This is the canonical release numbering for a packet first implemented under aliases 4761–4768. A later lane reused those numbers, so the mathematics was renumbered without changing the theorem payload.
Exact.
The 1,620 support-12 apartment thickenings give 28-edge Eulerian vectors in the 240-edge W33 line-intersection graph. Their binary span is exactly the even cycle hyperplane [240,200,4]₂. Its dual is [240,40,12]₂ = Cut(X)+<1_E>.
Exact literal equality.
Every thickening has exactly one partner with edge overlap 8. The 810 pairs collapse 18-to-1 onto 45 sixteen-line unions. Every union is the 4×4 rook graph L₂(4). Reconstructing the earlier protected supports gives exactly the same 45 subsets of the same 40 W33 line coordinates.
Exact.
Two rook-grid supports intersect in 7 or 4 lines. Intersection 7 gives SRG(45,32,22,24). For the 45×40 grid-line incidence matrix T:
TTᵀ = 12I₄₅ + 3A₄₅ + 4J₄₅TᵀT = 12I₄₀ + 3Adual + 6J₄₀.
Its rank is 25 over Q and 24 over F2.
Exact.
Each rook grid contains exactly 36 support-12 minima, canonically C(4,2)×C(4,2). The unique partner is simultaneous complement in both duad coordinates. The PGSp grid stabilizer acts through a transitive group of order 576 on these states; the partner is not in that image but centralizes it and doubles the rectangle action.
Exact.
The 240 line-edge coordinates and 240 point-edge coordinates are inequivalent under the canonical PSp action. Nevertheless the 45-grid row code is [40,24,6]₂, and its entire 240-word minimum shell is exactly Star(p) XOR Star(q) for the 240 collinear point pairs p~q. Its dual is [40,16,10]₂; its 252 minimum words split into 36 spreads and 216 kernel words.
Exact.
The 24 explicit Leech two-neighbor parity characters span 12 binary dimensions. Their full 12-dimensional relation space is exactly the extended Golay code G24. Hence all subsets of at most seven characters are independent and the first dependencies are the 759 Golay octads. The corrected sextet supplies 15 two-tetrad octad relations.
Exact obstruction.
The support-12 even-cycle parity quotient and the apartment deck line are both group-fixed mod-2 classes with no invariant rational oriented lift. But their homogeneous bases have PSp stabilizer orders 648 and 96, which are mutually nondivisible, so there is no equivariant base map in either direction and no canonical cohomological identification.
The full apartment-code numerical enumerator remains open beyond exact support 12. None of these finite graph/code/lattice structures is promoted to a physical identification without an additional explicit theorem.