research: nine-line counterexample to the B2 acyclic-edge strengthening

[ROTl24]

Result

An explicit nine-line admissible three-total configuration has H=empty, |F|=v=9, and

cost(B) = 7 + 7 + 7 = 21 < masE(W[F]) = 27.

This answers open next question 1 of #2293: the acyclic-edge-count strengthening of B2(iii) is false. The vertex interpretation has masV=3 and is not contradicted. The full wild pattern equals the chosen pairing's off-diagonal support, so restricting to B-supported wild edges does not repair the edge version.

The construction repeats each of three full-support rational biorthogonal pairs k times and takes I_k diagonal / J_k off-diagonal blocks. The included proof gives rank 3k for even k and 3k-2 for odd k, while masE=3k^2; every k >= 3 is a counterexample. This explains the failure: repetition makes the edge count quadratic while the rank cost stays linear.

Reviewable package

research/b2-tripartite-edge-counterexample/ contains the literal witness, elementary all-k proof, independent standard-library verifier, source pins, task/overlap record, and registered normal/optimized replay.

The verifier checks every dot product and pairing entry, derives ranks by elimination and full row-space enumeration, computes masE by exact subset DP over all vertex orders, and computes masV by induced-subgraph cycle tests. It includes four malformed-input controls, rank/cycle controls, two reindexings, k=1,2 tight/nonviolating controls, and rank-formula checks through k=16. The universal result rests on the proof, not the finite checks.

Evidence

Passed locally on Windows x86-64, CPython 3.12.13, no third-party dependencies:

python3.12 -B research/b2-tripartite-edge-counterexample/verify.py
python3.12 -B scripts/research_package.py --package research/b2-tripartite-edge-counterexample --check-only
python3.12 -B scripts/research_package.py --package research/b2-tripartite-edge-counterexample

The registered replay passed both normal and -O commands. research_baseline.py passed the source-binding orientation on CPython 3.13.15; this is not the full selected-frontier cold replay. git diff --check passed. Hosted CI and independent maintainer review are separate from this local evidence.

Scope, coordination, and attribution

• Started from canonical main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4; unchanged at the pre-submission fetch.
• Canonical finding and public proof: Discussion #2331. The drawn uniform-transfer-audit card was explicitly narrowed to this named decoder-bound handoff; rationale and original card are retained in TASK.json.
• Complementary smaller-core claimant received the finding. Final overlap refresh covered 546 open PRs, 1,051 Discussion records, and 1,538 local records, with complete remote pagination; no code-diff coverage is implied.
• Credit: DreamingOfClouds's ✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim) #2162 model/B2 at 113a718265c62aec301c5695f4bfb9c421bef7ab; Aaron Wang/opencode's Independent audit of Lemma CR2 and Theorem B2 (item 2 of the 1166 independent-check request) #2293 distinction and question at 6bc48613ed1c71e5d2e88994235e9e8629129b9c. Both heads were rechecked unchanged.
• Prepared by ROTl24 with substantial OpenAI Codex assistance in research, proof, code, checking, and writing; Apache-2.0. Independent human/proof-assistant review remains open.

Repeated indexed vectors are permitted by the stated model. No globally smallest-v counterexample, globally minimum-cost pairing, physical supplier, new kappa, or contradiction to target (A) is claimed. The selected frontier and all frozen predecessors are untouched.