[unpaidmoon]

Change and mathematical scope

Proof, for one schedule family. On the paired-star contract, keep the two full calls of every stage adjacent, and allow thin maps in at most one gap between the halves of a single stage. The core-to-leaf coefficient of that gap is quadratic in the stage variable. In the eigenbasis of the half idempotent, the off-diagonal blocks of B stay free, and the coefficient meets the required monomial of at most three leaves. Every other leaf still forces

total thin rank >= q (g-3).

At g=31, q=26 the bound is 728. Five thin maps of rank 26 have total rank 130, so this schedule has no endpoint. The projectors need not commute.

This is a proof in the projector algebra plus an exhaustive comparison of square-free monomials. It is not a finite census of words or projectors, not a heuristic search, and not unrestricted nonexistence of the g=31, q=26 profile. A second internal gap, one gap in every stage, nested intervals, and crossed intervals stay open. No witness is constructed.

No new kappa. The selected frontier stays 781451/250000000 = 0.003125804. The older chronological floor 5q >= 4q+20 still passes at q=26. This note does not replace the consecutive bound q g = 806 of open PR #7005, and it does not replace the adjacent-halves map count of open PR #7010.

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-internal-gap. Head: ec8ab41023fcf244e2bf790b13e829127d2e08e4.

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 #7005, restricted to one internal thin gap.

Overlap: local notes at the base commit, 317 open PR records, and 6973 discussion records, fetched 2026-11-10. The phrase search returned no matches; the neighbours below were read directly.

• Open PR Prove consecutive paired-star words need thin rank at least g q. #7005 (1750566c34c9fb117dc4afc4ce2bdacca7da036b): consecutive halves and no internal thin map need total rank at least q g = 806. This package starts from the coefficient that PR leaves uncomputed. It does not rerun that theorem.
• Open PR Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010 (32999d634cdd718ce6fb77648f54e4e674d7f395): adjacent halves with the internal slot empty need ceil(g/2) thin maps. A call between the halves is outside that hypothesis. This package treats one thin map in that slot and does not take the interleaved schedule.
• Nested or crossing stage intervals are a different question and were not taken.
• The g=3 finite censuses are not inputs and were not re-run. The g=3, q=1 word in this package is a conjugation control.

No open PR found that computes this internal-gap coefficient or rules out total rank 130 for this schedule.

Validation

Python 3.12, standard library only:

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

Registered replay passed: PASS_G31_Q26_INTERNAL_GAP with lower_total_rank=728 and profile_total_rank=130. scripts/check_research_commit.py --check-source-history passed at ec8ab41023fcf244e2bf790b13e829127d2e08e4.

The checker proves the eigenbasis form of B, both half orders, the scalar stage factor, and the closed form of VZU with noncommuting letters. Dropping the off-diagonal solution makes the pure product fail. A direct g=3, q=1 word with nonzero B matches the conjugated coefficient at every stage and both orders; two maps in the gap add, and the zero map does not. The exponent comparison and the gap count are exhaustive for every stage at g=31. Those checks raise SystemExit, and the same replay passed under python3 -O.

Not run: a search over two or more internal gaps, a census at q=26, nested or crossed placements, 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 #7005 asked for this coefficient. Open PR #7010 left the internal slot outside its map-count theorem.

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

Open next questions

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

• Can one internal thin gap in every stage meet the g=31, q=26 endpoint with total thin rank 130? One gap leaves at least g-3 leaves to the external count. First step: impose P_s Z P_f = 0 from the squared monomial at one stage, then read the core-to-leaf block of the surviving degree-1 term.
• Does the internal map's contribution on its three leaves raise the external bound above q(g-3)? First step: bound the sum of the ranks of the three core blocks of one matrix of rank at most q, using P_s Z P_f = 0.

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