[petalas]

Summary

Scoped written counting proof with exact finite corroboration, produced by Nick Petalas. For L of size Q and characteristic p, the complete automorphism order of B=L[x_i]/(x_i^(p^h_i)) is

N = p^(sum_i h_i)
r_e = number of variables of height e
k_e = N - p^(sum_j max(h_j-e,0))
C = sum_i k_(h_i) - sum_e r_e^2
|Aut_L(B)| = Q^C product_e |GL_(r_e)(L)|

The new step counts all nonlinear lifts of the permitted tangent matrices. It gives exact automorphism orders for nonsplit products with repeated full types, and hence complete covariant multiplication-tensor stabilizer and GL-orbit cardinalities. This is an algebraic side result; no new kappa or selected-frontier change is claimed.

Main predecessor: Armin Pourshafeie / apoursh, PR5663 at d72b2e42a4cf53ae85f5276da65c5cc84e17b253. Its intrinsic coefficient-field, type decomposition and autotopism normalization are cited premises. Credit its Ali Romman / PR5459 local height-flag and PR5637 arithmetic lineage, apoursh / PR5502 and PR5656, Nick Petalas / PR5252 normalization, David Ganz and the retained tensor-atlas authors. Monomial Frobenius, finite filtrations, parabolic group orders, finite-field Galois groups and orbit-stabilizer are classical; general literature priority is unclaimed. No predecessor executable or scientific receipt is replayed.

Advance CLAIM. The coverage-law-change card is explicitly changed to this self-posed symmetry-counting question. The original productive-credit acceptance gate is not claimed.

Evidence

Before, the cited interface gives determinant images and the number of special-linear orbits. This adds the complete group and orbit cardinalities through an exact nonlinear-substitution count.

• Independent literal quotient-ring maps give 21 automorphisms for F8[x]/x^2 over F2, 30 for F16[x]/x^2 over F4, 60 for the latter over F2, and six for F64 over F2.
• Every map in the repeated-product domain is checked: 882 maps for two copies of F8[x]/x^2. Its product with the equal-dimensional but different-type field F64 has 126 maps, with an explicit failed block exchange.
• A factorized census for F2[x,y]/(x^2,y^8) visits all 65,536 elements for each generator. The tangent fibers give exactly 33,554,432 automorphisms. This does not individually execute that many full matrices. Eighteen direct nonlinear witnesses check every basis product. An equal-height three-generator census checks the repeated tangent-block factor separately.
• The complete report checks 167,244 basis-product identities, nine rejecting mutations/domain controls and four semantic discriminators. A separate ternary fixture covers the odd-characteristic boundary. All guards remain active under optimization.
• Direct normal/optimized runs produce identical generated reports, bound to all eight package-file hashes. The two registered command argument lists were timed locally at 0.36 and 0.35 seconds, respectively, below their 120-second limits. The registered launcher also passes both commands.
• Metadata/source pins, committed bytes and source-history availability pass at a148f241a46db2779889751b8eb455af724d853b. Official export/input integrity passes. Both normal and optimized exported runs pass with Git unavailable, all exported files read-only and all nine exported file hashes/modes unchanged.

Reproduce from the repository root with CPython 3.12 and its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -E -W error -X dev -B research/nonsplit-frobenius-orders/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -E -O -W error -X dev -B research/nonsplit-frobenius-orders/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/nonsplit-frobenius-orders
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/nonsplit-frobenius-orders --check-source-history

Another CPython 3.12 path may replace the local interpreter. Expected markers are PASS_NONSPLIT_FROBENIUS_ORDERS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier generates its receipt from current bytes and writes no files. PROOF.md supplies the universal statement; finite execution corroborates it.

Limitations and unrun obligations

The theorem concerns the supplied finite algebra family. It uses PR5663's structural results; it does not independently validate them. The executable has small-domain guards, the literal binary oracle stops at dimension sixteen, and the complete large tensor orbit is not enumerated. Independent human and formal proof-kernel review are unclaimed.

No arbitrary-tensor recognition, field-embedding discovery, native/rounded execution, physical coefficient preparation, routing, dirty restoration, precision/tape bill or complete supplier/caller invoice is supplied. All five selected framework assumptions remain. No selected cold replay, Lean or Docker build is run. The selected frontier and frozen predecessor packages are unchanged.

Open next questions

Can a bijective encoding of the tangent matrices and their power-kernel fibers sample a uniform automorphism without a full algebra table? First prove the encoding and its uniformity. A table-free arithmetic contract, actual consumer and physical coefficient bill remain missing; no continuation is reserved.

Related open work

No other open PR studying these Frobenius-product symmetry cardinalities was found in the refreshed open-PR bodies and focused automorphism/truncated-algebra searches. The cited PR5459/5502/5540/5637/5656/5661/5663/5664/5668 interfaces are merged. Recent norm-witness and recognition-runtime contributions study different quantities and are not extended or audited. Sparse notes, arbitrary historical diffs and unpublished work remain overlap limits.

The completion refresh screened all fourteen other account open PRs created since the advance CLAIM, PR6571 through PR6585 excluding this PR. They concern scalar correction, phase metadata and isolation, Gaussian rounding, atom-power and ADDRESS tariffs, parity fanout, E8 geometry, local polynomial factors and separate replay optimizations. None studies these Frobenius-product symmetry cardinalities or supplies a competing value. No extra computations audited another contribution.

CI notes

Both hosted checks passed at head a148f241a46db2779889751b8eb455af724d853b in run 41617818808: plan in 4m44s and replay (research/nonsplit-frobenius-orders) in 1m23s. The completed checks were read once with gh pr checks; separate run metadata confirms completed success and the matching head.

Hosted replay executes the registered package checks. All local checks above also passed. Hosted success certifies the bounded replay and source bindings; independent mathematical acceptance of the universal proof remains separate. Physical realization, supplier admission and all five selected framework assumptions retain their stated status.

Merge danger

Door: two-way. Blast radius: isolated. Eight new files in one opt-in research package; shared code, frozen evidence and the selected result are unchanged.