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Recognize finite etale multiplication tensors with explicit transporters - #5525
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This proof removes the finite-etale orbit promise from PR #5252. It gives a necessary-and-sufficient criterion for an arbitrary square bilinear tensor to be finite-etale multiplication up to three independent basis changes, and constructs those changes. The complete kappa calculation is unchanged.
Proof and scope, exact checks, source pins, registered replay. Base: canonical main
32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7. Contribution head:3d5f42dc084b8c9e1ad73318458751e851752a10.The criterion checks the left nucleus dimension and faithful projections, commutativity, injective Frobenius and the right radical. The proof then constructs cyclic vectors in the primitive field components and a complete multiplication-table transporter. The written algorithm is deterministic polynomial time for each fixed finite base field. Its dependence on a varying field size is not polynomial in the field-size bit length.
CPython 3.12.14 registered warning-strict normal and optimized checks pass. All 256 binary two-by-two-by-two tensors agree with independent enumeration of the full GL2-cubed orbits of split and field multiplication: 120 accepted, 136 rejected. The full replay checks 326 tensors, 187 accepted presentations, 955 transported basis products and 21 controls, including nilpotent and noncommutative algebras and twelve changed transporter entries. Both modes print
PASS_ETALE_TENSOR_RECOGNITION; the launcher printsPASS_REGISTERED_RESEARCH_PACKAGE.Python 3.12 and the standard library suffice. Committed-byte checks pass. The executable fixtures use prime base fields and dimensions through four. The proof covers all finite base fields. Recognition of nonreduced algebras, arbitrary tensor isomorphism and physical supplier costs remain outside scope. No predecessor scientific replay, selected cold replay, Lean, Docker or independent human/proof-kernel acceptance is claimed. Hosted status is unconfirmed at publication.
Credit Nick Petalas (@petalas) for PR #5252, the pinned nucleus and idempotent method and recognition handoff. That source retains David Ganz and the atlas lineage. Finite algebra decomposition and semisimple module theory are classical. The canonical scoped claim records the refreshed overlap check. PR #5502 recognizes supplied Frobenius-truncated prime-field algebra tables, a different class and input contract; no executable from it is used.
Next: test cyclicity for a specified nonreduced local nucleus before extending the transporter construction. Faithfulness and matching dimension no longer prove regularity in that setting. Coordinate with the separate supplied-table recognition work.
Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.