[unpaidmoon]

Change and mathematical scope

Proof. No kappa change. The selected frontier stays 781451/250000000.

For one nullity-32 class in the partial F_4 census of two thin maps on the three-leaf paired star, the two Schur complements have ranks summing to at least 4. Equivalently, the two thin matrices have rank sum at least 8. The bound is sharp: the zero blocks give Schur ranks 2+2 and matrix ranks 4+4, and another point in the same family gives Schur ranks 0+4 and matrix ranks 2+6.

The class is resolutions (0,0,0), permutation index 0, thin gaps 2 and 4. Its F_4 points are the block family

Z = [[P, I, I+Q, 0], [I, R, 0, S], [I+T, 0, U, 0], [0, V, 0, W]]

with Z_h = Z + D, over every field of characteristic 2. The case analysis is in research/paired-star-nullity32-schur/PROOF.md.

The 2^20 enumerator still returns -3 on this class. After three row pairs the partial rank sum is at most 6, so the frontier fills before any branch is pruned. That return is an incomplete search, not a rank bound. The proof covers the points the enumerator does not finish.

The shape does not cover every capped class. Resolutions (0,0,0), permutation 4, gaps 2 and 4 have nullity 24 and component sizes 4,4,4,4,2,2,2,2, and the same enumerator returns -3. A mask pass with no rank search finds 1746 affine signatures, 24 of nullity 32, and one copy of this identity-block mask.

Remaining obligations: the other nullity-32 masks, the other capped shapes, and every larger field. This does not finish the F_4 census.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840. Predecessor solver 8889385a5bbe64df6f1378047b35d89e1aea355c (open PR #7006).

Task id joint-query-controller, scope refined. The card asks whether a changed caller or query demand removes real work. This package answers the first follow-up declared by PR #7006. It is not a caller, a native bill, or a change of kappa.

PR #7006 records the partial census and leaves this Schur question open. Merged PR #6940 is the F_2 two-map rank 8. PR #5747 is the paired-star contract. Open PR #7028 is the F_8 nullity-7 class and is not used. No open PR proves this Schur bound.

Validation

python3 -B scripts/research_package.py --package research/paired-star-nullity32-schur

Python 3.12 and gcc with OpenMP. The run prints PASS_NULLITY32_SCHUR and PASS_SHAPES.

Evidence classes:

• Proof: the Schur sum is at least 4 for this block family, over every field of characteristic 2.
• Finite identification: the named affine space is that family (private generators, fixed identity blocks, Z_h = Z_g + D).
• Exhaustive mask census of the 1746 signatures, with no rank search: 24 nullity-32 classes, one named mask.
• Incomplete search: both dynamic programs return -3. Those returns are not rank bounds.

The 65536-point grid and 20000 random tuples check the Schur identity. They are not the proof. Not run: a rank search of the other capped classes, F_8, and a proof assistant.

Credit

The contract is PR #5747. The F_2 two-map rank 8 is merged PR #6940. The partial F_4 census and the enumerator are open PR #7006. Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed.

Open next questions

• Do the other nullity-32 masks have Schur complements summing to at least 4? The mask pass finds 24 classes and one copy of this identity-block pattern. Exchanging the two thin maps preserves the sum, but the remaining masks are not that exchange. First step: reduce the 0x0f or 0xc3 free-support to 2×2 blocks and repeat the rank split on U and T.
• Does the nullity-24 class at resolutions (0,0,0), permutation 4, gaps 2 and 4 have rank sum at least 8? Its 2^20 run returns -3, and its components are not this elimination. First step: write the four 4-variable rows and four 2-variable rows as blocks.
• Can every capped class be given Schur complements summing to at least 4? That would replace the unfinished frontier for this F_4 census and still would not be a larger-field result.

Workflow feedback

The 2^20 cap on this class is an ordering artifact: three row pairs cannot reach rank sum 8, on a family the Schur elimination settles. #1200