[unpaidmoon]

Change and mathematical scope

Over F_2, at block size 2 and three paired leaves, there is a paired-star word with exactly three nonzero thin maps, of ranks 2, 2 and 2, whose product equals the target in F_2[D_0, D_1, D_2, e] / (e^2). Deleting any one of those maps misses the target. The chronological floor of 6 is therefore attained once three maps are allowed.

This is one explicit identity, not a census of other schedules and not a nonexistence claim. It is not a construction of a caller and not a kappa improvement. The selected frontier stays 781451/250000000 = 0.003125804 from PR #6030. Every two-map word still has total rank at least 8, and the commuting control still meets the endpoint with ranks 4 and 4.

The word keeps the original stage contract: exactly two full calls per stage, six full calls in all, thin entries in F_2. Stages 0 and 1 use A-bits 0001 and B-bits 0010. Stage 2 uses A-bits 0011 and B-bits 0000. Those A factors do not commute. The chronological order is 4, 2, 3, 0, 1, 5, and the nonzero maps sit in gaps 1, 3 and 5.

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 three-or-more-map question left by the two-map rank-8 census. It is not a caller and it does not change kappa.

Nearest overlap, rechecked against open upstream PRs and Discussion #5593:

• 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 shows that one F_2 thin map misses the endpoint.
• Open PR Record that two F2 thin maps on the three-leaf paired star need total rank 8 #6940 shows that every two-map F_2 word has total rank at least 8, and leaves three or more maps open. This package answers that question with a rank-6 witness. It does not replace the two-map write-up.
• Open PR Record that three full calls miss the paired-star endpoint. #6956 claims that three or more nonzero T-commuting projectors fail the pure determinant. That is the three-full-call extension. This witness uses exactly two full calls per stage and does not review that claim.
• Fork draft Shared-edge local circuit on PR168 v4 paired cubes (κ = 6.59180578557293e-4) #181 is a g=13, q=8 five-map statement with consecutive halves. This g=3 word does not lift there.

Validation

Python 3.12. From the repository root:

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

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome completed. The replay runs verify.py and then the same check under python3 -O -B. Both printed PASS_PAIRED_STAR_THREE_THIN_RANK with ranks 2+2+2, sum 6, three nonzero maps, 12 resolutions, 96 noncommuting ordered A pairs, and the commuting control at ranks 4+4. Failures are raised, so -O does not strip them.

The multiplier aborts if a product exceeds multi-degree (2, 2, 2). The checked word has no such term. No Lean proof, Docker replay, make frontier6030-verify, or selected-frontier replay was run.

Credit

The contract, the rank floor, and the commuting construction are from PR #5747 (Ali Romman), including the predecessor statements in PR #5673, PR #5631, and PR #5617. The one-map obstruction is open PR #6922. The two-map rank-8 bound is open PR #6940. Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed. Apache-2.0.

Open next questions

• Does a two-map word of total rank 6 or 7 exist at g=3, d=2 once block entries may lie outside F_2? Both published F_2 bounds leave that field open.
• Can a stage with three or more full calls, together with thin maps, meet total thin rank 6 or 7? Open PR Record that three full calls miss the paired-star endpoint. #6956 claims a pure-determinant obstruction for T-commuting extra projectors. This package does not review it.
• Do the unconstructed profiles g=13, q=8 with five rank-8 maps, and g=31, q=26 with five rank-q maps, exist when full projectors need not commute? This witness does not lift. Commuting projectors remain impossible for those map counts.

Workflow feedback (optional)

The seven-gap affine space on a pure-hit schedule has nullity about 260, so a GF(256) screen does not certify a rank. The useful check was direct multiplication of one explicit word. Feedback discussion: #1200