Exact Optimality of a Six-Cycle-Difference Graph Code on Six Vertices: A Computer-Assisted Integer Certificate

We determine DC6(6) = 4096 for families of graphs on six labelled vertices such that the symmetric difference of any two distinct members is not *exactly* the edge set of a Hamiltonian six-cycle. This excludes exact C6 differences, not arbitrary differences containing a six-cycle. The 60 forbidden differences generate a nine-dimensional subspace; the full 15-bit problem decomposes into 64 isomorphic 512-vertex Cayley components. A 64-word linear independent-set construction and a 65-exclusion certificate using 90 integer Rayleigh inequalities, eleven symmetrized three-point moment equalities, and exact integer Farkas arithmetic establish the optimum. The machine-reconstructed positive contradiction gap is 65245054658776971334908126197187583410176. Two independent implementations and a direct-coset control check support the computation. Audit: PROOF PASS (conditional, independent AI-assisted review); COMPUTATION PASS. This is an unrefereed preprint, not human peer review or a formal proof-assistant certification. The graph-code framework is due to earlier literature; the contribution is only the finite exact result and its certificate. Historical scholarly novelty remains INCONCLUSIVE, and no priority or first-discovery claim is made. Complete code, integer witness, prior bundle, audit logs, SHA256 manifest, and exact offline reproduction instructions accompany this record. Research Inquiries — Yonezu Research LabContact: yonezaemon1+research@gmail.com Associated GitHub repository: https://github.com/yonezaemon1-hub/no32-exact-six-cycle-graph-codes