[unpaidmoon]

Change and mathematical scope

Over F_2, at block size 2 and three paired leaves, no single thin map meets the paired-star endpoint, and every word with exactly two thin maps has total thin rank at least 8. The commuting control still meets the endpoint with ranks 4 and 4, so 8 is sharp for this contract. The chronological floor at this size is 6.

This is a finite obstruction. It is not a construction, not a native caller, and not a kappa improvement. The selected frontier is unchanged.

Three or more thin maps, entries outside F_2, and stages with three or more full calls remain open.

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 card asks whether a changed caller or query demand removes real work. This package answers the two-map rank question left by the paired-star floor in PR #5747, which is the concrete item under "New decoders and interfaces" for this session. It is not a caller and it does not change kappa.

Nearest overlap:

• PR Exclude the 31-leaf target with a chronological connector rank bound #5747 is the contract, the rank floor of 6, and the commuting rank-8 construction.
• Open PR Record that one F2 thin map misses the three-leaf paired star. #6922 is the one-thin census that left total rank 6 or 7 open. This package re-checks the one-map field screen and settles the two-map question. It does not replace that PR's write-up.
• Open PR Classify paired-root commutants and prove the four-thin boundary (no new κ) #6915 classifies paired-root commutants and a four-thin boundary. That is a different contract.

The census includes all 12 orthogonal resolutions. Ninety-six of the 144 ordered resolution pairs have noncommuting A factors, so this is not the commuting subcase.

Validation

Python 3.12, gcc, and OpenMP. From the repository root:

python3 -B scripts/research_package.py --package research/paired-star-two-thin-rank

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome negative-result. The replay is split into stage-0 resolutions 0..5 and 6..11 so each command stays within the package runner's time cap. Together they screen 1,244,160 orders. Both halves passed here: 159,984 pure hits, 568,752 consistent gap pairs, one_agree 0, and min_rank_sum 8. The second command runs under python3 -O -B; the checks raise, so -O does not strip them.

The polynomial control is independent of search.c. Deleting one thin map, or moving it into the other odd gap, misses the endpoint. On the diagonal resolution, every affine space of dimension at most 16 was also enumerated without pruning and agreed with the branch-and-bound, including the control gaps, whose unpruned minimum is 8.

A legal constant solution specializes into the searched affine spaces, and the search prunes a branch only when the rows already fixed have rank sum at least 8. A rank sum of 6 or 7 cannot be skipped.

Not run: native caller, Lean, Docker, make frontier6030-verify, and a selected-frontier replay. python3 -B scripts/research_selected_baseline.py returned PASS_RESEARCH_BASELINE_BINDINGS_ONLY only.

Credit

The paired-star contract, the rank floor, and the commuting map-count are from PR #5747 (Ali Romman), including the predecessor statements in PR #5673, PR #5631, and PR #5617. The two-map control is the construction in research/coupled31-chronology/COMMUTING.md, specialized to g=3 and q=1. The one-map question recorded in open PR #6922 is re-checked here at the two sample points.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

• Can three or more thin maps meet total rank 6 or 7 on this same g=3, d=2, F_2 contract? The two-map bound is now 8, and the chronological floor is still 6.
• Does a two-map word of total rank 6 or 7 appear once block entries may lie outside F_2, or once a stage may use three or more full calls?
• Do the unconstructed profiles g=13, q=8 and g=31, q=26 exist when full projectors need not commute? This g=3 result does not lift.

Workflow feedback

The package runner's 600-second command cap is why the census is split into two ranges. One suggested improvement is a documented way to register one logical replay that is allowed to run longer than 600 seconds when the check is a single finite census.