[petalas]

Summary

Scoped written algebraic proof with exact finite corroboration, produced by Nick Petalas. For every finite field K=F_q and split local algebra A=K plus J with J squared zero and dim J>=2, put n=dim A and g=gcd(n,q-1). Its multiplication tensor has exact ordered covariant stabilizer determinant image

Delta = {(u,v,w) : uw and vw are nth powers in K*}
|Delta| = (q-1)^3/g^2
number of SL_n(K)^3 orbits inside its GL_n(K)^3 orbit = g^2

Every allowed triple has an explicit attaining stabilizer. At q=5,n=4 the image is {(c,c,c^-1)} and the orbit count is 16. The proof covers every field size and dimension in the stated class, including characteristic two. No new kappa.

Ali Romman's #5540 supplies the recovered-algebra interface and leaves its determinant-character bridge open. That reconstruction theorem is cited, not rechecked. This contribution treats the non-Gorenstein square-zero class, outside the etale and truncated-polynomial classes. Unital isotopy, finite-field cyclicity and the group-action argument are classical. Credit Ali Romman and the retained apoursh, Nick Petalas, David Ganz and tensor/algebra lineage. Global literature priority is unclaimed.

Evidence

Before, the reconstructed general algebra did not have a supplied character answer for this class. After, PROOF.md gives two exact equations, all attaining maps and the special-linear orbit count. A new self-contained verifier reports PASS_SQUAREZERO_TENSOR_CHARACTERS.

• 42 field/dimension contexts, including F4 and F8; 5,448 literal tensor-action witnesses and 4,506 rejected requests outside the predicted subgroup.
• Three complete raw automorphism domains of 512, 65,536 and 19,683 matrices, accepting 6, 168 and 48 maps.
• All 28,224 pairs in GL3(F2)^2 give exactly 96 autotopisms, with the output map forced by the unit.
• All 833 elements in seven small algebras check unit-multiplication determinants; 15 controls reject altered factors, the wrong tensor convention and invalid domains.

Local CPython 3.12.15 normal and optimized runs passed in 4.636 and 4.778 seconds with identical generated hash-bound receipts. Both VERIFY.json commands and the registered runner passed; the runner took 9.338 seconds. Metadata/provenance, all seven committed bytes/modes/source-history checks, AST compilation and whitespace checks passed. Official export integrity and both read-only, Git-free direct replays with empty PATH passed without changing any exported file. No predecessor scientific verifier was run.

From the repository root:

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

Other machines may substitute their CPython 3.12 path. SOURCES.json pins the exact e545235 base and source interface as provenance. The selected #7000 frontier and five retained assumptions remain unchanged. Advance claim.

Limitations and unrun obligations

The universal statement rests on the written proof; independent human and formal proof-kernel review are unclaimed. The executable uses canonical charts in bounded fields and dimensions. It supplies no arbitrary-tensor recognizer, supplied-table radical extractor or special-linear transporter from an uncharted tensor. Nonsplit residue fields, higher radical products and semilinear equivalence remain outside the result.

There is no physical coefficient preparation, native caller, dirty-state tariff, precision/tape implementation, paid supplier or complete multiplication theorem. No selected cold replay, Lean or Docker run was performed. The formal-interface-lemma card is explicitly changed to this algebraic side result; its proof-assistant acceptance gate is not claimed.

Open next questions

• Construct a supplied-table extractor and explicit special-linear transporter for this exact class, beginning with radical and scalar-residue checks.
• Determine the determinant image with nonsplit residue fields, including the semilinear radical determinant.
• Name a physical producer/consumer and charge all coefficients, movement, precision and restoration before deriving a supplier consequence.

Related open work

No open PR studying this split radical-square-zero multiplication tensor or its stabilizer determinant image was found in the refreshed live title/body searches. Merged #5540 supplies the interface. Merged #5459 concerns characteristic-power truncated polynomial algebras, a different class. No additional computations were run to check those results.

Before DONE, this account's open PRs created since the CLAIM were refreshed. The nine other contributions are #7247, #7248, #7249, #7250, #7251, #7253, #7254, #7255 and #7256. Their scopes do not study this split radical-square-zero multiplication tensor or its determinant image; no competing class count was identified and no additional computation was run to check their results.

CI notes

CI did not certify this package. At unchanged head b80abf6285a93faee360a792942f06834f04c27a, run 43024030495, plan job 130534063268, failed in Onboarding helper tests. The checks were read once. The downloaded job log reports exactly one failing test among 760, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
FAILED (failures=1, skipped=24)

The assertion's full message quotes the document. At base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and omits that command. I checked those base bytes directly. This PR changes neither file. The package replay job was skipped because plan failed.

I also downloaded and checked all six comparison plan logs. Each has the same sole failing test and the same missing-command assertion:

• Other contributors: run 42905094314 and run 42903313790.
• Prove the exact rounded tensor-image block minimum (no κ) #7247: run 43023514635.
• Prove four independent full guards beat their product over every field #7248: run 43023633785.
• Prove the sharp rectangular native-fusion approximation floor (no κ) #7249: run 43023693429.
• Prove exact sparse tensor-annihilator coefficient valuations (no κ) #7250: run 43023829856.

All local mathematical checks recorded above passed. The repository-level failure remains for maintainers to repair; no fix, retrigger commit or job rerun was made. Hosted replay and independent human or formal proof-kernel acceptance remain unclaimed.

Merge danger

Two-way door. Blast radius: isolated. Seven new files in one opt-in research package; reverting it removes the result and replay registration. Shared code, predecessor packages and the selected frontier are unchanged.