[apoursh]

This resolves the supplied prime-field multiplication-table part of Ali Romman's recognition handoff in #5459. Given a common-basis table over F_p, the new package recognizes exactly the algebras F_p[x_i]/(x_i^(p^h_i)), recovers the height multiset, and returns a verified monomial change of basis and inverse. No new κ is claimed.

The criterion uses actual Frobenius kernels before projection to J/J²: with E minimal such that p^E ≥ n, take J = ker(F^E), V_e = (ker(F^e) + J²)/J² and d_e = dim V_e − dim V_(e−1). Acceptance is equivalent to codim J = 1 and p^(Σ e d_e) = n. The proof supplies necessity, sufficiency, constructive lifts and polynomial complexity for fixed p; the implementation recovers the unit and checks the algebra laws.

Verification

At head c09b7a57f81f941b7f9ee6882172eb21a683ded1, CPython 3.12.14:

• All four registered isolated normal/optimized commands pass, with identical paired JSON reports.
• 46 positive presentations through dimension 16 over primes 2, 3, 5 and 7; 3,217 independent basis-product checks.
• Nine nonmember controls, six malformed inputs, two coefficient-normalization controls and seven rejected witness mutations.
• Independent bitset/finite-span oracle: all 525 commutative fixed-unit tables in the listed tiny domains, plus 392 distinct four-dimensional F2 positive presentations from complete unit-fixing basis orbits.
• Committed-byte/source-history and export-integrity checks pass. All four direct commands reproduce the same reports from a read-only, Git-free export, with unchanged source bytes.

python3.12 -B scripts/research_package.py --package research/frobenius-algebra-recognition
python3.12 -B scripts/check_research_commit.py --package research/frobenius-algebra-recognition --check-source-history

Scope and collaboration

Eight new files in one isolated opt-in package; shared code, frozen predecessors and the selected result are unchanged. Base 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority 7. Task formal-interface-lemma, scope changed: this takes the explicit #5459 handoff as a written algebra-recognition theorem and implementation rather than a proof-assistant build. It removes the algebra-class promise for supplied tables, without claiming the original card’s formalization acceptance gate. The full original card, source pins and overlap coverage are recorded in TASK.json.

Canonical claim and completion: Discussion #5493. The existing handoff reply links the completed artifact. This is distinct from the separately claimed finite-etale recognition work.

Arbitrary three-factor tensor isotopy recognition, extension fields, physical suppliers and a new multiplication exponent remain open. Primality uses trial division; only fixed-prime polynomial complexity is claimed for the full implementation. Local checks are finite corroboration, not hosted CI, independent human acceptance or proof-kernel verification. No selected cold replay or proof-assistant build was needed or run.

Prepared by apoursh, with substantial OpenAI Codex assistance. Credit Ali Romman (#5459 at bef3575ddb19c44434a7e2460a353eac7c4d9387, #5316), Nick Petalas (#5252), David Ganz's atlas lineage and classical commutative algebra. No pending executable is imported, and no general literature priority is claimed. Apache-2.0.

Open next questions

The package’s next questions identify two follow-ups: recover a faithful regular nucleus action and compatible factor maps from an arbitrary tensor, and extend the calculation to supplied non-prime finite fields. The first would remove the remaining isotopy promise; the second would match the broader predecessor family. Neither tensor reconstruction nor the required semilinear field layer is supplied here.