[unpaidmoon]

Change and mathematical scope

Proof, for one legal block. Keep A = diag(I_8, 0) and replace the named off-diagonal block E_{0,8}+E_{9,2} by B = E_{0,8}+E_{8,0}. Over every field of characteristic 2,

(A+B) C = B,    C  = E_{0,0} + E_{8,0} + E_{8,8},
(A'+B) C' = B,  C' = E_{0,0} + E_{0,8} + E_{8,8},

with A' = I+A. In the after-stage-0 split, set the leaf-1 → leaf-0 block of the balanced correction to C or to C' according to which half stands second. The leaf-0 ← leaf-1 block of the normalized coefficient is then the zero matrix for every invertible specialization. The opposite block E does not enter that identity.

This is one named B, not a classification of legal blocks, and not an endpoint. At the tested F_4 points with E = 0, the opposite leaf block is nonzero.

No new kappa. The selected frontier stays 363005817/100000000000.

Task and overlap

Base commit e54523517d0f03e58352e647c403a5af3e525a57 (canonical CrocSwap main, re-fetched at submission and unchanged). Fork main is a different history, so this draft uses cursor/upstream-e54523517-pin-8cb4, which is that same commit, as its base. The upstream target is CrocSwap main.

Head 4e09248ca3a258ce7a204ec44e554ee3bf3a32a5. Package research/paired-star-transpose-corner.

Task card joint-query-controller, scope refined. The assigned question is open question 1 of PR CrocSwap#7097.

Nearest overlap:

• Open PR Prove an after-stage-0 leaf correction misses the g=13 block. CrocSwap/integer-mult-bounds#7097, head 97bc92e76e5954f46befbe5c6790b6f2107e0d3a, proves the miss for B_0 = E_{0,8}+E_{9,2} and asks this question. The new block differs in positions (8,0) and (9,2). Section 3 repeats the named obstruction as a control.
• GitHub issue search for E_{0,8} and for E_{9,2} each returns only Prove an after-stage-0 leaf correction misses the g=13 block. CrocSwap/integer-mult-bounds#7097. A discussion search for the column-space question returned no items. Local find_overlap.py --offline searched 10180 notes at this head, with a complete inventory, and reported no match; the remote overlap snapshot was not loaded.
• Open PR Prove one internal gap misses the g=13, q=8 paired star. CrocSwap/integer-mult-bounds#7026 is the internal gap. Open PR Prove the g=13 leaf-2 block stays nonzero when stage 0 is not first. CrocSwap/integer-mult-bounds#7096 is the stage-0-not-first schedule. This package keeps stage 0 first.
• Open PR Prove that adjacent paired-star halves need ceil(g/2) thin maps. CrocSwap/integer-mult-bounds#7010 is the adjacent-halves count. A thin map stands between the halves here.

Validation

Python 3.12, no third-party packages.

python3 -B scripts/research_package.py --package research/paired-star-transpose-corner

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

scripts/check_research_commit.py --package research/paired-star-transpose-corner --check-source-history --check-modes passed at 4e09248ca3a258ce7a204ec44e554ee3bf3a32a5.

The checker confirms the legal pair, both explicit solutions, the integer entry 2 that witnesses characteristic 2, the two column-space functionals for the named pair, the half relations, and the after-stage-0 word on the core plus leaves 0, 1, and one spectator. At both values in F_4, both choices of the second stage-0 half, and both later-pair orders, the matching block is zero and the empty correction reproduces D_1^{-1}(D_0+1) B.

Not run: a classification of every legal B, a search over the opposite block E, 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 CrocSwap#5747 (Ali Romman), including PR CrocSwap#5673, PR CrocSwap#5631, and PR CrocSwap#5617. The internal gap is PR CrocSwap#7026. The after-stage-0 question and the named obstruction are PR CrocSwap#7097.

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

Open next questions

• For this same B, with the matching forward block, does any constant opposite block E make leaf 1 ← leaf 0 zero? At the tested F_4 points with E = 0 that opposite block is nonzero. The first step is E equal to the transpose of C, at D_0 = t and one half order. One matrix is not S = I.
• Does B = E_{1,9}+E_{9,1} satisfy the same two equations for this fixed A? The first step is the same three-entry shape on coordinates 1 and 9. One further matrix is not a classification of the blocks [[0, Q], [R, 0]].

Workflow feedback (optional)

The useful step was one named transposition and the two column spaces. The registered replay finishes in under a second.

Feedback discussion: CrocSwap#1200