This packet sharpens the 576→192→96 story at the object level. It uses the repository’s explicit toroidal-knight→Q4 labeling, the canonical Hoffman W(D4) subgroup, and the natural action of W(D4) on antipodal pairs of tesseract square faces.
576 = 144 + 432 Klein/V4 class: 144 affine maps F2^4 → F2^2 Cyclic/C4 class: 432 non-affine C4 derivative defect: 192 squares: exactly 1 constant Q4 derivative 240 squares: 0 constant Q4 derivatives
The one-constant-direction cases split evenly: 48 for each of the four Q4 bit directions. No cyclic square has two or three constant derivatives.
10 Hoffman H-orbits → 22 W(D4)-orbits 4 H-orbits stay transitive 6 H-orbits split into triples
The outer quotient H/W(D4)≅C3 therefore acts literally as a triality fusion on the four-cell shortening layer. This is an orbit theorem only; it does not close the shortened minimum distance.
row → column → symbol → row
Inside the five-line PG(3,2) MOLS spread, the characteristic tomotope96 subgroup fixes the row, column, and symbol lines individually and swaps the two orthogonal-mate lines. The missing quotient C3 cycles row/column/symbol while fixing both mate lines.
24 tesseract square faces → 12 antipodal face pairs W(D4) / ⟨-I⟩ = 96 ≅ Γ(tomotope)
An explicit 12-point relabeling sends the natural W(D4) action on these antipodal square-face pairs exactly to the published tomotope permutation group.
On the same 12 face-pair objects, the tomotope/W(D4) quotient has orbital rank 5: three separate perfect-matching relations. The rotational quotient has rank 4: it preserves one matching and fuses the other two into a degree-two relation. Both share the same uncolored 3 K4 skeleton, but the natural permutation representations are not conjugate.
3 K4 components ↔ 01|23, 02|13, 03|12 kernel = (C2)^4 component action = S3 Γ(T) = (C2)^4 : S3
The three K4 blocks are precisely the three complementary 2+2 partitions of the four tesseract coordinate axes.
The Hoffman-shortened minimum remains d∈{28,32,36,40}; q11 footprint distance 121 remains open; and the all-odd footprint-rank equality remains open in general.