Passes 4495–4502. This page records exact finite results. It separates theorem, exhaustive computation, and open remainder explicitly.
For line signs σℓ=(-1)bℓ, apartment parity gives
w(Hᵀb) = (1620 - W₄(σ))/2.
The only coefficient kernel is global reversal. Fixing σ0=+1 leaves exactly 239 codewords. At each partial assignment the residual Walsh polynomial P=c0+Σ cM χM obeys the rigorous bound P ≤ c0+Σ|cM|. Branch-and-bound proves only the zero word has W₄>1296 and exactly 40 nonzero words have W₄=1296.
C_ap=[1620,39,162] and A_162=40.The exact low shell is
W_C(z) = 1 + 40 z^162 + 240 z^270 + 540 z^312 + 200 z^324 + …
The 10-dimensional protected module has composition structure 1|8|1. An exhaustive cyclic-submodule scan of all 1,023 nonzero vectors yields dimensions 1^1, 9^510, 10^512, so its invariant-submodule lattice is exactly 0<1<9<10.
For the irreducible middle eight-space V₈, symbolic cocycles were propagated over all 25,920 elements of PSp(4,3):
dim Z¹(PSp(4,3),V₈) = 10 Dim B¹(PSp(4,3),V₈) = 8 dim H¹(PSp(4,3),V₈) = 2.
The outer PGSp involution swaps a chosen H¹ basis. Its unique nonzero fixed class is their sum. After the invariant symplectic identification V₈*≅V₈, both adjacent extensions in 1|8|1 select that same outer-fixed nonzero class.
Let Bσ be the signed Hashimoto matrix, Tk=tr(Aσk), and Cn the signed sum over oriented primitive nonbacktracking prime classes of length n. Bass recurrence gives
tr B³ = T₃ tr B⁴ = T₄ - 11040 tr B⁵ = T₅ - 55 T₃ tr B⁶ = T₆ - 66 T₄ + 416640 tr B⁷ = T₇ - 77 T₅ + 1694 T₃ tr B⁸ = T₈ - 88 T₆ + 2420 T₄ - 9050400.
Hence
log Lσ(u) = 8 S u³ + (240+2W₄)u⁴ + C₅u⁵ + (160+C₆)u⁶ + C₇u⁷ + (1740+C₈)u⁸ + O(u⁹).
The 18,144 simple point-graph pentagons reduce by edge-line parity to exactly 2,160 induced P₃ supports of the dual W33 graph, each with multiplicity six, and 5,184 dual C₅ supports, each with multiplicity one:
U₅ = 6 W_P3 + W_C5*, C₅ = 2 U₅.
Under random independent line signs, Var(U₅)=82,944=288² and it is exactly uncorrelated with both the line sum S and W₄. Two frozen signings have the same S=-2 and W₄=8 but U₅=600 and -136. Thus u⁵ is the first coefficient carrying information not determined by S and apartment weight.
A verified hyperbolic basis sends the unique invariant plus-type quadratic on the eight-core to
q(x,z)=x₁z₁+x₂z₂+x₃z₃+x₄z₄ mod 2.
All five inner PSp generators and the outer projective similitude were conjugated into these Sp(8,2) coordinates and factored into binary symplectic transvections. The five inner words use six transvections each; the outer uses four. The frozen certificate records the corresponding four-qubit Pauli labels. This is a projective Clifford-label realization, not a claim that the 1,620-bit code is physically four qubits.
Z(β)=2 exp(1620β) W_C(exp(-2β))
now begins exactly as
2 e^(1620β)[1 + 40e^(-324β) + 240e^(-540β) + 540e^(-624β) + 200e^(-648β) + R(β)],
R(β)=Σ_{w>324} A_w e^(-2βw).
Open: the complete 239 weight enumerator is not claimed solved.
The 40 minimum words are exactly the 40 line-star rows. Their pairwise Hamming distance is 270 iff the corresponding dual W33 lines intersect and 312 otherwise. Therefore the minimum-weight shell intrinsically reconstructs the dual SRG(40,12,2,4). Because all 1,620 apartment columns have distinct incidence patterns, a code-coordinate automorphism acting trivially on this shell is trivial. Combining this injection with the natural PGSp(4,3) action and the repository's independent automorphism theorem gives Aut(C)=PGSp(4,3), order 51,840.
Every weight-3 dual word is a relation among three apartment columns. There are exactly 2,160. Their three four-line supports pairwise meet in two lines and their six-line union induces a triangular prism in the dual W33 graph; the apartments are exactly its three rectangular faces.
The span of these 2,160 minimum checks has rank only 1,215, below dim C⊥=1,581. So the minimum dual checks do not generate the full dual code.
Each prism's six-line indicator produces a primal codeword of weight 534. Applying A* to the same indicator collapses the 2,160 prisms exactly 9-to-1 onto 240 protected vectors, all of ambient weight 20. PSp(4,3) acts transitively; a stabilizer has order 108 and suborbits
1,1,2,2,18,18,18,18,27,27,54,54.
The action fingerprint is reported precisely to prevent a count-only identification with E8 roots, W33 edges, or any other 240-set.
These are finite coding, group-cohomology, graph-zeta, and Clifford-label results. Classical generalized-quadrangle coding work gives important incidence-code context, but this construction uses the 1,620 building apartments as coordinates. The absence of an exact match in a targeted literature search is not treated as proof of global novelty.