Prove bipartite commutator incidence tariff

[1337isnot1337]

Result

Proves the exact rank-one paid-call tariff for a bipartite family of central commutators in the class-two group H(m,n). For any finite bipartite edge set and edge multiplicities p_ij, the target with p_ij distinct dirty roles carrying z_ij = x_i y_j x_i y_j has minimum exactly 4 * sum(p_ij) in the full formal F2[H(m,n)] model, for every finite dirty-register count. The edgewise four-call construction attains the bound. The first cyclic case is K2,2: four distinct central outputs, exact cost 16.

The lower bound uses one UT3(F2) quotient and its contragredient per active edge. Each label's column and row frames must span the disjoint sum of its incident target spaces; rank-nullity then sums to 4 * sum(p_ij). This extends the shared-middle path tariff to arbitrary bipartite incidence patterns, including a cycle. It does not claim a supplier implementation, selected-moment improvement, or new kappa.

Evidence and replay

The stdlib-only exact verifier checks the 256-element H(2,2) group law, every quotient and dual homomorphism on all group pairs, the 16-call K2,2 word in the full formal algebra, a weighted six-target word, all 16 one-call omissions, a wrong-label control, and dirty spectators. The proof, package metadata/source pins, direct replay, and whitespace check pass locally under Python 3.14.7. Python 3.12 is unavailable locally; the registered Python 3.12 replay is left to CI. No hosted CI result is claimed.

Reproduce with:

python3 scripts/research_package.py --package research/heisenberg-commutator-incidence-seedb054 --check-only
python3 -I -B research/heisenberg-commutator-incidence-seedb054/verify.py

Pins and attribution

• Base/main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2
• Task seed: b054c63f8969592c7c1608ddeee294e9; bit-mechanisms, priority revision 4
• Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32
• Claim and correction: Larger κ: proposals that change the current cost structure #1167 (comment) and #18938817
• Predecessors: merged UT4 shared-middle tariff; open PR Research: prove the mixed UT4 z/uz endpoint needs fourteen calls #2175 (UT4 z/uz), Research: no 15-call two-target UT4 word on two roles; with #1680, exactly 16 calls for every r (no new kappa) #1816 (two central UT4 targets), and Research: exact UT5(Z) four-role minimum is five #2103 (UT5 four-role mixed-root target). These have different group/target incidence.

The pinned baseline returns PASS_RESEARCH_BASELINE_BINDINGS_ONLY and identifies the complex supplier as binding; this package does not affect that supplier or the selected bound. Prepared with substantial OpenAI Codex GPT-6 assistance in scope selection, derivation, implementation, verification and writing. No independent human review or proof-kernel verification is claimed. Apache-2.0.