Canonical theorem numbering is 4697–4704. The first implementation files were produced under collision aliases 4689–4696 before a parallel namespace overlap surfaced.
Exact.
The 63-point cross-shell homology is canonically H27⊕H36. The 39-dimensional apartment coefficient module is F2^40/<1>, but it is not isomorphic to that periodic sum: the two-dimensional equivariant Hom spaces have nonzero ranks 14 in one direction and 1 in the other.
Exact.
The corrected M24 sextet stabilizer contains a normal regular translation subgroup C2^6 on the 64 sextet transversals. With the chosen-transversal stabilizer K of order 2160, H=C2^6⋊K. K maps onto S6 with C3 kernel and has nonzero translation orbits 18+45.
Exact lattice certificate.
Starting from the Golay Construction-A A1^24 Niemeier lattice, use v=(3,1^23)/√2 and the index-two kernel M={(x,v) even}; adjoining v/2 produces an even integral 24-lattice. The frozen basis has determinant-one Gram matrix, all old coordinate roots are removed, the new coset has no norm-2 vectors, and norm 4 is attained. Hence the neighbor is the Leech lattice.
Exact, with embedding invariants.
All six Pass-4635 raw walk masses are closed in u=s−1 together with three local incidence invariants ρ,σ,τ. W33 and dual W33 both have order (3,3), yet have (ρ,σ,τ)=(0,16,1) and (4,0,3), giving apartment/star coefficients (712,180) and (728,252). Therefore no (s,t)-only law exists.
Exact exhaustive reduction.
Transitivity gives 12A_w=40B_w, so only C(39,11)=1,676,056,044 fixed-position subsets are enumerated. Exact rescaling yields all C(40,12)=5,586,853,480 support-12 subsets in 151 weight classes. Minimum: 608 with multiplicity 1,620. Maximum: 990 with multiplicity 4,320.
Open: full apartment-code numerical enumerator, supports 13–20.
The faithful Golay translation six-space has no nonzero invariant quadratic or bilinear form, excluding a direct identification with cubic O6−(2). The 1,620 support-12 minima are exactly the corner-star thickenings of the 1,620 apartments. Those thickening words span the [1620,38,270] even-coefficient apartment subcode, whose 240 minimum words are precisely the W33 edges.
Equal dimensions and matching cardinalities are not promoted without intertwiners or intrinsic inverses. The Leech claim is lattice-theoretic, with explicit basis/determinant/evenness/rootlessness evidence. No physical interpretation is inferred from these finite structures.