[aliromman]

This recognizes finite residue-field Frobenius products from arbitrary square tensors. For K=F_q, the class is every finite product of L_j[x_i]/(x_i^(p^h_i)), with arbitrary finite extensions L_j/K. The algorithm discovers the primitive factors, coefficient fields and height lists, then returns all three literal tensor factor maps. No new kappa is claimed.

Proof, exact evidence, contract and implementation, and source pins.

The key construction recovers the unique coefficient field as ker(Frob_(q^d)-I) in a local component with residue degree d. The module top has K-dimension d; the tangent-height test uses spans over the recovered field. The algorithm is necessary and sufficient for the declared class and deterministic polynomial time in tensor dimension for fixed K. It does not enumerate q^d field elements. Each coefficient-field table is checked without elementwise inverse enumeration, and every transported basis product is checked.

This complements apoursh's supplied-decomposition character work. This PR supplies recognition and coefficient-field discovery only. Character images, canonical cross-presentation labels and special-linear transporters are outside its final scope. The tensor atlas gains exact discovered model inputs; physical maps and the complete paid caller remain unpriced.

Python 3.12.14 registered normal and optimized replay passed with identical reports. A separate coefficient-index orbit generator agrees on all 72,353 two-dimensional tensors over F2/F3/F4, accepting 174/4,320/62,460. There are 52 further presentations, 1,392 basis products and 272 controls over F2/F3/F4/F9, with degree-two and degree-three residues, nilpotents, repeated factors, false fields, incorrect maps and nonregular modules. Committed-byte and portable-export checks cover all 13 package files; the publication handoff records the Git-free replay and source-corruption control.

python3 -B scripts/research_package.py --package research/residue-product-tensor-recognition
python3 -B scripts/check_research_commit.py --package research/residue-product-tensor-recognition

Expected marker: PASS_RESIDUE_PRODUCT_TENSOR_RECOGNITION. The standard library suffices. The checkout is shallow; full-history preflight is not claimed. Exact predecessor bytes are retained in this branch's first commit. Independent human or proof-kernel acceptance, hosted CI, a selected cold replay and a physical supplier are not claimed.

Base is canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7. The final overlap refresh covers 351 open PRs, 4,969 Discussion records and the complete tracked main note inventory; no arbitrary-diff or unpublished-work coverage is claimed. The canonical scope separates this discovery task from the concurrent character claim. Only the new package changes; the selected frontier is unchanged.

Credit Ali Romman's PR #5661, exact 5d722279f4292c63361f1e958ff4ed5a6ec638aa, for split-product recovery and the retained helper/check routines; PR #5637 for finite-field arithmetic and semilinear kernels; @apoursh for PR #5502 and PR #5656; Nick Petalas for PR #5252; and David Ganz and the retained atlas lineage. The source manifest identifies every consumed copy and its exact upstream object. Coefficient-field lifting is classical; no literature-priority claim is made. Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.