MF-199RECEIPTED
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.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.
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. Sources: Xu, Chen, Guehne 2020 (PRL 124, 230401) · Lisonek, Badziag, Portillo, Cabello 2014 (PRA 89, 042101)
Published 2026-09-06
MF-197RECEIPTED
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)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.
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. Sources: Zhang, Szeider 2023 (CP 2023)
Published 2026-09-06
MF-196RECEIPTED
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) = 0Every 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).
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. Sources: Arasu, Gordon, Zhang 2021 · Gordon, circulant weighing matrices table
Published 2026-09-06
MF-182RECEIPTED
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.A greedy column-heap carry-save recursion reproduces all known exact multiplicative-complexity values and boundary constants for multi-operand addition.
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.
Published 2026-09-04
MF-181RECEIPTED
Exact multiplicative complexity of the resolved-carry family K_m
MC(K_m) = 2m+1 for every m ≥ 1The 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.
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.
Published 2026-09-04
MF-176RECEIPTED
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)=4The 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
MF-168RECEIPTED
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) = 7Inversion 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.
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.
Published 2026-09-04
MF-156RECEIPTED
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) + 1The 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.
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.
Published 2026-09-04
MF-154EXHAUSTIVE CHECK
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_4The two-round Keccak nonlinear layer chi_5^2 requires exactly 7 AND gates, resolving the previous bracket of 6 to 7.
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.
Published 2026-09-04