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Exact reachability of quadratic lifts of a split radical pencil - #5320
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Mister-Meeseeks merged 1 commit intoOct 23, 2026
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This scoped proof classifies all 4,096 quadratic lifts of [[x,y],[y,x]] over F2[x,y]/(x,y)^3 in the two-role right-vector alphabet. A second-order residue class gives exact cost two for 2,048 lifts and excludes the other 2,048 at every word length. It gives both a failure of quotient lifting and reachable targets without an exact common base-field zero. The complete kappa calculation is unchanged.
Proof and scope, exact checker, replay declaration, and source provenance are pinned to the reviewed commit. Base: canonical main
ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7.CPython 3.12.14 registered normal and optimized replays passed: all 4,096 lifts, 2,048 literal constructions, 1,048,576 covariance checks and all 1,024 nonunit determinant pairs. Both print
PASS_CUBE_ZERO_LIFT_OBSTRUCTION; the launcher printsPASS_REGISTERED_RESEARCH_PACKAGE. All five package files match their committed bytes, and the base is an ancestor of the package commit.Use Python 3.12 and the standard library. No external runtime inputs or third-party Python packages are required. Hosted CI is pending at the final check (2026-10-23T17:19:49.153969+00:00): run 39993842015. No predecessor scientific suite, selected cold replay, native replay, compiler, Lean or Docker build was run. The optional full-history preflight is unavailable in this shallow checkout. Finite checks corroborate the written proof; they do not constitute independent human or proof-kernel acceptance.
Classify another first-order pencil or the determinant-zero cube-zero stratum before proposing a general reachability theorem. Source preparation, native operations, precision, movement, and supplier/outer admission remain separate obligations.
Credit Nick Petalas's PR #4764 and pinned cube-zero proof for the higher-nilpotency question and separate cost-three example. His PR #4276 and pinned exact two-factor criterion supply the finite criterion used for corroboration. His PR #4776 and pinned pencil proof supply the square-zero split-pencil result and determinant lineage. The square-zero theorem, PR #4745, applies only under its original algebra hypothesis. Wentao Li's PR #4099 supplies the right-vector alphabet, and ToikY's PR #4334 supplies the common-chart context. Classical determinant and local-ring arguments retain their credit.
Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.