[unpaidmoon]

Change and mathematical scope

Proof, under a stated schedule hypothesis. In the paired-star contract, a word whose two full calls of every stage are adjacent needs at least ceil(g/2) thin maps. The projectors need not commute, the field may be any characteristic-2 field, and the thin maps may have any rank.

At g=13 the bound is 7, so five thin maps cannot realize the g=13, q=8 profile inside this schedule. Five rank-q maps meet the paired rank floor of 40 and still fail when the halves stay adjacent. The same count shows five adjacent-halves maps are too few at g=31 as well. A word that places a call between the two halves of a stage is outside the hypothesis and remains open at both profiles.

The sharp consecutive-halves word from research/coupled31-chronology/COMMUTING.md still meets ceil(g/2) for every legal off-diagonal block B, including B for which projectors of different stages do not commute.

This is not a construction of the profile, not a native caller, and not a kappa improvement. Selected κ stays 781451/250000000 from PR #6030. A failed search is not used as nonexistence. The g=3 censuses do not lift.

Task and overlap

Base commit 7c49fe32c55de02c2c873655c804b16e4c3fef0a (canonical CrocSwap main). This fork draft uses cursor/crocswap-main-7c49fe32-1dbb as its base because that branch is the same commit; the fork's own main is older. The upstream target is CrocSwap main.

Task card joint-query-controller, scope refined. The assigned question is the unconstructed g=13, q=8 profile from discussion #5593 and the open questions of PR #5747: whether that profile exists when full projectors need not commute. The result answers the consecutive-halves subcase.

Nearest overlap:

• PR Exclude the 31-leaf target with a chronological connector rank bound #5747 is the contract, the rank floor, and the commuting map-count. Commutation there forces B=0. This package drops that hypothesis and requires adjacent halves instead.
• Open PR Record that one F2 thin map misses the three-leaf paired star. #6922 is the F2 three-leaf one-thin census. It leaves g=13, q=8 and g=31, q=26 open.
• Open PR Record that two F2 thin maps on the three-leaf paired star need total rank 8 #6940 is the F2 three-leaf two-map rank bound. It also leaves both profiles open and says the three-leaf result does not lift.

Those censuses fix g=3, block size 2, and entries in F2. They are not a proof at thirteen leaves or eight ports. GitHub search is not a full-text proof that no other note mentions the same schedule hypothesis.

Validation

Python 3.12, no third-party packages.

python3 -B scripts/research_package.py --package research/paired-star-g13-q8-five-map

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome partial. Both the python3 -I -B replay and the python3 -I -O -B replay print PASS_PAIRED_STAR_G13_Q8_CONSECUTIVE. Checks raise, so -O does not strip them.

The g≤5 grids on {2,4,8} are complete polynomial checks for those words. The g=13 run checks the interpreter against the Laurent formula at two points; the general identity is the written proof. Deleting one map from the sharp word misses the endpoint. The chosen nonzero B has stage projectors that do not commute.

Not run: a proof assistant, a search for an interleaved word, a native or supplier replay, and scripts/research_selected_baseline.py. The selected record is provenance only.

Credit

The paired-star contract, the rank floor, and the commuting map-count are from PR #5747 (Ali Romman), with the predecessor statements in PR #5673, PR #5681, PR #5631, and PR #5617. The sharp word is the construction in research/coupled31-chronology/COMMUTING.md. The three-leaf censuses that do not lift are open PR #6922 and open PR #6940.

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

Open next questions

• Does an interleaved word meet the g=13, q=8 endpoint with five rank-q thin maps when full projectors need not commute? Fix one pattern that separates at least one stage's halves, impose the thin maps forced by setting every D_j=1, and test the normalized coefficient at two field points before any native bill. No such word is supplied, and the consecutive-halves theorem does not apply.
• The interleaved g=31, q=26 profile is a different question. Only its adjacent-halves subcase follows from the count above. It is not investigated here.
• A uniform proof that one thin map fails at g=3 for every q would say whether the smallest interleaved escape exists. The F2 census in PR Record that one F2 thin map misses the three-leaf paired star. #6922 does not lift.

Workflow feedback (optional)

The registered replay of this proof finishes in about two seconds, inside the package cap. The useful split was between a schedule proof and a finite census: the three-leaf censuses were already open as #6922 and #6940, and repeating them would not have answered g=13.

Feedback discussion: #1200