Exact cocycle and twisted-boundary verification for Family 199

A verified finite algebraic interface and an explicit homotopy in OpenAI mathematics Family 199, the proposed Auslander-Reiten counterexample. The original 179-entry cochain certificate is preserved unchanged: two independent polynomial implementations verify all 104,976 radical quadruples, 324 symbolic twisted-boundary pairs, 8,000 associativity triples and 400 trace pairs. The displayed closed bar cycle evaluates to q^3 f. Version 1.1.0 adds a uniform choice for the homotopy G in source Lemma lift:G: remove the final bar letter, place its z_H value in the terminal coefficient, and apply h_H inverse to the underlying old terminal coefficient. A written all-degree cancellation proves pB=dG+Gd from the previously certified finite boundary identity. It checks vertex balancing, twisted module linearity, the suffix comparison convention, and the unaugmented degrees zero and one. This makes a recursive projective-lift choice directly computable; no defect in the original existence proof is alleged. A new sparse-tensor checker separately expands both sides on all 17,050 composable generators in degrees zero through four, over F_2[q,H,H^-1]. Fifteen tests include 7,600 G and 7,600 B endpoint-action equalities, boundary cases, differential-squared checks and a wrong-prefix comparison. Four corrupted formulas reject in ordinary and optimized Python. The checker explicitly reuses the predecessor arithmetic and algebra data; it is not an independent algebra implementation. A separately tasked AI reviewer checked the new proof and implementation and found no blocker. The characteristic is two and H is invertible. The specified suffix comparison and certified finite boundary identity are inherited. Nonzero q is unnecessary for the homotopy identity but necessary for the prior nonvanishing claim. Finite replay supports the written proof; neither is kernel verification or human peer review. The remaining cone components, stable-category passage, explicit resolution, all-degree Ext vanishing, Tachikawa analogue and full counterexample remain outside scope. No new counterexample, proof-gap repair or global novelty is claimed. The self-contained archive preserves public source cbc0195c69c1d70a6c2161920c00285fb2b22b89, including the unchanged predecessor contribution 9ce7229089c23e26630850b3c7165a77e9297b85. OpenAI supplied the construction, cochain, boundary equation and recursive lift. Pico, operated by Håkon Åmdal, supplied the earlier algebra verification. Alexander Richard Towell contributed with substantial OpenAI Codex assistance in derivation, implementation, testing, review coordination and publication. Original public disclosure: PR23 at 09:45:10 UTC on 23 October 2026; the homotopy supplement was publicly verified later that day. Upstream acceptance remains pending, separately from this archival publication. A DOI is a persistent citation and archive, not a correctness, novelty or legal-priority certificate.