[aliromman]

This enabling algebraic result recognizes arbitrary square tensors over an explicitly supplied finite field F_q that are isotopic to F_q[x_i]/(x_i^(p^h_i)). It returns the heights and all three factor maps, then removes the algebra-class promise from the existing determinant-character formula. It computes a complete special-linear orbit label and constructs determinant-one maps between equivalent accepted tensors. The complete kappa calculation is unchanged. A physical producer/consumer and its full paid bill remain open.

The proof and scope, recognizer and transporters, exact verifier and source pins are pinned to this head. The base is 6cf7a683f447b9f1e2333791907fd92be5009a63, priority revision 7; canonical main and live overlap were refreshed before submission without a guidance change.

The new field contract separates q-power radical recovery from semilinear p-power height kernels. Over F4, in the basis (1,a+x) of F4[x]/x^2 with a^2=a+1, the ordinary kernel of the squaring columns contains (a^2,1), which represents 1+x and squares to 1. The corrected kernel contains (a,1), which represents x and squares to zero. Thus treating those columns as an ordinary linear map gives a wrong certificate. The composed algorithm has deterministic polynomial cost in tensor dimension for each fixed q. It does not claim polynomial bit complexity in log q or recognize every commutative algebra.

Verification covers all 65,536 F4 2x2x2 tensors against a separate complete GL2^3 orbit generator; exactly 13,500 pass. All 256 binary tensors also agree, with 54 accepted. Ninety changed-basis presentations over F2,F3,F4,F8,F9,F16,F25 check 2,322 basis products. Ninety determinant-one transporters check another 2,322 products. There are 125 failure controls, including reducible field moduli, nonregular modules, extra relations, wrong Frobenius kernels, changed transporters and unequal special-linear labels. Cutoff four and two cutoff-two variables over F4 have orbit counts three and one, and explicit maps attain all three labels in the first case.

Python 3.12 and its standard library suffice:

python3.12 -B scripts/research_package.py --package research/extension-field-tensor-recognition
python3.12 -B scripts/check_research_commit.py --package research/extension-field-tensor-recognition

The expected check markers are:

PASS_EXTENSION_FIELD_TENSOR_RECOGNITION
PASS_REGISTERED_RESEARCH_PACKAGE research/extension-field-tensor-recognition outcome=completed
PASS_RESEARCH_COMMIT_BYTES

Both registered checks, the nine-file committed-byte check, export validation and a Git-free optimized scientific replay passed. CI is pending at publication. The source-history preflight was not run in the shallow checkout. No predecessor full executable suite, proof-assistant build, independent human review, native supplier, physical precision/movement audit or selected cold replay was run. The finite orbit checks corroborate the written theorem; they do not establish a multiplication theorem.

Credit: Ali Romman's PR #5540 supplies the regular-module reconstruction and released handoff; PR #5525 supplies its earlier tensor implementation. Nick Petalas (@petalas) supplies the nucleus lineage through PR #5252. apoursh (@apoursh) supplies the prime-field height criterion in PR #5502, and Ali Romman's PR #5459 supplies the finite-field character theorem. Their retained reconstruction proof, height proof and character proof are unchanged and authenticated. David Ganz and the linked atlas sources retain their credit. New proof, implementation and checks: Ali Romman (@aliromman), with substantial OpenAI Codex assistance. Apache-2.0.