[unpaidmoon]

Change and mathematical scope

Proof, for one call order and one legal frame. No new kappa.

On the paired-star contract, the full-call order that applies every first half in stage order and then every second half in the same order does not meet the endpoint at g = 31, q = 26 for the legal frame A = diag(I_q, 0) and B e_q = e_0 on every stage. The stage projectors do not commute. The pure product differs from product_j (I + x_j E_j) by the monomial x_0 x_1 on an invariant plane. Thin maps do not change the pure factor, so no thin maps of any rank repair this order and frame.

The five rank-26 maps forced by setting every D_j = 1 are explicit. They meet the normalized coefficient at that point and miss it at the two field points (2, 4, 1, ..., 1) and (3, 5, 1, ..., 1). The same projectors meet the pure target when the halves of each stage stay adjacent. This is not the consecutive rank bound and not the adjacent-halves map count.

Selected κ stays 781451/250000000 from PR #6030. A failed search is not used as nonexistence. The result is this order and frame only.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main). This fork draft uses cursor/upstream-main-17469fff-b813 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 interleaved g = 31, q = 26 profile left open by PR #7010. The first step was one pattern that separates stage halves, the thin maps forced by D_j = 1, and the normalized coefficient at two field points.

Nearest overlap:

• PR Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010 proves the adjacent-halves count, including the adjacent subcase of g = 31, q = 26. It names this interleaved profile as open and does not investigate it.
• PR Prove consecutive paired-star words need thin rank at least g q. #7005 proves the consecutive rank bound R ≥ g q and leaves internal-gap and other nested or crossing orders open. This package does not redo that bound and does not classify those schedules.
• PR Exclude the 31-leaf target with a chronological connector rank bound #5747 is the contract, the rank floor, and the commuting map-count. The commuting count already excludes five maps when every full projector commutes. The frame here does not commute.

The g = 3 censuses fix three leaves, block size 2, and entries in F_2. They are not a proof at thirty-one leaves or twenty-six ports.

Validation

Python 3.12, no third-party packages.

python3 -B scripts/research_package.py --package research/paired-star-g31-q26-interleaved-five

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome negative-result. Both the python3 -I -B replay and the python3 -I -O -B replay print PASS_PAIRED_STAR_G31_Q26_INTERLEAVED. Checks raise, so -O does not strip them.

The 6-by-6 difference is the exact product in GF(2)[x_0, x_1]. At both GF(16) points, all 36 entries of the q = 26 pure product on that plane match the 6-by-6 product, and the normalized coefficient of the forced maps is not I. Setting the nonzero entry of B to zero makes the crossed pure product match. Deleting the five maps misses the D_j = 1 point. A q = 1 block engine agrees with deriv · pure^{-1}.

Not run: a proof assistant, a search over other orders or frames, 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), including PR #5673, PR #5631, and PR #5617. PR #7010 left this interleaved profile open. PR #7005 left internal-gap and other nested or crossing schedules open.

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

Open next questions

• Does this same all-first-then-all-second order meet the endpoint for some other legal frame whose stage eigenbases are not the same? Form the two-stage cross product for one pair of legal frames that are not conjugate to the same (A, B), and compare one off-diagonal block with the target before any native bill.
• A word that puts only thin maps between a stage's halves keeps the pure factor. That internal-gap schedule is the question left by PR Prove consecutive paired-star words need thin rank at least g q. #7005, not this order.

Workflow feedback (optional)

The useful split was the pure factor from the normalized coefficient. At D_j = 1 the crossed order looks successful, because every x_j vanishes. The mismatch appears only at a second field point, which is why that point was worth computing before any native bill.

Feedback discussion: #1200