[ToikY]

Commutativity alone does not make the right-vector query cost equal to delta on invertible targets. This answers Open question 3 of #4276: over F2^3, the component target (I, [[0,1],[1,1]], [[1,1],[1,0]]) costs exactly 3, with delta 2, even though free GL2(F2) maps are allowed anywhere.

The scoped proof is sharp: over K^k, for a finite field of size q, all invertible targets cost at most 2 when k<=q, while examples cost 3 with delta 2 for every k>=q+1. It proves the common-unit-pivot criterion and constructs a complete three-factor word for invertible targets over finite products of field K-algebras. The all-finite-field statement uses a written quadratic-extension argument. The software witnesses cover only the listed prime fields. Singular targets and general commutative algebras remain outside scope.

An independent literal BFS covers all 216 invertible F2^3 targets: (cost,delta,count)=(0,0,6),(1,1,54),(2,2,144),(3,2,12). All 4,752 paid and 1,296 free edges, 132 conjugations, 216 matrix witnesses and 13,824 arbitrary-input cases pass. A separate projective/explicit-word oracle agrees on every cost/delta record, checks 8 prime-field sharp witnesses, 512 F3 first-column triples and 2,550 literal upper words. Both Python 3.12 normal/optimized warning-strict commands freshly execute both algorithms and compare complete typed artifacts. Source-history/public-byte preflight and committed Git-free export pass with all 16 files unchanged. Separate AI proof/code/integration review found no correctness defect; witness-order notation and the normal warning flag were clarified before freezing. No independent human or formal-kernel acceptance is claimed.

Reproduce: python3.12 -B scripts/research_package.py --package research/right-vector-commutative-products. Only this isolated package is added. Physical preparation, routing, control, precision, restoration and stock costs are UNKNOWN; the formal word does not admit a native caller, supplier or multiplication theorem. No new kappa, merge or frontier change. Maintainer edits disabled.

Canonical main refreshed before submission remains 12fa8c5c844be21bfefb38a19503fbd867a11244, priority7. All 442 open/draft bodies plus an update-ordered delta and live Discussions/source handoff checked; broader history/code/Discussion coverage partial, no novelty inference. Canonical investigation. The original bit-new-register-model card is explicitly changed to this named commutative-algebra boundary question; no physical construction campaign. Exact source PR4276 head59c7bac0122ef2b80793c30a3d52329db6bdf17d and retained input ancestor326b25e5da8330cc3a44f84e8dcde0e4ff50c5e0 are authenticated, with unchanged source notices.

Remaining: products of local algebras/residue-field hypotheses; any actual complete physical invoice; independent human/kernel review. README/TASK give bounded follow-ups and stop rules. Credit Nick Petalas #4276/#4265/#4243, Wentao Li #4099 and the #2513 lineage. Their criteria and classical projective/field/determinant facts retain their credit. New scoped deductions, independent engines and integration by ToikY with substantial OpenAI Codex assistance and separate AI review contexts. Apache-2.0.