[petalas]

Summary

Scoped constructive proof with exact finite corroboration, produced by Nick Petalas. This answers the local uniform-sampling question in #6578: all L-linear automorphisms of L[x_i]/(x_i^(p^h_i)) have an explicit bijective integer encoding, without constructing the full algebra multiplication table.

integer index
  -> invertible same-height tangent blocks
  -> free power-kernel monomial coefficients
  -> generator images

The inverse reconstructs the same index. The field-operation bound is O(sum_e r_e^5 + rN), with polynomial integer bit cost. A uniform integer index gives an exactly uniform automorphism. This is a rejection-free decoder given such an index, not a fixed-length fair-bit sampler.

The power-kernel, tangent and counting theorems from Nick Petalas's #6578 are cited premises. Credit apoursh #5663 and Ali Romman #5459 for the structural lineage and the classical ordered-basis and positional-integer methods. No predecessor scientific executable is imported or replayed. No new kappa or selected-frontier change is claimed. Advance claim.

Evidence

Before, #6578 gives symmetry cardinalities and leaves an explicit uniform encoding open. After, PROOF.md supplies the inverse maps and resource proof, and a new self-contained verifier reports PASS_FROBENIUS_AUTOMORPHISM_CODEC.

• 5,039 exhaustive local indices in eight contexts, including extension fields F4 and F9; all round-trip, are distinct, obey their literal polynomial relations and have independently checked nonsingular substitution maps.
• 404 exhaustive GL indices in six contexts, independently checked by Leibniz determinants; thirty selected indices in six larger or differently ordered contexts.
• 359 literal basis-product checks in the exhaustive contexts, additional product checks at selected larger maps, and sixteen rejecting controls. Every reported index and control is executed; larger spaces are not exhaustively enumerated.
• CPython 3.12.15 on macOS 27.0.1 arm64. Exact normal and optimized registered command arguments passed locally in 2.35 and 2.46 seconds, with identical generated SHA256-bound receipts. The registered launcher passed both commands in 4.78 seconds. Each command has a 120-second limit.
• Metadata/provenance, all nine committed bytes and modes, source-history availability, AST compilation, function-size inspection and whitespace checks passed at 11b1aa6e16ef680b2bc3ab8b665b6902e227ec4c.

From the repository root:

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

Another CPython 3.12 path may replace the local path. All dependencies are standard-library modules. Source texts, catalogue and selected record are provenance-pinned at canonical e54523517d0f03e58352e647c403a5af3e525a57. The executable has no external scientific inputs and writes no files.

Limitations and unrun obligations

The universal proof uses the named predecessor premises. The executable is guarded to fields of size 2,3,4,5,7,8,9, four generators, heights at most six and dimension at most 64; exhaustive fixtures are smaller. Independent human and formal proof-kernel review are unclaimed.

Coefficient-field discovery, semilinear automorphisms, nonsplit products, tensor-character transport and arbitrary-tensor recognition are not supplied. There is no native implementation, physical coefficient-preparation tariff, consumer, dirty-restoration bill, complete signed invoice or multiplier theorem. All five selected assumptions remain. No selected cold replay, predecessor scientific suite, Lean or Docker build was run.

Open next questions

• Join the local codec to a supplied product decomposition with component permutations and residue-field Galois actions. First prove the joint data bijection; a concrete semilinear arithmetic contract remains missing.
• Identify a physical changed-basis consumer and price one decoded substitution, including coefficients, alignment and restoration. Abstract uniform sampling alone gives no supplier saving.

Related open work

• #7252 determines tensor determinant images for split square-zero algebras with radical dimension at least two. That is a different algebra class and quantity; its proof and construction are not inputs here.
• #7254 constructs invertible maps under two prescribed subspace capacities. This codec addresses polynomial-algebra substitutions and does not use its quotient/lift construction. No common numerical optimum is compared.

The final live check covered the nine other account PRs opened since the advance claim, #7261 through #7269. None studies this local Frobenius-algebra codec or provides a competing codec value. The two adjacent relations above remain unchanged; no extra scientific computation was run to check another contribution.

CI notes

CI did not certify this package. At unchanged head 11b1aa6e16ef680b2bc3ab8b665b6902e227ec4c, run 43030419612, plan job 130555716866, failed in Onboarding helper tests. I read the checks once and inspected the job log. Exactly one test failed among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

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 both base files directly. This PR changes neither file. The package replay job was skipped because planning failed.

I downloaded and checked all five comparison logs below. Each has the same sole failing test, missing-command assertion, 760-test total and 24 skips:

• Other contributors: run 42905094314 and run 42903313790.
• Recent comparison runs: 43029932349, 43030221243 and 43030268355.

All local checks recorded above passed. No fix, retrigger commit, push or job rerun was made. Hosted package replay and independent human or formal proof-kernel acceptance remain unclaimed.

Merge danger

Two-way door. Blast radius: isolated. Nine new files in one opt-in research package. Shared code, frozen predecessor evidence and the selected frontier are unchanged.