The 240 protected edge vectors span the unique 9D invariant layer V9. The quotient is now explicit:
Every vector in ker A* has even coefficient parity, so π is well defined. Any single line-star A*eᵢ supplies the missing tenth direction.
The 240 even weight-20 vectors are exactly the protected edge shell; the 40 odd minimum vectors are the 40 line-stars.
One local center line-star plus nine spokes from the 13-line Borel cell form a basis of all H10. Sampling ambient coordinates 0,1,2,3,4,5,7,8,10,11 gives an invertible 10×10 binary restriction. Its frozen inverse reconstructs all 40 ambient protected bits from those ten samples and was exhausted over all 1024 protected vectors.
The first decoded coordinate is π; the remaining nine are local spoke coordinates. The separate eight-state Borel quotient is spectrally complete but intentionally forgets identity within each orbit, so it is not this full decoder.
All five maximal subgroup types remain nonsplit. Standard subgroup structure reduces every remaining proper subgroup class above order 162 to representatives of orders 360,324,288,216,216,192,192; exact section systems for all seven are inconsistent. Hence no subgroup of order greater than 162 splits the apartment extension, while the order-162 chamber/Borel does.
An independent elliptic-quadric build of Q^-(5,7)=GQ(7,49) gives 2,752 points and 17,200 lines with exact binary ranks rank N=2451 and rank(NᵀN)=2150. Together with q=3, both anchors match
but the all-q rank formula remains explicitly conjectural. The exact general odd-q theorem proved here is A*²=0 over F₂.
For W33, primitive-six degree-two Walsh data obeys M6=48(J-I)+204A*. Therefore it determines A*, and hence H10=im(A*), the 9D even-coefficient layer, and the parity quotient. The same cross-check explains why the prism→edge collapse is special to t=3: for odd t a 3-rung fan equals its (t-2)-rung complement modulo the line-graph kernel.
Fixed-vector duality. The parity functional is π(x)=B(x,1), pairing with the unique PSp-fixed protected vector, so V9=1⊥.
Self-reconstruction. Join two of the 40 odd minimum vectors when their XOR has weight 20. The resulting graph is exactly dual W33, SRG(40,12,2,4); adjacent differences reproduce the 240 protected edges.
108 local bases. In the 12 neighbors of the center line, omit three. If the omitted triple is a triangle, edge+isolated, or independent, the remaining nine columns have ranks 7, 8, or 9 respectively, with counts 4, 108, 108. Adding the center to the 108 independent cases yields exactly 108 full local H10 bases.
Boundary. These are finite GF(2), finite-group, graph-zeta and representable-matroid statements. The closed Q(5,q) rank formula is still a conjecture beyond exact q=3 and q=7 anchors. No physical charge, spacetime locality, hardware decoder threshold, or particle identification is inferred.