[unpaidmoon]

Change and mathematical scope

Proof, for one balanced placement. In the paired-star contract at g=13 and q=8, subtract a constant matrix supported on the two leaf-0/leaf-1 blocks from the internal sum, add that same matrix after both halves of stage 0, and keep every later stage an adjacent pair. The total thin sum stays the swap K forced by every D_j=1. For the exhibited legal pair, the leaf-0 ← leaf-1 block equals

D_1^{-1} (D_0+1) (B + (A_Q+B) C)

and is nonzero whenever D_0 ≠ 1. The two linear functionals φ(v)=v_2+v_9 and ψ(v)=v_0+v_8 are the obstructions. The same schedule with any one of the five named rank-8 bundles moved into that gap leaves the nonzero block D_t^{-1}(D_0+1) B π_{S^c}.

A different rank-8 coordinate map, on coordinates {0, 2, 8, 9}, clears every leaf-to-leaf block while the D_j=1 sum stays K. That is a witness for the block only. At the two tested points the defect has rank 32, so no second balanced matrix of rank at most 8 repairs the endpoint there.

This is not an exhaustive census of rank-8 supports. A failure of the leaf-supported insertion is not unrestricted nonexistence of a cancelled block. The coordinate witness is not an endpoint and is not one of the five bundles.

No new kappa. The selected frontier stays 781451/250000000 = 0.003125804 from PR #6030.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at submission and unchanged). Fork main is a different history, so this draft uses cursor/crocswap-main-17469fff-1751, which is that same commit, as its base. The upstream target is CrocSwap main.

Head 97bc92e76e5954f46befbe5c6790b6f2107e0d3a. Package research/paired-star-g13-after-stage0-block.

Task card joint-query-controller, scope refined. The assigned question is open next question 1 of PR #7026.

Nearest overlap, refreshed 2026-11-10 (python3 scripts/find_overlap.py --refresh, 339 open PRs, 7013 discussion records, local inventory complete):

• Open PR Prove one internal gap misses the g=13, q=8 paired star. #7026, head 99b5ed6d80ca3d97cf5f7176bf8b3557d67662bb, proves the internal gap and asks this after-stage-0 question. It does not evaluate the split. Canonical note: discussion Common advancing stages and rational full-query callers #5593, comment 19514179. Handoff pointer: Handoffs and requests for independent checks #1166, comment 19514180.
• Open PR Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010, head 32999d634cdd718ce6fb77648f54e4e674d7f395, proves the adjacent-halves count. No thin map stands between the halves there.
• Open PR Prove consecutive paired-star words need thin rank at least g q. #7005, head 1750566c34c9fb117dc4afc4ce2bdacca7da036b, is the g=31, q=26 consecutive bound. This package does not enter it.
• Open PR Record that three F2 thin maps meet total rank 6 on the three-leaf paired star #7011, head 9a63d6d99158f5ced1058df403d3e3b9847fcc17, is a three-leaf F_2 census.

No open PR in that snapshot evaluates this after-stage-0 split at g=13, q=8.

Validation

Python 3.12, no third-party packages.

python3 -B scripts/research_package.py --package research/paired-star-g13-after-stage0-block

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome negative-result. Both the plain replay and the python3 -O -B replay print PASS_PAIRED_STAR_G13_AFTER_STAGE0_BLOCK. Checks raise, so -O does not strip them.

scripts/check_research_commit.py --package research/paired-star-g13-after-stage0-block --check-source-history --check-modes passed at 97bc92e76e5954f46befbe5c6790b6f2107e0d3a.

The checker confirms the legal pair, the two column-space functionals, agreement of the fast updater with dense conjugation on a 3-leaf split word, the leaf-supported formula, the five bundles, the coordinate witness, the D_j=1 swap, and defect rank 32 at the two field points in both half orders.

Not run: a search over other legal blocks B, a native or supplier replay, a proof assistant, and scripts/research_selected_baseline.py. The selected record is provenance only.

Credit

The paired-star contract and the chronological rank floor are from PR #5747 (Ali Romman), including PR #5673, PR #5631, and PR #5617. The internal gap that asks this question is PR #7026. The adjacent-halves count is PR #7010. The consecutive bound at the other profile is PR #7005.

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

Open next questions

• For a legal off-diagonal B other than E_{0,8}+E_{9,2}, with the same A=diag(I_8, 0), does (A+B)C=B or (A'+B)C=B have a solution? Keep A fixed, choose one other matrix B with AB+BA=B and B nonzero, and test the column spaces of A+B and A'+B.
• Does one balanced constant matrix of rank 16 clear the coordinate witness on {0, 2, 8, 9} at D=(2, 3, ..., 14)? Balance the rank-16 coordinate map on {1, 3, 4, 5, 6, 7, 10, 11} across the same gap and recompute the defect rank at that point in the order A-then-A. One matrix is not a classification.

A stage that is not first, and the g=31, q=26 internal gap, are different questions.

Workflow feedback (optional)

The useful split was one declared gap and the matrices fixed by the D_j=1 sum. The registered replay finishes in about twenty seconds.

Feedback discussion: #1200