[unpaidmoon]

Change and mathematical scope

Over F_2, let A be the rank-1 idempotent with rows (1, 1, 0, 0). An ordered triple of idempotent matrices (A, B, C) satisfies (I + x A)(I + x B)(I + x C) = I + x I_4 if and only if A B = B A = 0 and C = I + A + B. There are exactly 58 such triples, one for each idempotent endomorphism of ker(A).

The same equivalence is a coefficient proof for every idempotent triple in M_n(F_2), any n. The count 58 extends, by the same kernel bijection, to every rank-1 idempotent in M_4(F_2). That extension is not a second census: the 802-by-802 multiplication was run only for this named first factor. Among the 802 idempotent middle factors, 177 satisfy B A = 0, and 119 of those have a non-idempotent third factor I + A + B whose gate product is still the target. Those 119 are not idempotent triples.

This is a proof of the equivalence and an exhaustive finite census of M_4(F_2), of the 802-by-802 idempotent pairs after this A, of the subspaces used in the counts, and of the 512 endomorphisms of the kernel. It is not a truncated search, not a native caller, and not a kappa improvement. The selected frontier stays 781451/250000000.

Each of the 58 triples embeds on stage 0 as a factorisation of I + x E_0. The nine-factor three-stage word is not rebuilt. The coefficient of e is not changed.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, fetched at session start and again before submission; it had not moved).

Task card: joint-query-controller, scope refined. The card asks whether a changed query demand removes real work. The actual question is open question 2 of PR #7027: 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? The first factor is the named matrix with rows (1, 1, 0, 0).

Overlap was refreshed at 2026-11-10T10:12:49Z with python3 scripts/find_overlap.py --refresh (339 open PR records and 7013 discussion records, complete pagination). The only open statement of this question is PR #7027, which leaves it open and classifies E_11 instead. That E_11 census and the T-commutant valuation identity are not rerun. The constant-R insertion and the g = 13 and g = 31 profiles are untouched. Fork cursor/* branches and open fork PRs were listed; none classifies this matrix.

Validation

Python 3.12, standard library only.

python3 -B scripts/research_package.py --package research/rank1-idempotent-triple-1100

Both the ordinary run and the -O run print PASS_RANK1_IDEMPOTENT_TRIPLE_1100. Checks raise SystemExit, so optimized mode does not drop them. The direct command is python3 -B research/rank1-idempotent-triple-1100/verify.py.

Checked counts: 802 idempotents with ranks 1, 120, 560, 120, 1; 643204 ordered idempotent pairs after the named matrix; 58 solutions, all orthogonal, matching the 58 kernel extensions; 177 idempotents with B A = 0, of which 119 have a non-idempotent third factor; 58 stage-0 embeddings equal to I + x E_0. The explicit triple (A, 0, I + A) is also multiplied by hand in PROOF.md. No Lean, Docker, native, or selected-frontier replay was run.

Credit

The question is open question 2 of PR #7027, research/paired-star-valuation-one-factor, head 1ba2ac5adfe58fbe9dba34d50d4e5d30633ce3d0. That package classified the E_11 slice and proved the orthogonal product identity. The paired-star coordinates and E_0 are from PR #5747 (Ali Romman), as recorded in research/paired-star-three-full-calls. The T-commuting projector obstruction is PR #6956. Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed. Apache-2.0.

Open next questions

• For this same idempotent A, which of the 65,536 matrices B, not necessarily idempotent, satisfy (I + x A)(I + x B)(I + x (I + A + B)) = I + x I_4? The linear coefficient forces the third factor for every B. The counts 58 and 177 classify only idempotent middle factors. The first step is to let the 16-bit integer run from 0 through 65535, set the third factor equal to I + A + B, and test the product. Nothing external blocks that scan. One rejected B is not a classification. This is not the E_11 slice, not the constant-R insertion, and not a g = 13 or g = 31 word.

Workflow feedback (optional)

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