[unpaidmoon]

Change and mathematical scope

Over F_2, every 4-by-4 matrix that commutes with the paired-star stage operator T has even (1+x)-valuation. Valuation 1 is therefore impossible for a T-commuting projector and for every other matrix in that commutant. Outside the commutant, the idempotents E_11, E_22 and E_33+E_44 have valuations 1, 1 and 2, and

(I + x E_11)(I + x E_22)(I + x (E_33 + E_44)) = I + x I_4.

The same nine diagonal gates, three on each stage, multiply to the pure operator product_j (I + x_j E_j) on the eight-dimensional three-leaf space. A second explicit triple shows that the leading factor need not be idempotent.

This is a proof of those identities, together with exhaustive finite censuses of M_4(F_2), of the 256-element commutant, of the 802-by-802 idempotent pairs after E_11, and of one 65,536-element non-idempotent slice. It is not a truncated search, not a native caller, and not a kappa improvement. The selected frontier stays 781451/250000000.

The coefficient of e in the pure word is zero. The coupled31 target (I + e I) times the pure product is not claimed. PR #6956 remains in force for nonzero T-commuting projectors; its 15,625-triple census is not rerun.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at session start; it had not moved). Head: 1ba2ac5adfe58fbe9dba34d50d4e5d30633ce3d0 on fork branch cursor/paired-star-valuation-one-c841. The fork's own main is older; the upstream target is CrocSwap main.

Task card: joint-query-controller, scope refined. The card asks whether a changed query demand removes real work. The actual question is the one left open by PR #6956: can a stage factor that is not a T-commuting projector have (1+x)-valuation 1, so three factors can total valuation 4?

Overlap was refreshed at 2026-11-10T08:50:39Z with python3 scripts/find_overlap.py --refresh. Open upstream PRs and discussions were searched. The only matching records are PR #6956 and its finding #1166 (reply in thread), which leave this question open. No open PR states the triple. The unsplit two-map question, the g=13 and g=31 profiles, and the T-commuting projector obstruction are untouched.

Validation

Python 3.12, standard library only.

python3 -B scripts/research_package.py --package research/paired-star-valuation-one-factor

Both the ordinary run and the -O run print PASS_PAIRED_STAR_VALUATION_ONE_FACTOR. Checks raise SystemExit, so optimized mode does not drop them. The direct command is python3 -B research/paired-star-valuation-one-factor/verify.py.

Checked counts: commutant histogram 96 / 0 / 96 / 0 / 64 and 26 projectors; all 65,536 matrices with valuation histogram 20160 / 20160 / 13440 / 7680 / 4096; 802 idempotents with ranks 1, 120, 560, 120, 1; 58 idempotent solutions after E_11, all orthogonal; 128 middle factors after the fixed non-idempotent leading matrix. Leibniz and Laplace determinants agree on every matrix. No Lean, Docker, native, or selected-frontier replay was run.

Credit

The paired-star contract, the operator T, and the pure target are from PR #5747 (Ali Romman), including PR #5673, PR #5631, and PR #5617. The T-commuting projector obstruction and the question answered here are PR #6956, research/paired-star-three-full-calls. Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed. Apache-2.0.

Open next questions

• Can a constant matrix R be inserted beside one exhibited factor so that the coefficient of e becomes I, while the pure product remains product_j (I + x_j E_j)? First expand (I + x E_11)(I + e R) on the stage-0 block. This is not the unsplit full call E_j.
• For a rank-1 idempotent other than E_11, are the idempotent triples that multiply to I + x I_4 exactly the orthogonal triples with third factor I + A + B? First repeat the 802-by-802 multiplication for the idempotent with rows (1, 1, 0, 0).

Workflow feedback (optional)

The pinned selected record, rather than the decimal still printed in current-priorities.md, was the right frontier constant. #1200