For each W33 line ℓ, gℓ=Hᵀeℓ is its weight-162 apartment signature and xℓ=Neℓ is its weight-4 line incidence vector. Under the Pass-4469 quotient bridge, [gℓ]→[xℓ]. The forty H10 classes are distinct, span H10, and their polar Gram matrix is the dual W33 line-collinearity graph.
The fixed isotropic defect class f is the unique H10 class with B(xℓ,f)=1 for all forty line classes. Thus xℓ→xℓ+f gives a disjoint anisotropic W33 twin with the same polar graph.
With R=ker N, I=im A*, K=ker A*, U=R∩I, S=R+I, and J=<1>, exact dimensions are (J,U,S,K)=(1,9,16,30). Hence
and the radical has invariant profile 8 | (6⊕1) | 14. The 8-, 6-, and 14-dimensional factors pass exhaustive cyclic-span irreducibility tests for two PSp(4,3) generators; the 1-space is trivial.
The lexicographic line indices
form an H10 basis. Their intersection graph is P4 ⊔ 3K2, with six intersecting pairs. Six is optimal: a nonsingular alternating ten-vertex graph with only five edges would be an induced 5K2, while exact search gives induced-matching number four for W33.
If g_i are the selected apartment signatures and G their 10×10 polar Gram matrix, then ten software parity bits recover every protected class:
All 1,024 classes were verified. Ten output bits are information-theoretically minimal for an injective linear readout of a 10-dimensional quotient.
The CSS code, forty minimum logical lines, H10=1|8|1 filtration, and plus-type quadratic form are prior repository results. The new content is the explicit apartment/incidence bridge and the consequences above. Searches of relevant primary incidence-module and generalized-quadrangle code literature did not reveal this exact construction; that is recorded only as “no occurrence found in the searched sources,” not as proof of global novelty.
Physical boundary. The ten readout bits are XOR post-processing of already acquired apartment data; each selected line functional touches 162 apartment bits. This is not a ten-measurement syndrome-acquisition theorem, not an optical cost optimum, and the fixed-layer transvection is not being promoted to a physical gate.