[ToikY]

This answers Nick Petalas's PR #6682 unequal-cutoff question: classify every unital K-linear automorphism of K[x,y]/(x²,y³) over every finite field K. In basis (1,x,y,xy,y²,xy²), the complete normal form is

X=a*x+alpha*x*y+beta*y²+gamma*x*y²
Y=c*x+d*y+delta*x*y+epsilon*y²+zeta*x*y²

Here a,d are nonzero; beta=0 unless characteristic 2 and c=0 unless characteristic 3. Every other coefficient is free and every allowed lift is invertible. The order is (q-1)^2 q^6 in characteristics 2 or 3, otherwise (q-1)^2 q^5. The full six-dimensional determinant is a^3 d^6, with exact image the cube subgroup H3. The proof derives the relations and complete graded blocks; this is a fixed (2,3) classification.

Executable evidence is separately bounded:

• Complete F2/F3/F4/F5 censuses contain 89,844 actual maps. An independent raw-radical oracle evaluates 379,132 relation-valid pairs, then validates every submitted accepted-map row in normal and optimized modes. The additional 180 F7/F13 witnesses are not larger censuses.
• Every typed common field, accepted-set digest, determinant count and witness agrees. Canonical common SHA256: 90961ea96ee98b5b77102fbcf660fb5a01dbdfcaa96c74100c989f56ebc5513c. Actual decompressed map data, rather than gzip container bytes, determine equality.
• Primary four strict modes, oracle six modes, 21 oracle diagnostic controls and 19 intended external rejection checks pass. Both engines' actual code and first retained receipts remain unchanged from their blind firsts.
• At scientific commit b6a3955e80d9b787d0109eec1776efdf95d6f48a, all four registered commands, 23-file byte/mode/source-history preflight, official export integrity and four read-only Git-free, empty-PATH cold commands pass. Final documentation commit25500a2d75b54fa44ed0d2156384d45cf27aa553 and the exact public head also pass all23committed byte/mode/history checks; final official export/input1 and four unchanged read-only cold commands pass.

The external mutation harness initially failed to change pretty oracle JSON; the clean oracle correctly passed. Its preserved no-op attempt was corrected only in the harness, with explicit mutation preconditions. A precommit registered invocation correctly rejected the untracked map sidecar; the declaration was retained and replay passed after committing it. Neither was an engine failure. VALIDATION.md and RUN-RECORDS.json preserve exact first clocks and these distinctions.

Credit: Nick Petalas #6682/#6578/#5252; Ali Romman #5316/#5459; Armin Pourshafeie #5656/#5663; David Ganz and retained tensor-atlas lineage. Exact source pins, original headers and classical radical/filtration/determinant-method credit are retained. The pair (2,3) is outside the prior same-characteristic-power family.

No all-cutoff family, tensor recognition/stabilizer implementation, semilinear count, physical field preparation, caller saving, κ or frontier change is claimed. No merge; maintainer edits and auto-merge remain disabled. Hosted CI succeeded on original run 41724318493: 711 helper tests (24 skipped), all four registered package commands, both full-map comparisons and both staged/unstaged drift checks passed. Synthetic test commit d8e4eff58f61d128a18d2080e1ab5f4dd80efe42 has the exact verified tree and expected main/PR parents; no actual merge occurred. The mode, source-history and export checks above were local. Public head: 64e08a1b1a73f80c814f2a6954c693c26bebf358.

Substantial OpenAI Codex assistance with separate AI primary/oracle implementations and static review; independent human or formal proof-kernel acceptance is unclaimed.