The repeated eight-core and a nonsplit apartment code

Pass 4485 — literal self-gluing

Write M=F₂⁴⁰, A*=NᵀN, K=ker A*, I=im A*, R=ker N, U=R∩I, and J=<1>. The apartment code is M/J, its radical is K/J, and the protected quotient is M/K.

Therefore the protected middle factor is literally U/J. The same U/J already sits inside the apartment radical. This is one quotient space appearing twice, not a dimension coincidence.

Pass 4486 — same module, different form

Inside the radical, the apartment polar form and Hamming quadratic vanish identically on U/J. Transported through the protected quotient, the same module carries a rank-8 alternating form and a plus-type quadratic:

Thus the representation survives in the radical while its nondegenerate metric structure appears only through the protected quotient.

Pass 4487 — exact weld to the parallel E8 / Pauli core

The independent Pass-4477/4478 construction was put into the same 40-line coordinates. Its fixed vector maps to the all-ones class J; its perpendicular 9-space maps exactly to U. In the resulting quotient basis, the parallel 8×8 Gram matrix equals the lift-defined protected Gram matrix entry by entry, and the two quadratic refinements agree on all 256 classes.

So the E8/2E8 and real four-Pauli finite phase-space packet uses exactly this ambient U/J. This is a finite quadratic/representation identity, not an E8-dynamics or hardware claim.

Pass 4488 — the 39-space is genuinely nonsplit

The exact sequence is

An equivariant section S would satisfy ΠS=I and G_E S = S G_V. For four generators this is a binary affine system with 390 unknowns and 1,660 equations. Its ranks are

so no PSp(4,3)-equivariant section exists. Even more sharply, Hom_PSp(H10,C_ap) is one-dimensional. Its unique nonzero map has rank 9 and is exactly

It lands in I/J inside the radical, has one-dimensional kernel, and projects back to zero in H10. Symmetry therefore permits only a 10→9 return channel into the radical—not a clean invariant 10-dimensional complement.