[aliromman]

This closes the three-factor tensor-reconstruction handoff in PR #5502. The new regular-module criterion recognizes commutative unital multiplication from an arbitrary square prime-field tensor, including algebras with nilpotents. Composing the recovered table with the pinned Frobenius-truncated classifier returns the height list and explicit independent maps for both inputs and the output. Kappa is unchanged.

Proof and scope, exact checks, source pins, registered replay. Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7. Contribution head: 831b41da0fc09e3e4cb983c03662da5df7970f68.

The new condition tests the top of each local nucleus module modulo the nilpotent radical. Both modules must have one residue-field generator. Matching dimension and faithfulness alone are insufficient: the faithful dual module of F_p[x,y]/(x,y)^2 is an explicit rejecting control. The proof is necessary and sufficient and gives deterministic polynomial complexity at each fixed prime. It does not solve isomorphism of arbitrary recovered algebras.

CPython 3.12.14 warning-strict registered normal and optimized checks pass. All 256 binary 2x2x2 tensors agree with independent complete GL2-cubed orbit enumeration: 174 general commutative tensors and 54 Frobenius-truncated tensors. Forty additional presentations cover primes 2,3,5,7,13 and dimensions through nine. The checks verify 1736 literal transported basis products, 76 controls and three immutable source copies. Both modes print PASS_NONREDUCED_TENSOR_RECOGNITION; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE.

python3.12 -B scripts/research_package.py --package research/nonreduced-tensor-recognition

Python 3.12 and the standard library suffice. Committed-byte checks pass. The pinned PR #5502 class theorem remains a credited premise; its complete prior scientific suite is not rerun. The new written proof has no independent human or proof-kernel acceptance. No extension-field input layer, variable-characteristic bit-complexity claim, selected cold replay, physical supplier or complete multiplication theorem is supplied. The optional full-history preflight could not run because this checkout is shallow; declared source bytes and the direct canonical-main base were checked separately. Hosted status is unconfirmed at publication.

Credit apoursh (@apoursh) for PR #5502, its pinned table classifier, class proof and explicit reconstruction handoff. Credit Ali Romman (@aliromman) for PR #5525 and its pinned nucleus routines, and Nick Petalas (@petalas) for PR #5252. Source copies retain the David Ganz, finite-algebra and atlas lineage. Scoped claim.

Next: add an explicit extension-field contract that separates q-power decomposition from the semilinear p-power height flag, or apply the recovered heights to the pinned determinant-character formula. General recovered-algebra isomorphism and physical costs remain open.

Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.