This page records executable finite code/group/module/lattice results. It deliberately separates exact statements from open frontiers.
Exact.
The two-periodic cubic incidence complex has homologies of dimensions 24 and 15. Against the point-side W33 modular layers of the same dimensions, the unique equivariant maps have ranks 9/1 and 14/1. Equal dimensions therefore do not identify the modules.
Exact, with a correction to a stale Pass 4615 JSON certificate.
The unique zero-coordinate assignment is (21,20,19,18,22,17). The sextet orbit has 1771 members; its stabilizer has order 138240, acts as S6 on the six tetrads with kernel 192, and the actual zero-transversal stabilizer has order 2160. That section group is faithful/transitive on the 18 active points and the 45 section octads.
Exact exhaustive census.
All 2,311,801,440 support-11 subsets are evaluated. The shell has 153 distinct weights, from 614 (multiplicity 12,960) through 1026 (multiplicity 1,080).
Open: supports 12–20 and the complete 2^39 numerical enumerator.
Exact local reduction.
Apartments admit four contributing length-eight line-multiplicity signatures; K1,3 supports admit two. Collision is exactly equality of those six raw masses. In GQ(2,2), 224+64=96+192=288, hence coefficient 36 on both species. GQ(2,4) gives 60 versus 36; W33 gives 712 versus 180.
Open: closed all-(s,t) formulas for the six signature masses.
Exact obstruction.
The six-dimensional Golay section gives determinant 4096. The 64 cosets of G24/C6 extend it with index 64 to the full Golay Construction-A lattice, determinant 1. The 48 coordinate roots survive, so the endpoint is the rooted A1^24 Niemeier lattice, not Leech. A further neighbor/shift step is required.
The 64 selected sextet transversals are uniquely equivariantly identified with the 64 C6 codewords, transporting an F2^6 affine structure. The six-dimensional glue quotient G24/C6 is isomorphic, under the section group, to the binary permutation module on the six tetrads. Finally, the cross-shell and within-shell 63-point polar differentials are commuting square-zero rank-12 maps with the same image; their characteristic-two sum has rank 6 and gives the [[63,51,3]] simplex/Hamming CSS code.
No physical particle, spacetime, or Leech identification follows from these finite structures. The corrected Golay coordinates, module non-isomorphism, exhaustive support-11 mass, local C8 signatures, and lattice determinant/root counts are the claims certified here.