[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. For every finite field K of characteristic p, d >= 2 and n >= 2 which is not a power of p, the multiplication algebra A = K[x_1,...,x_d]/(x_i^n) has monomial tangent maps. Its complete automorphism determinant image and ordered covariant tensor-stabilizer image are

N = n^d, E = n^(d-1)n(n-1)/2, S = n^(d-2)n(n-1)/2
D = H_E <(-1)^S>,  H_r = {a^r : a in K*}
Delta = {(u,v,w) : vw in H_N, uw in H_N, uvw in D}.

The proof gives three character equations and the exact number of special-linear orbits. The variable-swap sign is necessary: at K=F3 and n=d=2, the exact covariant image is {(1,1,1),(2,2,2)}, while scaling alone misses the latter triple. Nonlinear lifts contribute no further character.

This settles a non-Frobenius part of the nilpotent-algebra question in PR5252. Credit Ali Romman / PR5316 and PR5459, Armin Pourshafeie / PR5656 and PR5663, David Ganz and the retained tensor-atlas lineage for the distinct predecessor interfaces. The radical filtration, elementary binomial identity and unital isotopy normalization are classical. No predecessor result is audited and no global literature-priority claim is made. No new kappa or selected-frontier change.

Evidence

Before, the cited character interfaces cover one-variable truncations or multivariable characteristic-power cutoffs. This adds the different non-Frobenius tangent restriction and its complete determinant character.

Local CPython 3.12.15 normal and optimized checks pass with identical generated hash-bound reports:

• 138 literal attaining automorphisms in fifteen contexts, including F4 and F8; 232,714 basis-product identities.
• Complete 729-pair and 65,536-pair generator-image domains, accepting 72 and 2,048 automorphisms. Each accepted nonlinear lift has the determinant of its tangent substitution.
• 310 literal unit multiplication maps and all 3,243 determinant triples in the declared contexts.
• All 64 coefficients of a nonlinear covariant tensor fixture; raw-input and missing-transpose controls change fifteen and eight coefficients. Five invalid domains reject; four named semantic discriminators pass.

From the worktree root:

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

Another CPython 3.12 path may replace the prefix. Expected markers are PASS_NONFROBENIUS_RECTANGULAR_CHARACTERS no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The two registered argument lists took 0.375 and 0.482 seconds locally, below their 120-second limits. These are execution durations, not a performance comparison.

The registered runner, metadata/pins, exact committed bytes/modes/source ancestry, official export/input check and both read-only Git-free exported runs pass at 143491876f44d2bbf83bf52ae6c9fe5e9496a03c. All seven exported files retain their bytes. The direct verifier uses only the standard library and writes nothing. The advance CLAIM matches the delivered scope.

Limitations and unrun obligations

The quantified proof covers every finite field in the stated equal-cutoff family. The executable is bounded finite corroboration. Characteristic-power and unequal cutoffs, orbit recognition, chart/transporter recovery and physical coefficient preparation remain outside the result. It supplies no native caller, routing/restoration/precision/tape invoice, supplier admission or complete multiplication theorem. The assigned card is explicitly changed to this algebraic side result; its physical-mechanism acceptance gate is unclaimed. Independent human and formal proof-kernel review are unclaimed. No selected cold replay, Lean or Docker build was needed or run.

Open next questions

• Determine the tangent restrictions and exact character for unequal non-Frobenius cutoffs, starting with K[x,y]/(x^2,y^3). A complete lift argument for the changed relations is missing.
• Recognize the promised equal-cutoff tensor orbit and recover a multiplication chart. Begin with a unital table and its radical generators; no certified recognition or physical coefficient bill is supplied here.

Related open work

• PR6578 counts automorphisms and tensor symmetries in the Frobenius-power family. This computes determinant images for non-Frobenius relations and does not extend or replay that count.
• PR6422 constructs least-subfield norm witnesses for the residue-field determinant interface. This needs only constant units in a split local algebra and computes no minimum witness field.
• PR6464 reconstructs unital multiplication tables from a promised tensor without field enumeration. This supplies the determinant character once the different non-Frobenius algebra class is promised; recognition and chart recovery remain separate.

No competing open result on this non-Frobenius determinant image was found in the refreshed body and main/general-family searches. No additional computations checked another contribution.

The completion refresh listed 212 account open PRs and read the six other open submissions created since the advance CLAIM at 2026-11-03T18:19:17Z: PR6677, PR6678, PR6679, PR6680, PR6681 and PR6683. They concern reader coefficients, scalar preparation, clean checkpoints, finite-replay implementations and mirror windows. None studies this non-Frobenius determinant image or reports a competing value. The three related multiplication-tensor entries above remain applicable. No extra scientific computation checked another contribution.

CI notes

CI did not certify this result. At exact head 143491876f44d2bbf83bf52ae6c9fe5e9496a03c, run 41694097281 was cancelled by the workflow's ten-minute timeout while plan was still executing actions/checkout. The plan job ran from 2026-11-03T18:28:10Z to 18:38:14Z. Python setup, onboarding tests and package selection were skipped; the scientific replay job was skipped. No package check ran in CI. This is a checkout infrastructure timeout, not a package test failure.

Two other recent pull-request runs show the same cancelled checkout: run 41696887381 and run 41702891238.

Verification is local only: the normal and optimized exact checks, both registered commands, metadata/source pins, committed-byte/mode/source-history preflight, official export integrity and both read-only Git-free exported replays listed above passed at this head. These local checks support the scoped result; hosted CI, independent human review and formal proof-kernel acceptance are unclaimed.

Merge danger

Door: two-way. Blast radius: isolated research package. Six new files and an opt-in bounded replay. Shared code, predecessor evidence and the selected frontier are unchanged.