[unpaidmoon]

Change and mathematical scope

Proof, for one schedule family. On the paired-star contract, put the stage blocks in order 0, ..., g-1 and leave thin maps in the gap between the two halves of every stage. Other thin maps may sit only in the gaps before, between, and after those blocks. The squared stage monomial must vanish. The surviving degree-1 core-to-leaf block is a sum of two sandwiches, so its rank is at most twice the rank of the internal map. Counting the leaves each map can meet gives

total thin rank >= q g / 2.

At g=31, q=26 the bound is 403. Five thin maps of rank 26 have total rank 130, so this schedule has no endpoint. Either half may be applied first, and the off-diagonal blocks of B stay free.

This is a proof in the projector algebra plus an exhaustive comparison of monomial exponents. The separate q=1 rank comparison is an exhaustive census of 4 by 4 matrices over F_2: under P_s Z P_f = 0, the degree-1 block never outranks Z on that domain, while one matrix of rank 1 has degree-1 rank 2 when the square is allowed. The bound 403 uses the proved factor 2 and does not use the census. This is not a heuristic search, not a census of words, and not unrestricted nonexistence of the g=31, q=26 profile. No witness is constructed.

No new kappa. The selected frontier stays 781451/250000000 = 0.003125804. This note does not replace the one-gap bound q(g-3) = 728 of open PR #7025, and it does not replace the consecutive bound q g = 806 of open PR #7005.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840 (upstream CrocSwap/integer-mult-bounds main, re-fetched at session start and before submission; both reads were this commit). Package: research/g31-q26-every-stage-gap. Head: d176f84f39ff5678b2bd3dabb53c47099a31584b.

Catalogue card joint-query-controller, priority revision 8. Scope status: changed. The card asks about a changed query demand. The assigned question is open question 1 of open PR #7025.

Overlap: local notes at the base commit, 339 open PR records, and 7013 discussion records, fetched 2026-11-10. The every-stage phrase search matched only the #7025 finding that leaves this schedule open, plus this package.

• Open PR Prove one internal paired-star gap still needs thin rank at least q(g… #7025 (ec8ab41023fcf244e2bf790b13e829127d2e08e4): one internal gap, in at most one stage, needs total thin rank at least q(g-3) = 728. This package starts from the degree-1 block that note names and does not rerun that theorem.
• Open PR Prove consecutive paired-star words need thin rank at least g q. #7005: consecutive halves and no internal thin map need total rank at least q g = 806. This package does not rerun that theorem.
• Nested or crossing stage intervals are a different schedule and were not taken.
• The g=3, q=1 word in this package is an assembly control. The q=1 matrix census is not an input to the bound 403.

No open PR found that rules out total rank 130 for an internal gap in every stage.

Validation

Python 3.12, standard library only:

python3 scripts/research_package.py --package research/g31-q26-every-stage-gap --check-only
python3.12 scripts/research_package.py --package research/g31-q26-every-stage-gap

Registered replay passed: PASS_G31_Q26_EVERY_STAGE_GAP with lower_total_rank=403 and profile_total_rank=130. scripts/check_research_commit.py --check-source-history passed at d176f84f39ff5678b2bd3dabb53c47099a31584b.

The checker proves the scalar stage factor, both core directions of two adjacent halves, the identity of the degree-1 block with the two sandwiches, and the squared coefficient P_s Z P_f. Dropping a diagonal block of B makes the pure product fail. A direct g=3, q=1 word matches the predicted core-to-leaf polynomial for every stage, both half orders, and every B12, B21 in F_2; two maps add, and the zero map does not. With B = 0, the leaf-to-core identity in every stage cancels every core-to-leaf block. At q=26 that same matrix has rank 52, square zero, and degree-1 block I_52. The exponent weights are exhaustive at g=31. Those checks raise SystemExit, and the same replay passed under python3 -O.

Not run: a search over nested or crossed placements, a proof that the square condition improves the factor from 2 to 1 for every q, the leaf-to-core rank count, native transport, Lean, Docker, and a selected-frontier replay. make frontier6030-verify was not run.

Credit

The paired-star contract, the chronological rank floor, and the commuting map-count are from PR #5747 (Ali Romman), including PR #5673, PR #5631, and PR #5617. Cinna Davis supplied the controller/simplex lineage those packages retain. Open PR #7025 asked for this every-stage count and proved the one-gap bound. Open PR #7005 proved the consecutive bound.

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

Open next questions

Linked from research/g31-q26-every-stage-gap/TASK.json:

• Under P_s Z P_f = 0, is the own-leaf degree-1 core-to-leaf block of rank at most rank(Z) for every half width q? The proved estimate is twice that rank. The exhaustive q=1 census finds no square-free excess. A general factor of 1 would raise this bound from q g / 2 to ceil(2 q g / 3). First step: show that the images of pi_core Z P_f and pi_core P_s Z lie in one common subspace of dimension rank(Z).
• Do the leaf-to-core blocks of the same every-stage word force total thin rank strictly above q g / 2? The bound 403 uses only the core-to-leaf side. First step: read the leaf-to-core block of the surviving degree-1 term under P_s Z P_f = 0, then repeat the weight count.

Workflow feedback

docs/research/current-priorities.md on this main is still labelled priority revision 8 and quotes an older kappa. The selected record 781451/250000000 is the one used here. Fork main is not upstream main, so this draft is based on a pin of 17469fff78c22941280f04d39dc5a1faee09c840 rather than on fork main. Feedback discussion: #1200