DONE: recognize Frobenius-truncated prime-field algebra tables

[apoursh]

DONE — supplied prime-field algebra-table recognition, exact head c09b7a57f81f941b7f9ee6882172eb21a683ded1. The scoped claim is released; PR #5502 is open for review from the personal apoursh fork.

Public package and reproduction instructions. The written iff theorem recognizes F_p[x_i]/(x_i^(p^h_i)) from a common-basis multiplication table, recovers the height multiset and constructs a verified monomial change of basis. It uses actual Frobenius-kernel lifts followed by a dimension equality; Frobenius on the tangent quotient alone is insufficient.

All four registered CPython 3.12.14 normal/optimized checks pass with identical paired reports: 46 positive presentations, 3,217 independent basis-product identities, nine nonmembers, six malformed inputs, two normalization controls and seven witness mutations. A separate bitset/finite-span oracle checks all 525 commutative fixed-unit tables in the listed tiny domains and 392 distinct four-dimensional F2 target presentations from complete unit-fixing basis orbits. Exact committed-byte/source-history and export-integrity checks pass. The read-only Git-free export replays all four commands with the same reports and unchanged source bytes. These are local checks, not hosted CI, independent human acceptance or proof-kernel verification.

This completes only the supplied prime-field table part of Ali Romman’s PR5459 handoff. Arbitrary tensor isotopy recognition, extension fields, physical supplier admission and a new κ remain open. Next: recover a faithful regular nucleus action and prove compatible factor maps before invoking this recognizer. Canonical upstream was refreshed and remains 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority 7.

Credit Ali Romman (PR5459/5316), Nick Petalas (PR5252), David Ganz’s atlas and classical commutative algebra. Prepared by apoursh with substantial OpenAI Codex assistance; Apache-2.0. No general literature-priority claim.

Original advance claim

CLAIM — recognize Frobenius-truncated algebras from supplied prime-field multiplication tables.

Taking the supplied-table part of Ali Romman's PR5459 recognition handoff, pinned at bef3575ddb19c44434a7e2460a353eac7c4d9387. Base 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority 7; selected κ is unchanged.

Question: for a checked commutative associative unital F_p algebra, do the lifted Frobenius-kernel flag in m/m² and a dimension equality characterize F_p[x_i]/(x_i^(p^e_i)), with an explicit monomial change of basis? The proof/implementation will use actual lifts in ker(F^e), not merely the induced map on m/m². Verification will include transformed unequal-height tables, nonmembers, full basis-product identities and an independently enumerated small-table domain.

This is distinct from the separately claimed finite-etale tensor-recognition work. Arbitrary three-factor tensor recognition, extension fields, physical supplier costs and a new κ are outside this bounded scope. Evidence is currently LOCAL-ONLY. Expected artifact: research/frobenius-algebra-recognition, written criterion plus standard-library exact checker. Stop after one proved/counterexample decision and replayable witnesses. Claim expires 2026-10-25T06:30:00Z.

Credit Ali Romman (PR5459/5316), Nick Petalas (PR5252), David Ganz's atlas and classical finite local algebra/Nakayama arguments. Prepared by apoursh with substantial OpenAI Codex assistance; Apache-2.0. Live handoff, relevant all-state PR keywords and Discussion search checked; sparse local and unpublished-work coverage remain limited.