135 = 45×3
Three disjoint D4 frames lie over every canonical W33 octad.
D4/F4 triality
and the 45-axis carrier
Passes 3701–3714 · exact finite verification · semantic certificate 17e8e1…c644
45·12/2 = 270
Octad disjointness is SRG(45,12,3,3); the other 720 pairs meet in two W33 points.
192 → 576 → 1152
W(D4), its even-frame triality normalizer, and the full W(F4) outer stabilizer.
rank 24
The 45 centered octad vectors form a two-distance tight frame with Gram spectrum 43200²⁴+0²¹.
Three-qubit context theorem
The 135 maximal chamber K4s are exactly the 135 Lagrangian three-spaces of W(5,2), restricted to the 36 anisotropic points of a minus quadratic form. Their incidence matrix has binary rank 29.
1 + 63z¹⁶ + 63z²⁰ + z³⁶
The seven-dimensional dual is precisely the restriction of all affine-linear functions B(a,x)+c to q(x)=1.
Weyl-group tower
| Object | Order | Exact role |
|---|---|---|
| W(D4) | 192 | One frame stabilizer, C2³⋊S4 |
| Even-frame triality group | 576 | Normalizer of the three frames over one octad |
| W(F4) | 1152 | Full outer octad stabilizer, conjugate on 24 short roots |
The central quotient of the order-192 group is non-split: its cocycle equation has rank 95 and augmented rank 96.
Exact firewalls
0 common sheets
Each ternary generator has 108 inverting outer involutions, but no involution inverts all four.
16 > 7
Every colour-split rank-eight factorization exceeds the noncommuting-edge budget of a string diagram.
no 2A
The 45 canonical octad axes never acquire Peirce spectrum {0,1/4,1/32} anywhere in the complete two-product plane.
symmetry breaks
II(24,24) contains every even rank-24 graph section, but no U4(2)-equivariant positive unimodular one.
Polarization theorem
For the indefinite Gram matrix [[G,I],[I,0]], the graph of an integral matrix B has Gram matrix G+B+Bᵀ. Every even integral rank-24 lattice is therefore a primitive graph section. The verifier constructs E8³ explicitly. The standard Leech existence theorem implies a noncanonical rootless section, but this packet does not freeze a Leech basis.
Still pending: a thin regular polytope cover, serialized Monster words, a canonical Leech section, Majorana/Griess/VOA realization, remote CI/PDF evidence, and physical validation.