[unpaidmoon]

Change and mathematical scope

Proof, not a κ change. On the paired-star one-gap schedule of open PR CrocSwap#7025, one internal matrix Z of rank at most q cannot cancel the external remainder on its three helped leaves.

A successful word needs P_s Z P_f = 0: the degree-2 part of the internal product is that operator, supported on the stage, and the full-call bracket between the halves has no square. Under that equation each of the three core blocks has rank at most q, and their ranks sum to at most 3q. Each helped leaf therefore still demands external rank at least q, and the heavier of the even and odd demand sums is at least q(g-1) for every gap placement.

At g=31, q=26 this is external rank at least 780, which is 2q = 52 above the dropped-leaf count q(g-3) = 728. The inequality system is sharp at stage 1: an explicit gap tuple meets every leaf demand at total q(g-1), and lowering any positive gap breaks a leaf. That tuple is not a word.

The selected frontier stays κ = 781451/250000000 = 0.003125804 (PR CrocSwap#6030). This does not construct a word of external rank 780, does not cover several rank-at-most-q maps in the same gap, and does not replace the consecutive-rank theorem of CrocSwap#7005.

Task and overlap

• Base: 17469fff78c22941280f04d39dc5a1faee09c840 (upstream CrocSwap/integer-mult-bounds main, re-fetched before push; unchanged).
• Package: research/paired-star-internal-three-blocks.
• Task card: joint-query-controller, catalogue priority revision 8, scope_status: changed. The card premise is not this schedule; the assigned question is the three-leaf internal contribution of Prove one internal paired-star gap still needs thin rank at least q(g… CrocSwap/integer-mult-bounds#7025.
• Neighbor: Prove one internal paired-star gap still needs thin rank at least q(g… CrocSwap/integer-mult-bounds#7025 (ec8ab41023fcf244e2bf790b13e829127d2e08e4, research/g31-q26-internal-gap) gives external count q(g-3) after one gap leaves at least g-3 leaves outside. This package starts from that count and bounds the three core blocks.
• Not redone: the one-gap count of Prove one internal paired-star gap still needs thin rank at least q(g… CrocSwap/integer-mult-bounds#7025 and the consecutive-rank theorem of Prove consecutive paired-star words need thin rank at least g q. CrocSwap/integer-mult-bounds#7005. Not taken: every-stage rank, g=13 remainders, interleaved or nested/crossing schedules.
• Source pins are files at the base commit only. No external head pin. Provenance: certificates/selected-result.json, docs/research/agent-task-catalog.json, research/coupled31-chronology/PROOF.md.
• docs/research/current-priorities.md is still labelled revision 8 and quotes an older κ. The selected record above is the one used.
• Fork main is f9701fe9b, not upstream main. This draft is based on the pin cursor/upstream-17469fff-base-6479 at exactly 17469fff so the diff is only the new package.

Validation

Evidence class: proof over any characteristic-2 field, for one matrix of rank at most q. The q=1 census is an exhaustive check of that 8-by-8 matrix domain (all rank-at-most-1 matrices, all scalar (B12, B21), both half orders). It is not a census of words. The explicit family and the gap-tuple check are inequality witnesses, not words.

python3 -B scripts/research_package.py --package research/paired-star-internal-three-blocks

Python 3.12, no dependencies. Both registered commands (python -B and python -O -B) print:

PASS_INTERNAL_THREE_BLOCKS proof g=31 q=26 d=52 external_rank=780 dropped_leaf_rank=728 rank_sum_cap=78 q1_attained=18720

scripts/check_research_commit.py --ref HEAD reports PASS_RESEARCH_COMMIT_BYTES for the six package files. --check-source-history was not completed: this checkout is shallow. The declared pins are the parent commit 17469fff, and there is no external head pin.

Failure control: mutating the stage-1 equality tuple by decrementing one positive gap makes a leaf inequality fail, and the checker rejects that mutation.

Credit

The one-gap schedule, the three blocks, and the count q(g-3) are from open PR CrocSwap#7025. The consecutive-rank theorem is CrocSwap#7005. Earlier paired-star work includes PR CrocSwap#5747 (Ali Romman) and CrocSwap#5673 / CrocSwap#5631 / CrocSwap#5617 (Cinna Davis).

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Two rank-at-most-q matrices in the same gap, each with P_s Z P_f = 0: can every helped core block sum to rank 2q? First step: add two copies of the rank-q witness on complementary core coordinates. If they do, Prove one internal paired-star gap still needs thin rank at least q(g… CrocSwap/integer-mult-bounds#7025's q(g-3) still applies and this lift does not.
2. Does a gap tuple of total q(g-1) plus one rank-q internal matrix meet I_d on every core-to-leaf block at g=31, q=26? First step: stage 1, place the witness, and solve the even external gaps for a complement. This is not the every-stage question.

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CrocSwap#1200