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Construct common three-factor words for all matrix-algebra component targets - #5156
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Mister-Meeseeks merged 2 commits intoOct 23, 2026
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PR4805 constructs three factors for a single full matrix algebra but chooses free charts per target. Those choices do not directly assemble across components sharing one scalar control word. This package removes that incompatibility: every target over any finite product of full matrix algebras has a common
e1,e2,e1three-factor word, with no free endpoint maps or auxiliary roles.Extend the top block-row space to full row rank, apply the retained complement/graph construction in a fixed chart, and absorb the potentially singular completion into the first legal paid factor. This covers singular and zero components as well as invertible ones. The primary
(M2(F2))^kcontract follows for every finite k. PR4780's six scalar rank-one components, embedded as scalar I2 blocks, give a singular tuple with exact cost three via the inherited common-chart two-factor criterion.The proof retains Nick Petalas's PR4805 complement/graph credit and helper source, PR4276's criterion, PR4780's scalar-product construction, and the PR4749/#4243, ToikY/#4334, Wentao Li/#4099/#2513 lineage. Classical basis extension is not claimed as new. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.
Validation: 1,286 bounded constructor executions with an independent full-matrix product/rank oracle, all 226 binary rank-at-most-one 4x4 targets, four prime fields and block sizes1..4, 96 dirty-input executions and 30 controls. The six-component sharp tuple explicitly distinguishes individually passing charts from a shared chart. Registered Python3.12 normal/-O replay, source-history/committed-byte check, official18-file export integrity and actual Git-free cold replay pass. Cold controls reject Boolean/float receipt types, wrong digest, retained-source drift and lost singular absorption; all restored bytes remain unchanged. Root separately reviewed the exact upper-bound proof; this is agent review, not human/kernel acceptance.
Formal overwriting algebra only: native coefficient preparation, physical routing, overwritten-data recovery, precision, stock and complete supplier costs remain unknown. No kappa or selected frontier change. No historical exhaustive suite, Lean or native replay. Singular radical extensions and a fully priced actual consumer remain open.
Claim: #1166 (comment)