Fischer closure, Hadamard completion, and the six-bit orthogonal core

Passes 3722–3728 turn the 120 Norton triples into a complete group, code, cover, and axial-rigidity certificate.

36 involutions → 120 Fischer lines → O₆⁻(2)

The Miyamoto involutions form a 3-transposition system. The generated group has order 51,840; its even-product subgroup has order 25,920 and the exact prior U4(2) element-order census.

One explicit 36-port Hadamard transform

For the spread graph adjacency matrix A, K=2A-J satisfies KKᵀ=36I. Therefore H=K/6 is a symmetric orthogonal involution, with E15=(I+H)/2 and E21=(I-H)/2.

A [36,6] code contains both 27 and 36

The binary triple-incidence kernel has weights 0¹+16²⁷+20³⁶. Its 27 singular vectors recover SRG(27,10,1,5); its 36 nonsingular vectors are the Miyamoto supports and recover SRG(36,15,6,6).

The 120-block graph is a nonregular three-cover

It has degree 27, spectrum 27¹, 9²⁰, 3¹⁵, −3⁸⁴, forty fibers of size three, quotient SRG(40,27,18,18), full S3 holonomy, and no nontrivial deck transformation.

Axial rigidity

The multiplication operators generate all M15 modulo 1,000,003; the algebra is simple and has zero rational derivations. It remains explicitly non-Majorana/non-Griess.

Evidence boundary. No serialized Monster words, executed Monster character restriction, physical optical implementation, or remote PDF result is claimed here.