[unpaidmoon]

Change and mathematical scope

Proof of one pure-product step, plus one pure-product witness. On the paired-star contract, the pure product of one nested pair of stage intervals equals the pure target if and only if the outer off-diagonal block B is zero. No commutation hypothesis is required, and the inner block may be nonzero. If that outer block is nonzero, the coefficient of x y on the core-to-leaf block is that matrix. Thin maps do not appear in the pure part, so they cannot repair the failure. The failure survives any number of adjacent stages placed entirely outside the nest. At g=31, q=26 this schedule has no endpoint.

With both off-diagonal blocks zero, a crossed pair equals the pure target if and only if the two idempotents commute. That commuting case remains the separate ceil(g/2)=16 map-count; this note does not replace it.

There is an explicit equal-half witness, the direct sum of thirteen F_2^4 seeds, with core dimension 52 and half-rank 26, whose projectors do not commute and whose crossed pure product equals the target. Concatenating a noncommuting nest with outer B=0 and this cross also matches, as does one adjacent stage on either side. This is a witness for the pure product only. The coefficient of e was not computed, so it is not an endpoint witness and not a construction of the profile.

No new kappa. The selected frontier stays 781451/250000000 = 0.003125804. Five rank-26 maps still have total rank 130; that number is not a new lower bound. A pure-product failure is scoped to the stated schedule. A pure-product match is not existence of the endpoint.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840 (upstream CrocSwap/integer-mult-bounds main). Package: research/g31-q26-nested-crossed-pure. Head: d346ad3a0e38d474ab401295bc20713fe9f1c000.

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 2 of PR #7005: the pure product of one crossed pair and one nested pair, before any q=26 search.

Overlap: local notes, 317 open PR records, and 6972 discussion records were searched at 2026-11-10; no match for this product. Nearest work:

• Open PR Prove consecutive paired-star words need thin rank at least g q. #7005 proves the consecutive-halves rank bound and leaves nested and crossed intervals open. This note answers that pure-product question and does not use the rank bound.
• research/coupled31-chronology (PR Exclude the 31-leaf target with a chronological connector rank bound #5747) is the contract, the chronological floor, and the commuting map-count. Commutation there forces B=0 and needs 16 thin maps. The crossed B=0 locus here is consistent with that count and does not replace it.
• Open PR Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010 is the adjacent-halves map-count. Words with a call between the two halves stay outside it. This note does not retake that count.
• The internal-gap rank question and the interleaved five-map profiles are different questions and were not computed.

No open PR found that evaluates this nested or crossed pure product.

Validation

Python 3.12, standard library only:

python3 scripts/research_package.py --package research/g31-q26-nested-crossed-pure --check-only
python3.12 scripts/research_package.py --package research/g31-q26-nested-crossed-pure

Registered replay passed: PASS_G31_Q26_NESTED_CROSSED_PURE with endpoint=open and kappa=781451/250000000. scripts/check_research_commit.py --check-modes passed at d346ad3a0e38d474ab401295bc20713fe9f1c000. The source-history ancestry check was not run: this checkout does not have full Git history. Blob pins were checked against the base commit.

The checker expands the projector algebra and compares the nested and crossed errors with the closed forms. Dropping AB+BA=B makes the one-stage factor fail. It then multiplies the explicit q=26 matrices: the nested obstruction, including a nilpotent outer B; the noncommuting crossed witness; their concatenation; and one adjacent stage with nonzero B on either side of a bad nest and of the compatible product. A dual-number thin map changes the coefficient of e and not the pure part.

Not run: a search over the coefficient of e, a classification of crossed pairs with arbitrary B, a census of call orders, native transport, Lean, Docker, and a selected-frontier replay. make frontier6030-verify was not run.

Credit

The paired-star contract 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 posed this pure-product question and proved the separate consecutive-halves rank bound.

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

Open next questions

Linked from research/g31-q26-nested-crossed-pure/TASK.json:

• For a nest with outer B=0, or for the crossed witness, with every other stage adjacent and outside that block, can thin maps of total rank 130 cancel the coefficient of e at g=31, q=26? First step: that core-to-leaf coefficient after dividing by the pure scalar product. The pure match does not answer it.
• Which crossed pairs with a nonzero off-diagonal block match the pure target, beyond the direct-sum witness? The B=0 locus is the commutator. First step: the core-core coefficient of x y for general B.

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