135 K4
Maximal four-cliques in the 36-spread chamber graph.
Exact closure of nine fronts linking U4(2), W33 spreads, the tomotope, universal amalgamation, lattice glue, equivariant products, and moonshine.
Maximal four-cliques in the 36-spread chamber graph.
Chamber–K4 incidences.
A non-split central K4 stabilizer quotient marked-isomorphic to Aut(tomotope).
The four-parabolic geometry is thick, not polytopal.
Exact non-scalar discriminant cancellation.
The entire equivariant commutative Frobenius envelope.
The canonical chamber/spread bridge from Passes 3663–3669 supplies two relations. Chamber intersections of order 12 give SRG(36,15,6,6); intersections of order 18 give its complement SRG(36,20,10,12). The first relation has exactly 135 maximal K4s, and U4(2) is transitive on them.
A maximal K4 stabilizer has order 192, center order two, and census 1¹ 2⁴³ 3³² 4⁸⁴ 6³². Its central quotient has order 96 and census 1¹ 2²⁷ 3³² 4³⁶. Four quotient involutions satisfy exactly the tomotope edge-generator pair orders, and breadth-first word transport gives a marked isomorphism.
Exact sequence. 1 → C2 → Stab(K4) → Aut(T) → 1. The extension is non-split because C2 × Aut(T) has a different order census.
The four three-generator parabolics each have order 648. Their pair intersections are five copies of order 24 and one of order 54; every triple intersection has order three; the total core is trivial. The resulting coset geometry has 40 objects of each type and 25,920 chambers.
Diamond obstruction. Every panel contains three chambers. The natural exchange motion is C3 rather than an involution, so this is a thick chamber system and not an abstract polytope.
For the positive rank-24 projector Gram matrix G, the block form [[G,I],[I,0]] is integral, even, unimodular, and has signature (24,24). The scalar obstruction is therefore bypassed exactly, but the canonical closure is the indefinite lattice II(24,24), not the Leech lattice.
Character inner products give dim Hom(Sym²V,V)=2 and dim Hom(Λ²V,V)=1. Projected coordinatewise multiplication and projected triangle-neighbour multiplication span the complete commutative envelope and are both Frobenius.
Associativity no-go. The complete associator ideal is the unit ideal modulo 1,000,003. No nonzero rational member of this two-dimensional envelope is associative. This does not rule out a nonassociative Griess-type product, but it confines any candidate to this exact plane.
| Pair | First A coefficient | First B coefficient |
|---|---|---|
| 2A / 2B | 4372 | 276 |
| 3A / 3B | 783 | 54 |
| 5A / 5B | 134 | 9 |
The exact eta-quotient formulas extend the scalar J+24 completion to the six class-dependent channels selected by the prime support {2,3,5} of U4(2).
The W33 graph and the chamber-complement graph each carry a 15-dimensional eigenvalue −4 sector. Both are irreducible U4(2) modules, but their character inner product is zero. They are spectral doppelgängers, not equivalent representations.
The strict mmgroup harness requires four order-three Monster elements with pair orders 3,6,6,6,6,6, four order-648 triple closures, full closure order 25,920, and an independently hashed class-fusion certificate. The candidate slot is still pending.
Not promoted: Monster words, a unique class fusion, the degree-81 restriction multiplicity, positive-definite Leech glue, a Griess/VOA identification, or remote PDF/CI evidence.
Semantic certificate: 2d5fe418bb2bbee0eb479e42d22dd47d31c37f7d5f8be6b1732acf0dc1db8b78