrier for unrestricted 6-AND XAG decision of M_2(F_2)`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.`	Direct sums, wedges and the p14 frontier	OPEN	OPEN QUESTION	2026-09-04
ML-089 limit	Resolution tax τ_4 in its smallest open form`τ_4 = MC(K_4) − MC(J_5) ∈ {1,2}; MC(K_4) = 9, MC(J_5) ∈ [7,8]`	Adders, counters and the heap law	OPEN	OPEN QUESTION	2026-09-04
ML-090 limit	Search wall for the smallest 7-universal tournament at order 13`13 ≤ t(7) ≤ 15; n = 13 undecided under SAT and lazy pattern-core CEGAR which exceeds 1800 s timeout at 32 enforced patterns` Prior art: The bounds 13 <= t(7) <= 15 were established by Zhang and Szeider (CP 2023). Sources: Zhang, Szeider 2023 (CP 2023)	Exact answers in open problems	OPEN	OPEN QUESTION	2026-09-06
ML-091 limit	Partial degree-cube search wall for Zarankiewicz number z(16,17;3)`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` Sources: Afrasyab 2026 · Hou 2026	Exact answers in open problems	LIVE-PARKED	OPEN QUESTION	2026-09-06
ML-092 limit	Solver wall in SAT-based exact median network synthesis`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` Prior art: A verification check was unable to reproduce the published n = 9 optimality claim from Smith (1996). Sources: Dobbelaere, median networks table	Exact answers in open problems	OPEN	OPEN QUESTION	2026-09-06
ML-093 limit	Proof-system mismatch for the circulant weighing matrix cell CW(112,36)`DRAT resolution and RoundingSat cutting planes fail on CW(112,36) and known-nonexistent CW(n,36) for n ∈ {40, 44, 50, 56}.` Sources: Arasu, Gordon, Zhang 2021 · Tan 2026	Exact answers in open problems	OPEN / PROVED DEAD	RECEIPTED NEGATIVE RESULT	2026-09-06
ML-094 limit	Exact B2 circuit size of MOD3 on 6 inputs and solver scaling wall`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` Prior art: Knuth's conjecture predicts a circuit size of 12 for MOD3,0 on 6 inputs. Sources: Kulikov, Pechenev, Slezkin 2022 (MFCS)	Exact answers in open problems	OPEN	OPEN QUESTION	2026-09-06
ML-095 limit	Open status and catalogue corrections for resolution hardness h_11`h_11 ≥ 28 remains undetermined; reproduced h_8 = 19, h_9 = 22; candidate census corrected to 626,973 with missing RSMU(8,10) identified` 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. Sources: Peitl, Szeider 2021 (JAIR) · Peitl, Szeider, short-proof code	Exact answers in open problems	OPEN	OPEN QUESTION	2026-09-06
ML-096 limit	Bounds and search limits on the constant-weight code size A(13,4,5)`A(13,4,5) ∈ [123, 129]` Sources: Brouwer, bounds for constant-weight codes	Exact answers in open problems	OPEN	OPEN QUESTION	2026-09-06
ML-097 limit	Limits of plain CDCL with cardinality totalizers for Turán numbers ex(n, K_4^(3))`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` Sources: Turan 1941	Exact answers in open problems	EMPIRICAL-WALL	RECEIPTED NEGATIVE RESULT	2026-09-06
ML-098 limit	Failure of SMS standalone LRAT certificate verification for Kochen-Specker n = 18`Standalone lrat-check of smsg -v 18 --lrat-output fails due to unintegrated --sym-break-clauses outside the LRAT chain.` Sources: SAT Modulo Symmetries (Kirchweger, Szeider)	Exact answers in open problems	PROVED DEAD	CERTIFIED PROOF NEGATIVE RESULT	2026-09-06
ML-099 limit	Exclusion of 19- and 20-vector Kochen-Specker sets in C^6`No 19- or 20-vector Kochen-Specker set exists in C^6 (both excluded via MF-195, MF-199)` Sources: Xu, Chen, Guehne 2020 · Lisonek, Badziag, Portillo, Cabello 2014	Exact answers in open problems	CLOSED	RECEIPTED NEGATIVE RESULT	2026-09-06

Snapshot 2026-09-06. Generated from the division's registers; never hand-edited.