[unpaidmoon]

Change and mathematical scope

Over F_2, insert one matrix R beside the stage-0 gate I + x E_11 in the nine diagonal gates of PR #7027. The scalar e is central and e^2 = 0. The pure product remains product_j (I + x_j E_j) for every such insertion.

No matrix R with entries in F_2[x, y, z] makes the coefficient of e equal to I. The obstruction is diagonal: that coefficient's i-th diagonal entry is R_ii times the corresponding diagonal entry of the pure product, and each of those pure entries is a nonunit.

The coupled31 operator (I + e I) times the pure product is a different equation. For this one slot it holds if and only if R = I. That is a witness for the operator identity. It is not a T-commuting full-call word, and it does not make the coefficient equal to I.

Every one of the 65,536 constant matrices supported on the stage-0 block fails both equations. This is a proof for the polynomial statement and an exhaustive finite census for the stage-supported constant matrices. It is not a truncated search and not a kappa improvement. The selected frontier stays 781451/250000000.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at session start and again before submission; it had not moved). Head: e4ddcd2fde8c46b7e528e8806ab72d9bf7690978 on fork branch cursor/paired-star-constant-r-0aa2.

Task card: joint-query-controller, scope refined. The card asks whether a changed query demand removes real work. The actual question is the first open question of PR #7027.

Overlap was refreshed at 2026-11-10T10:05:04Z with python3 scripts/find_overlap.py. The question is open in PR #7027 and in #5593 (comment). The handoff pointer is #1166 (reply in thread). No open PR solves the insertion. The unsplit two-map word, the other-idempotent census, and the g=13 and g=31 gaps were not taken. PR #6956 was not rerun.

Validation

Python 3.12, standard library only.

python3 -B scripts/research_package.py --package research/paired-star-constant-r-insertion

Both the ordinary run and the -O run print PASS_PAIRED_STAR_CONSTANT_R_INSERTION, then PASS_REGISTERED_RESEARCH_PACKAGE with outcome negative-result. Checks raise SystemExit, so optimized mode does not drop them. The direct command is python3 -B research/paired-star-constant-r-insertion/verify.py.

Checked counts: local coefficient I_4 has 0 hits among 65,536 constant matrices, on either side of E_11; the local pure target (1 + x) I_4 has exactly one hit, R = I_4; the stage-supported embedding has 0 hits for coefficient I_8 and 0 hits for the pure product, on either side. All 64 matrix units were multiplied directly. 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 is PR #6956, research/paired-star-three-full-calls. The valuation-1 triple and the question answered here are PR #7027, head 1ba2ac5adfe58fbe9dba34d50d4e5d30633ce3d0. 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 two constant matrices, placed in two different gaps of this same diagonal word, make the coefficient of e equal to the pure product? A single gap admits only R = I for that equation, but two gaps add and the off-diagonal scales depend on the gap. Any finite number of constant insertions already fails to make the coefficient equal to I. The first step is the off-diagonal system for insertions immediately after stage-0 E_11 and stage-0 E_22.
• For the non-idempotent leading factor N exhibited in PR Record a valuation-1 factor outside the paired-star T-commutant #7027, does one constant matrix beside I + x N make the stage-0 coefficient of e equal to I, or the eight-dimensional coefficient equal to the pure product? The diagonal identity does not apply. The first step is to expand (I + x N)(I + e R) and multiply by the other two non-idempotent factors.

Workflow feedback (optional)

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