Finite Nonlinear Separations for Exact Four-Cycle Graph Codes

Abstract

This unrefereed preprint studies codes of labelled simple graphs in which the symmetric difference of two distinct codewords is forbidden to be exactly the edge set of one four-cycle. The condition concerns an exact four-cycle difference, rather than every difference containing a four-cycle. For every n >= 4, the largest linear code has cardinality 2^(binom(n,2) - ceil(log2(n))). The exact unrestricted optimum is 192 at n = 5, compared with the linear optimum 128, and 6144 at n = 6, compared with the linear optimum 4096. For n = 6, a finite Fourier-positive kernel gives the matching upper bound; a 96-word seed translated over 64 cosets attains it. The accompanying unchanged v1.4 evidence bundle contains the explicit witness, proof memo, standard-library Python verification scripts, saved verification results, and SHA-256 checksums. The pairwise verifier checks 18,871,296 unordered pairs and finds zero prohibited differences. Independent computational cross-checks have passed. These checks do not constitute formal peer review. To the best of the author's knowledge, these particular finite values and the all-n linear formula were not found in the literature examined. This is not a priority or first-discovery claim. Historical novelty remains INCONCLUSIVE. Information about earlier or overlapping work is welcome at the research contact below. The proof memo in the evidence bundle records a pre-audit stage and includes an outdated statement that external recomputation was pending. The authoritative v1.4 manuscript and publication gate record the completed independent computational cross-check. The original audited bundle is preserved unchanged. GitHub repository: https://github.com/yonezaemon1-hub/no26-exact-four-cycle-graph-codes Research Inquiries — Yonezu Research Lab Contact: yonezaemon1+research@gmail.com