86,240
Independent entries in the 1,540 row-pair permutations of the 56×56 residual grid.
PASSES 3506–3512 · EXACT SEVEN-FRONT PACKET
The 57-hole is scheme-blind—and Moore matchings carry curvature.
A gauge-reduced Moore-57 solver surface, literal Golay/Witt descendant chains, a complete Krein audit, a W33–Gewirtz polynomial compiler, and two deliberately high-risk but exact experiments.
Moore-57 permutation CSP
172,480
Directed CP-SAT integer variables with inverse channels, versus 4,743,200 one-hot Boolean variables.
6,271
Base permutation, inverse, vertex-star, and gauge constraints before lazy curvature cuts.
27,720
Row triangles inspected for fixed points of non-Abelian permutation holonomy.
Two exact descendant chains
Clebsch → Petersen
Higman–Sims → M22 → Gewirtz
Higman–Sims → M22 → Gewirtz
F₂⁵ even words
The Clebsch second shell at zero is the ten weight-two words with disjoint-support adjacency: Petersen KG(5,2).
1 + 759 + 2576 + 759 + 1
The extended binary Golay weight distribution is regenerated from a cyclic generator.
S(3,6,22)
Fixing two Golay coordinates yields 77 hexads, with every triple covered exactly once.
100 → 77 → 56
Incidence extension gives Higman–Sims; its second subconstituent is M22; avoiding a point gives Gewirtz.
The 57-vertex hole passes every ordinary scheme test
SRG(57,14,1,4): q₁₁=(38,1273/49,1140/49)
q₁₂=(0,540/49,722/49)
q₂₂=(18,342/49,111/49)
q₁₂=(0,540/49,722/49)
q₂₂=(18,342/49,111/49)
All entries are nonnegative, and both absolute bounds pass. The known nonexistence therefore lives beyond the rank-three Bose–Mesner algebra—in local compatibility or star complements.
W33 ↔ Gewirtz: what actually transplants
A² + 2A − 8I = 0 ⇒ f(A)=aA+bI
U=(-I-A)/3 and U²=I
U=(-I-A)/3 and U²=I
| Level | Transfers automatically? | Examples |
|---|---|---|
| Polynomial-only | Yes | Adjacency identities, two-channel calculus, complement reflection |
| Multiplicity-sensitive | No | Trace, determinant, projector ranks |
| Geometry-sensitive | No | Lines, incidence factorizations, codes, automorphism groups |
The surviving 57-cell symmetry shadow
|PSL(2,19)| = 3420
Conditional oddness of Aut(M57) kills the full group and every common A5 action.
19:9 = 171
The odd Borel point stabilizer lies below the peer-reviewed odd-order bound 375, so parity alone does not exclude it.
Bonkers experiment I: permutation curvature
15 matchings + 20 holonomies = all 2³
In an exact Hoffman–Singleton edge chart, every row-pair matching and every triangle holonomy is a fixed-point-free involution. The high-risk M57 branch asks for cycle type 2²⁸ throughout. This is an ansatz, not a necessity theorem.
Bonkers experiment II: Golay puncture functor
[23,12,7] → [24,12,8] → S(3,6,22) → M22 → Gewirtz
Higman–Sims, M22, and Gewirtz are reconstructed as one executable coordinate-fixing, incidence-extension, and point-avoidance pipeline—not isolated parameter coincidences.
Machine certificate
PASS_7_FRONTS
15ab9b21e744e755917e0b4c40ec806b43c5463b8a11cfbfac7bdbb4a9c8dfe7
15ab9b21e744e755917e0b4c40ec806b43c5463b8a11cfbfac7bdbb4a9c8dfe7