[apoursh]

Change and mathematical scope

This answers the nonsplit-character question left open by #5656 and #5661. For a supplied product of Frobenius-truncated local algebras whose residue fields are finite extensions of K=F_q, the package proves the exact automorphism and covariant tensor-stabilizer determinant images, constructs determinant witnesses, and validates supplied three-factor certificates and their complete SL orbit labels. No new kappa is claimed.

For component L_j[x_i]/(x_i^(p^h_ji)), put d_j=[L_j:K], N_j=p^(sum h_ji) and E_ji=N_j(p^h_ji−1)/2. The automorphism image is (K*)^g, where g is the gcd of q−1 and the E_ji, also including (q−1)/2 when q is odd and an even residue degree occurs or an identical odd-degree type repeats. The full covariant image is {(u,v,w):(uvw)^((q−1)/g)=1}. The proof identifies the intrinsic coefficient field, separates norm and Frobenius contributions, and proves attainment.

The residue degree must not multiply E_ji. Over K=F9, take beta=1+t and L=K[u]/(u²−beta). On L[x]/x³, scaling x by u has determinant beta⁷ of order eight; the incorrect extra-degree formula misses every nonsquare. A separate example checks the Frobenius sign when all residue-linear automorphism determinants are one.

Proof · Implementation and scope · Exact results · Validation.

Task and overlap

Base: canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7. The canonical claim records handoff-first selection. Live discussion checking found the split-product recognizer already claimed; that scope is now #5661 at 5d722279f4292c63361f1e958ff4ed5a6ec638aa. This follows its separate nonsplit boundary. Fresh all-state PR searches found no competing nonsplit-character submission; unpublished work and arbitrary historical diffs remain outside coverage.

TASK.json explicitly changes the formal-interface-lemma card to a written algebraic theorem with finite evidence. This adds twelve files in one opt-in package, including unchanged authenticated predecessor sources. Common explicit residue presentations and canonical component order are required by the executable. It does not discover decompositions or field isomorphisms, recognize arbitrary tensors, price a physical supplier, or construct SL transporters between arbitrary tensors.

Validation

At final head d72b2e42a4cf53ae85f5276da65c5cc84e17b253, CPython 3.12.14/macOS arm64:

• Six complete tiny matrix domains: 40,487 matrices, 22,854 invertible matrices, 14 automorphisms and 2,851 normalized autotopisms.
• 150 complete small nilpotent substitutions; 6,561 independently computed F81 products, 80 norm determinants and 88 larger substitutions.
• 512 covariant determinant triples: 128 accepted and 384 rejected, with literal coefficient checks and a missing-transpose control.
• Fourteen changed/malformed certificates and eight presentation/type mismatches rejected.
• Registered isolated normal/optimized replay, committed-byte/source-history checks and official export integrity pass. The final proof-formatting commit changes no executable, input or result bytes.
• Both direct commands reproduce the complete committed RESULT from a read-only Git-free export with PATH=/nonexistent. All 15 exported files retain their bytes and modes. Five actual source mutations reject before model import. Archive SHA256: 94142ebcc0eb3ef5717dfa88cf2bf500809957299269fce2dc60f5f0f4e06159.

python3 -B scripts/research_package.py --package research/nonsplit-frobenius-characters
python3 -B scripts/check_research_commit.py --package research/nonsplit-frobenius-characters --check-source-history

These finite checks support the separate theorem; they are not independent human or proof-kernel acceptance. No selected cold replay, native supplier, Lean or Docker run was performed. Main still has the shared planning-link defect repaired separately by #5659; hosted CI for this PR is unconfirmed.

Credit

Credit Nick Petalas #5252 for the etale norm/sign interface; Ali Romman #5459 for local characters, #5637 for the unchanged field/matrix implementation and #5661 for the split recognizer; apoursh #5502/#5656 for height and product-character work; David Ganz and the retained atlas lineage. Exact original commits and source notices are retained. Norms, normal bases and finite-field structure are classical. New work by apoursh with substantial OpenAI Codex assistance, including separate implementation, proof and finite-check agents. Apache-2.0; no global literature-priority claim.

Open next questions

Recover coefficient-field sections and local maps for arbitrary nonsplit tensors; validate explicit identifications between different residue presentations; or supply a named native producer/consumer with a full paid invoice. These remain separate obligations, detailed in README and TASK.json.