[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. For every odd finite field K, split local commutative unital algebras with J^3=0 and one-dimensional socle J^2 have one isomorphism class when r=dim(J/J^2) is odd, and two when r is even. In the even case the determinant square class of the recovered multiplication form distinguishes them. The same signature classifies their multiplication tensors up to isotopy. No new kappa is claimed.

supplied table -> radical and multiplication form -> canonical signature/chart
matching signatures -> explicit algebra isomorphism

This resolves one specified class of the recovered-algebra isomorphism question in Ali Romman's merged #5540. Its reconstruction theorem is cited, not audited. The new result is the specialization and complete constructive interface; finite-field symmetric-form theory is classical. Only research/cube-zero-gorenstein-isomorphism/ is added.

Evidence

Before, #5540 recovers commutative algebra tables and leaves their general isomorphism problem open. After, PROOF.md gives a complete classification for this restricted cube-zero class, and a standard-library implementation constructs exact maps from arbitrary-basis supplied tables.

Normal and optimized local checks report PASS_CUBE_ZERO_GORENSTEIN_ISOMORPHISM:

• 51,200 nonsingular symmetric forms with complete similarity certificates.
• Independent complete GL/similarity orbit generation: all 18/100/294 binary forms over F3/F5/F7, and all 468 ternary forms over F3, using 48/480/2016/11232 invertible matrices respectively. Here binary and ternary describe form dimension, not field characteristic.
• 96 arbitrary-basis algebra presentations and 4,128 transported basis products; 96 unit-image transporter mutations and 12 input/class rejection controls.
• Package metadata/provenance, committed bytes/modes/source-history and official export integrity pass. The optimized direct replay also passes from the read-only Git-free export with PATH=/nonexistent; all eight exported files, including its generated export inventory, remain unchanged.

Every verifier run emits a generated receipt bound to all seven package files by SHA256. No predecessor scientific verifier is imported or replayed.

From the worktree root, CPython 3.12.15 and its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/cube-zero-gorenstein-isomorphism/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/cube-zero-gorenstein-isomorphism/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cube-zero-gorenstein-isomorphism
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cube-zero-gorenstein-isomorphism --check-source-history --check-modes

Other machines may substitute their CPython 3.12 path. Each registered command was timed locally: normal 6.114 seconds and optimized 6.127 seconds. Both full registered commands also pass through the package launcher. Warning-strict execution and AST compilation pass; no external static analyzer was available. No historical suites were run.

Limitations and unrun obligations

The proof covers all positive r and odd finite fields. The executable table API is bounded to odd primes 3 through 31 and dimensions 3 through 10. Characteristic two, nonsplit residue fields, larger socles and arbitrary algebra isomorphism remain outside the result. The predecessor tensor reconstruction is a credited premise rather than new coverage.

The emitted maps have no native coefficient, frame, storage, movement, dirty-restoration, precision or complete recursive tariff. No physical caller, supplier admission, selected-frontier change or complete multiplication theorem follows. Lean, Docker and selected cold replays were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Extend this specific interface to characteristic two, beginning with alternating versus nonalternating multiplication forms. A separate proof and field contract are needed.
• Price one emitted map in an actual native caller, including legal coefficients, dirty inputs, every conversion and final restoration. No such tariff is supplied.

Related open work

The completion refresh read the titles and bodies of all 12 other open account PRs created since the advance CLAIM. None studies this cube-zero Gorenstein isomorphism signature or computes a competing class count. No additional computations were run to check those contributions.

• Prove a fourth dirty role keeps the S3 xyx double at six calls (no new κ) #5947 proves a four-role call minimum over F2[S3]. It concerns a noncommutative register-word endpoint in characteristic two, rather than isomorphism of the split commutative local algebras classified here. No shared numerical result is compared.

No open PR studying this cube-zero Gorenstein isomorphism signature was identified in the live targeted title/body searches. #5540 is the merged predecessor. #5934 concerns Gaussian tensor-annihilator heights, a different tensor and quantity; no shared numerical result is compared and no extra computation was run to check it.

The live overlap refresh included open PRs, recent account contributions, handoff claims, active Discussions, main-tree object/formula searches and merged searches. Cached local coverage is sparse and was supplemented by Git-object searches; it is not a novelty certificate. Advance CLAIM. The delivered scope matches that claim. The formal-interface-lemma card is explicitly changed to this written algebraic side result, without claiming its proof-assistant acceptance gate.

Credit

This contribution was produced by Nick Petalas. Apache-2.0. Credit Ali Romman #5540 at 831b41d for the recovered-table interface and isomorphism handoff, apoursh #5502, Ali Romman #5525, Nick Petalas #5252, David Ganz and the retained tensor/algebra lineage. Symmetric-form reduction, finite local-algebra structure and unital isotopy are classical. No predecessor credit or source file is changed.

CI notes

Both hosted checks passed at exact head f1c2c18b650b8e46590815b09daa6120f50e380d in run 40531240894: plan in 4m38s and replay (research/cube-zero-gorenstein-isomorphism) in 1m47s. The completed check list was read once; the run is completed/success and its head matches the PR head. The registered replay executes both warning-strict normal and optimized commands.

This certifies the bounded registered checks, not independent human or formal proof-kernel acceptance of the all-dimension written proof. All local checks listed above also passed at this commit. No CI failure remains.

Merge danger

Two-way door. Blast radius: isolated. Seven new files in one opt-in research package. Reverting the package removes the new proof, implementation and replay registration; shared code and the selected frontier are unchanged.