Passes 4461–4471 · exact finite-combinatorial frontier · W(3,3), protected H10, and general GQ(s,t)
Assign one sign σℓ∈{±1} to each of the 40 geometric lines of W(3,3). Each K4 line owns six point-collinearity edges, and the 40 six-edge supports partition all 240 edges. If N is point–line incidence, the signed adjacency is
The first moments are exact: tr(Aσ²)=480 and tr(Aσ³)=24 Σℓ σℓ. Any 20/20 balanced line signing therefore has zero cubic trace.
The point graph has exactly 1,620 induced quadrangles. Each uses four distinct geometric lines; those four lines form a 4-cycle in the dual line graph, and inserting the line vertices gives the corresponding Levi 8-cycle. These are three shadows of the same rank-two-building apartment.
Here W₄ is the sum of the four-line sign products over the 1,620 apartments, and f₄ is the fraction with negative product. The fourth moment’s entire variable part is therefore an apartment-parity observable.
Under independent random line signs, the apartment products are distinct quartic Walsh characters. This gives a closed-form control distribution rather than a Monte-Carlo benchmark:
The null is not exactly Gaussian: standardized skewness is about 0.198761598 and excess kurtosis about 0.769135802.
Let H be the 40×1,620 line/apartment incidence matrix and A* the dual W33 line-collinearity adjacency. Exact pair counts give
Modulo two, HHᵀ=A*. More strongly, rankF₂(H)=39 and ker(Hᵀ)=⟨140⟩. Thus all 1,620 apartment parities recover the 40 line signs up to the unavoidable global reversal σ↦−σ.
The apartment-parity image is a binary [1620,39] linear code. The dual adjacency form has rank 10 over F₂, so the code radical has dimension 29 and the quotient is a nondegenerate 10-dimensional alternating, hence symplectic, vector space.
A one-line code generator has weight 162. The XOR of two line generators has weight 270 when the two lines intersect and 312 when they are disjoint.
For a finite GQ(s,t), set n=(s+1)(st+1), L=(t+1)(st+1), and d=s(t+1). A sign on each geometric line gives:
The exact number of induced quadrangles is Q=n s²t²(t+1)/8. If H is line–quadrangle incidence in that GQ, a line lies in r=(s+1)s²t²/2 quadrangles, intersecting line pairs in α=s²t, and disjoint line pairs in β=s(s+1)/2, so
The Pass 4433 helper claimed to list every simple 4-cycle once. On W33 it actually emits 3,480 records: 1,740 underlying simple C₄s, each twice. Those 1,740 split into the 1,620 induced apartments plus 120 cycles wholly inside the geometric K₄ lines.
Line-internal cycles always have positive parity under a line signing, so Pass 4433's raw fraction was a diluted apartment fraction:
This does not invalidate its standardized z-score against a random line-signing baseline; multiplying every sample by the same positive factor leaves z unchanged. In general, flegacy = Q/(Q+3L C(s+1,4)) fapartment. The dilution factors are 243/245 for GQ(3,9) but only 81/137 for GQ(9,3), so cross-GQ raw legacy fractions should not be compared without undilution.
The 10-dimensional quotient is not just numerically equal to the older CSS logical-label shadow. If C=ker(Nᵀ) is the established [40,15,8] sentinel code and C⊥=im(N), then incidence itself gives the canonical chain
The kernel and pairing are literal:
All 2¹⁰ quotient classes were checked. This identifies the apartment-parity quotient with one protected H10 logical-label copy of the established [[40,10,4]] CSS code; the full logical Pauli label space remains 20-dimensional.
Both quotients carry the natural Hamming quadratic refinement q(v)=wt(v)/2 mod 2 and both have exactly 528 singular classes including zero, so both are plus-type O+(10,2). The raw incidence map is symplectic but not quadratic: the two refinements differ by one nonzero linear functional.
Nondegeneracy represents that functional by a unique isotropic class a. Under the incidence bridge its H10 image is exactly
the fixed one-dimensional layer already present in the older H10=1|8|1 module filtration. A single transvection Ta(v)=v+B(v,a)a repairs the quadratic on all 1,024 classes. Because the defect line is invariant and one-dimensional over F₂, the correction is equivariant. This is a comparison-map correction, not a claim of a new physical gate.
Point–line incidence always satisfies NᵀN=(s+1)I+A*. Comparing this with the general apartment Gram above and reducing modulo two gives the sharp condition
So W(3,3) passes. GQ(3,9) also passes in the displayed line orientation. The dual parameter orientation GQ(9,3) fails. The bridge is therefore a characteristic-two orientation effect, not a duality invariant. This exact criterion is adjacent to—but does not explain—the empirical s/t line-signing success sweep in Pass 4457.
Evidence boundary. These statements are finite-geometric identities, exact quotient/isometry calculations, and explicit corrections. They do not establish a physical gauge field, do not make any optimized signing statistically special without comparison to the exact null, do not assert that an optical experiment automatically measures all apartment parities, and do not turn the quadratic comparison transvection into a hardware gate. The executable witnesses and frozen JSON certificates now run through Pass 4471.