[unpaidmoon]

In the nine diagonal gates of the PR CrocSwap#7027 word, constant F_2 matrices in the gap after stage-1 E_22 and the gap after stage-2 E_11 make the coefficient of e equal the pure product for exactly 2^30 pairs. The matched positions are the off-diagonal pairs inside {4, 5} and {1, 2, 3, 6, 7}. Every such coefficient has diagonal entries in {0, P_ii}, and none equals I. No kappa change. Selected κ stays 363005817/100000000000.

Change and mathematical scope

Proof, in F_2[x, y, z][e]/(e^2), of the constant solution set for those two slots. The middle diagonal is diag(1+z, 1, 1, 1, 1+y, 1+y, 1, 1). The count 2^30 agrees with PR CrocSwap#7215, and the solution set does not: (1, 4) is free there and blocked here, while (1, 2) is blocked there and free here. This is not a caller, a native bill, a word for g=13 or g=31, or a selected-frontier change. Polynomial entries and every other slot pair remain open.

Task and overlap

Base: canonical main e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8. Live selected record certificates/selected-result.json is 363005817/100000000000. current-priorities.md on that base still prints 0.00310045; this package does not edit it.

Card: joint-query-controller, scope changed. The card asks whether a changed query demand removes real work. This package takes the first open question of PR CrocSwap#7215, which is the constant classification of the two later gaps in the same nine-gate word.

Ordered eligible list for session r270, after excluding structured CrocSwap#1166 handoffs whose next tasks are already delivered and excluding open-next bullets that a later unpaidmoon PR answers or refines:

1. PR Prove the weight-1 no-edge image bound is 6 at pattern dimension 4. CrocSwap/integer-mult-bounds#7138: a heavy-covector no-edge pool of pattern dimension 4 acting irreducibly in dimension 6 or more.
2. PR Record a length-7 support witness for three old signed frames. CrocSwap/integer-mult-bounds#7214: another support solution of R+S+T+U = 4I-E-F-G with at least four odd-c labels. PR Prove a no-edge pattern-dimension-4 pool has total image at most 6. CrocSwap/integer-mult-bounds#7139's first two bullets are answered by Exhibit an irreducible no-edge image of dimension 7 at pattern dimens… CrocSwap/integer-mult-bounds#7237 and Show two dimension-6 no-edge pools outside the three printed coordina… CrocSwap/integer-mult-bounds#7221, and its third bullet only points back at question 1 of Prove the no-edge irreducible joint-image bound is 3. CrocSwap/integer-mult-bounds#6921.
3. PR Classify constant insertions after stage-0 E_22 and stage-1 E_11. CrocSwap/integer-mult-bounds#7215, first open question: constant F_2 pairs after stage-1 E_22 and stage-2 E_11. This package.

Nearest records: PR CrocSwap#7215 classifies the previous pair and leaves this one. PR CrocSwap#7103 classifies the adjacent stage-0 pair. PR CrocSwap#7191 classifies polynomial entries in that stage-0 pair. PR CrocSwap#7039 is the one-insertion diagonal obstruction. GitHub search on 2026-11-11 found no PR whose title states these two slots. The degree-at-most-1 question left by CrocSwap#7215 for its own slots is not taken.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B scripts/research_package.py --package research/stage1-e22-stage2-e11-gaps

Both the normal and -O commands print PASS_STAGE1_E22_STAGE2_E11_GAPS, with matched_off_diagonal 22, blocked_off_diagonal 34, solution_pairs 2^30, witness_both_nonzero 3, and coefficient_equals_I 0. The registered metadata check passed. The 2^30 matrices are not enumerated: each of the 64 positions has a four-element bit equation, solved exhaustively, and the equations separate. Controls reject (1, 4), an unequal matched bit, a mixed matched-plus-blocked pair, and (I, I). Predecessor replays, Lean, Docker, and the selected cold replay were not run.

Credit

The paired-star contract is PR CrocSwap#5747 (Ali Romman), with PR CrocSwap#5673, PR CrocSwap#5631, and PR CrocSwap#5617. The nine-gate word is PR CrocSwap#7027. The one-insertion obstruction is PR CrocSwap#7039. The adjacent stage-0 classification is PR CrocSwap#7103. Polynomial entries in that pair are PR CrocSwap#7191. The question taken here is the first open question of PR CrocSwap#7215, head e1656a671fffddcd0b9ad07d500825686ff852ae.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed. Apache-2.0.

Open next questions

1. Do degree-at-most-1 entries in these same two gaps give solutions of C = P beyond the 2^30 constant pairs? At the blocked position (1, 4), R_{1,4} = 1 and S_{1,4} = 1+y solves that one entry. First step: keep a legal constant diagonal and legal matched bits, substitute that pair, and multiply against P. Nothing external blocks it. This is not the polynomial question PR Classify polynomial two-gap insertions with exact dimension 64d+50 (no κ) CrocSwap/integer-mult-bounds#7191 answered for the stage-0 gaps, and not the same question PR Classify constant insertions after stage-0 E_22 and stage-1 E_11. CrocSwap/integer-mult-bounds#7215 left for its own slots.
2. Which constant pairs make C = P after stage-2 E_22 and after the stage-2 leaf gate? The E_22-then-next-E_11 series stops at stage 2. First step: the diagonal of the single gate between those slots.

Workflow feedback (optional)

CrocSwap#1200