Paired cubic axes fuse to simplex/Hamming and enter Golay through the hexacode

This page records exact finite-code and finite-geometry results from the Pass 4592–4615 continuation. It deliberately separates proven code embeddings from group-order or physical interpretations.

1. Two self-orthogonal axes become one 63-coordinate code

The cubic line/double-six incidence matrix has rank six over F₂. Its row and column images are C36=[36,6,16] and C27=[27,6,12]. Both evaluate the same vector x in a six-dimensional minus quadratic space. Concatenating the evaluations for the same x gives

C63 = [63,6,32], W(z)=1+63z³².

This is the binary six-dimensional simplex code; its dual is the Hamming code [63,57,3]. Its 63 coordinates are the 63 nonzero vectors of F₂⁶. The old 27+36 split is exactly the nonzero-singular / anisotropic partition of the minus quadratic form.

2. Exact hexacode-to-Golay bridge

Choose a compatible F₄³ structure on the same six binary dimensions. The standard hexacode is [6,3,4]4 with weight enumerator 1+45z4+18z6. Concatenate each F₄ symbol with the binary inner simplex [3,2,2]. The resulting binary code is [18,6,8] with 1+45z8+18z12.

The repository's explicit cyclic extended Golay construction contains a six-generator subcode that vanishes on six positions. Restricted to the other eighteen coordinates, it is mapped word-for-word to the concatenated hexacode by an explicit frozen coordinate permutation.

paired cubic axes → [63,6,32] simplex → chosen F4 structure → hexacode → [18,6,8] → G24.

3. The Golay ambient code reconstructs the MOG sextet

The 45 weight-eight words of that six-dimensional section are Golay octads. On the eighteen active positions every point occurs in twenty of the 45 octads. Pair frequencies are ten on exactly eighteen pairs and eight on the other 135. The frequency-ten graph is six disjoint triangles, recovering the six binary inner-code triples.

The subcode is identically zero on six Golay coordinates. Exhausting all 720 bijections from those six positions to the six recovered triples gives exactly one assignment for which the resulting six tetrads form a Golay sextet: every union of two tetrads is a Golay octad.

4. Other exact closures in the same packet

All 847,660,528 support-ten apartment-code coefficient subsets were evaluated exactly; 147 weights occur, from 582 to 1080.
An explicit eleven-factor composition series is frozen inside the 186-dimensional dual-middle submodule.
Primitive C8 alone fails to select apartments in GQ(2,2), but succeeds exactly in GQ(2,4) and GQ(3,3); the general symbolic criterion remains open.
The 135 apartment fibers carry a symmetric five-class association scheme. Its valency-two relation is 45 disjoint triangles, and the quotient relation gives SRG(45,32,22,24).
Every unary PSp(4,3)-equivariant set map between protected V8 and cubic U6 is zero. The known exterior-square bridge survives because it is two-input.
Forgetting the minus-quadratic color on the 63-point carrier gives SRG(63,32,16,16), the symplectic graph associated with Sp(6,2).
The cubic incidence maps R and R^T form a two-periodic differential. The associated CSS codes are [[36,24,3]], [[27,15,3]], and, after fusion, [[63,51,3]].

Evidence boundary

The simplex fusion is canonical from the two cubic axes. The F₄/hexacode step requires an additional compatible F₄ structure. The result is an explicit code embedding into G24, not an embedding of O⁻(6,2) into M24. The full apartment weight enumerator, the all-generalized-quadrangle C8 criterion, and the complete unlabeled Loewy/Ext lattice remain open. No finite-code object on this page is promoted to a physical particle, field, or hardware channel.