[unpaidmoon]

Change and mathematical scope

On the g=3, d=2 paired-star full-call contract, the lexicographically first consistent gap pair of F₂-nullity 7 is one affine class of two thin maps. Its scalar extension to any field of characteristic 2, including F₈ = F₂[x]/(x³+x+1), has rank(Zg)+rank(Zh) at least 8.

The class is resolutions (0, 0, 3), permutation index 4, and gaps 2 and 5. Chronological order, rightmost factor first, is calls 1, 2, 0, 3, 4, 5. The matrices are the seven-parameter shape in research/paired-star-f8-nullity7/PROOF.md. The bound is a case analysis of that shape: a column partition of Zh and a 6×6 minor of Zg. It is attained by two F₂ points, of ranks (4, 4) and (6, 2).

This is a proof for one affine class. It is not an exhaustive census of every F₈ class, and it is not unrestricted nonexistence of a rank-6 or rank-7 two-map word. The F₈ point set of this class is finite, of size 8⁷ = 2,097,152. Both enumerators report minimum 8 and no point below 8. That enumeration corroborates the proof on F₈; the proof is what covers every characteristic-2 field for this same shape.

No κ change. The selected frontier stays κ = 781451/250000000.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main). This fork PR is based on the pin branch of that commit so the diff is only this package.

Task card: joint-query-controller, scope refined. The card asks whether a changed caller or query demand removes real work. This package answers the single small-nullity F₈ question left by the partial F₄ census, on the same two-map full-call contract.

Nearest overlap:

• PR Exclude the 31-leaf target with a chronological connector rank bound #5747 is the contract and the commuting rank-8 control.
• Upstream PR Record that two F2 thin maps on the three-leaf paired star need total rank 8 #6940 settles two thin maps over F₂ and leaves entries outside F₂ open.
• Upstream PR Record a partial F4 census of two thin maps on the three-leaf paired star #7006 is the partial F₄ census. It proves 1,562 classes over F₄ and leaves 184 classes unresolved, and it explicitly leaves F₈ open. This package takes one nullity-7 class, which that census already settled over F₄, and extends that one class.
• The nullity-32 Schur-complement question inside the capped F₄ classes is not this class.
• Merged Record that one F2 thin map misses the three-leaf paired star. #6922 is the one-thin census. Merged Record that three full calls miss the paired-star endpoint. #6956 changes the full-call count. Open Record that three F2 thin maps meet total rank 6 on the three-leaf paired star #7011 uses three thin maps over F₂. This package keeps exactly two thin maps and two full calls per stage.

Overlap was checked against open upstream PRs, discussion #5593, discussion #1166, and fork cursor/* paired-star branches. No open item settles one small-nullity class over F₈.

Validation

python3 -B research/paired-star-f8-nullity7/verify.py

Python 3.12 and gcc, no OpenMP. The command checks the field, the Zh block formula on every F₈ point, the load-bearing constant ones, the polynomial control identity, and the lex-first derivation of the shape. It then compiles search.c and requires the two enumerators to agree.

Recorded F₈ census: 2,097,152 points, minimum rank sum 8, exactly two points at 8, none below. Checksum 6631801623909619557. The same class over F₄ has 16,384 points and minimum 8. Registered package command: PASS_REGISTERED_RESEARCH_PACKAGE research/paired-star-f8-nullity7 outcome=negative-result.

Evidence class: a scoped proof for one affine space, plus an exhaustive finite census of that space over F₈. The census is not a heuristic search. It does not speak for the other 1,745 classes.

No native caller, Lean proof, Docker replay, or selected-frontier replay was run. The source-history preflight was not required: every provenance pin is on the base commit.

Credit

The contract and the rank floor are PR #5747. The F₂ two-map census is research/paired-star-two-thin-rank (upstream PR #6940). The partial F₄ census is upstream PR #7006, fork commit 8889385a5bbe64df6f1378047b35d89e1aea355c.

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

Open next questions

research/paired-star-f8-nullity7/README.md and TASK.json.

The smallest useful follow-up is the next small-nullity class on this same contract, over F₈. The column partition used here does not transfer. A nullity-8 class has 8⁸ points, so the same direct enumeration does not fit this replay.

Workflow feedback (optional)

The F₄ package named this class as the extension to try. Keeping the replay on one nullity-7 shape made a field-independent rank argument visible; enumerating all of F₈ would have hidden it.

#1200