[krzysztofwojt]

PR #5182 excluded exactly one six-cycle at the fifteen-owner equality boundary but left counts 2–5 open. This package resolves that remaining branch: the possible six-cycle counts are exactly 0 and 4, and every graph in the nonzero branch is isomorphic, preserving the two parts, to the explicit witness.

The precise hypotheses are a simple cubic bipartite graph on 15+15 vertices, no four-cycle, and at most one simple six-cycle through each vertex. The witness is connected, has 45 edges and four vertex-disjoint six-cycles. It disproves the proposed implication from these abstract equality hypotheses to girth eight.

The independently reviewable addition is:

• A written reduction around one distinguished six-cycle to three small matchings, their attachments, and a Boolean 3-by-3 residual block.
• An exact enumeration of all 4,309 reduced cubic completions: 16 survive, and 16 explicit vertex permutations identify the complete survivor set with the witness.
• A separate unreduced enumeration of 134,469 cubic completions using walk counts, with literal cycle checks of all 432 survivors; a third implementation checks the witness, the inherited zero-cycle example and the signed defect by all-root DFS.

Universal coverage rests on the written reduction, not solely on program agreement. The witness remains directly checkable independently of the classification proof. The zero-cycle example is inherited; uniqueness of that branch is not claimed here.

Coordination and exact sources. This takes the explicitly released PR5182 handoff, under the advance canonical claim. Scientific base: ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c; publication head: 339bfb5cc26185104984afd4647e5c1edeb11d8f. The pre-submission canonical-main checkpoint is ff3cc58e370d0f678e529246f8c73e8094e18fcf (priority revision 7). Guidance, task catalogue and selected record are unchanged; #5182 was merged at its cited head, and there is no package-path collision. The original scientific source pins are retained.

The final paginated overlap refresh covered 305 open PRs at 13:18:24 UTC and 4,470 Discussion records at 13:19:35 UTC on 2026-10-23, plus all 6,998 in-scope local HEAD notes. Fresh all-state topic searches and recent account-claim checks found no competing classification. #3147 remains adjacent work; the account's active #5230 semilinear-tariff scope is unrelated and excluded. Arbitrary historical diffs and unpublished work remain coverage limits.

Validation — CPython 3.12.12, standard library only. All six registered commands pass: each of the three implementations in ordinary and -O mode with -W error -B. The exact committed-byte/source-history check and official export integrity checks pass. All six commands also pass from the committed export outside Git, with PATH=/nonexistent and read-only directories; normal/optimized outputs match byte for byte, and all exported hashes remain unchanged. Ten isolated semantic mutations are rejected, including omitted negative cases, a weakened C4 predicate or cycle cap, a missing residual block or isomorphism, a one-root DFS, and omitted witness cycles in each implementation.

python3.12 -W error -B scripts/research_package.py --package research/centre-equality-six-cycles
python3.12 -W error -B scripts/check_research_commit.py --package research/centre-equality-six-cycles --check-source-history
python3.12 -W error -B scripts/export_research_package.py export --package research/centre-equality-six-cycles --ref HEAD --output /tmp/centre-equality.tar
python3.12 -W error -B scripts/export_research_package.py check /tmp/centre-equality.tar

Limits and credit. No actual route labeling, scalar coefficient solution, physical cost/restoration budget, supplier adoption, improved fifteen-owner floor or new κ is claimed. No selected-frontier file or frozen predecessor is changed. The next unreserved question is whether this unique nonzero-cycle branch admits the actual routes and coefficients. No literature priority for the abstract graph, independent human review or proof-kernel formalization is claimed.

Credit Sagar Thakrar #3291 (5d027c3c8724aec4501e0e2876388bebefa00ef7) for the sharp floor and duad example, and Nick Petalas #5182 (a4a8956cb9c769cddd593d00b4c294d307b5fac5) for the equality reduction, disjointness and signed defect. README/TASK retain the earlier centre-route and selected-source lineage, including #3147; this is complementary to its broader route-incidence work. Prepared for Krzysztof Wójt with substantial OpenAI Codex assistance, including separate Luna supporting implementations. Apache-2.0.