This packet resolves the relation among the Hoffman 13-cover stabilizer, the two natural order-192 tesseract/D4 groups, the tomotope automorphism group, the 576 Latin squares of order four, and the fixed 4×4 toroidal-knight Q4 substrate.
H576 ≅ W(D4) : C3 |W(D4)| = 192 H576 / W(D4) ≅ C3
The C3 cyclically permutes the three normal V8 sectors of W(D4), giving an explicit D4 triality fingerprint.
W(D4) / Z(W(D4)) ≅ Γ(T) 192 / 2 = 96
W(D4) acts on the 13 Hoffman cover cells with orbit structure 1+12. Its center is exactly the action kernel. On the 12 moving cells, the induced order-96 permutation representation is explicitly conjugate to the published four-generator tomotope action.
The orientation-preserving tesseract group and W(D4) both have order 192, but they are not isomorphic. The rotation group has 48 elements of order 8; W(D4) has none. Nevertheless both central quotients have structure (C2)^4:S3 and order 96, so the tomotope is the common quotient where the distinction disappears.
same-symbol knight edges 0 → 96 squares same-symbol knight edges 8 → 192 squares same-symbol knight edges 16 → 288 squares 576 = 96 + 192 + 288
Under the common board symmetry preserving both the row/column grid and this Q4 edge set, together with symbol relabeling, the 576 squares refine into five orbits of sizes 48, 48, 96, 192, 192.
The Pass5300 Hoffman/Latin order-288 isomorphism is an affine F2^4 module theorem. It does not identify their natural degree-12 actions: their point stabilizers are nonisomorphic. Similarly, equal order 192 does not identify W(D4) with tesseract rotations.
The exact 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.