Evidence boundary. The q=5 apartment-code minimum-distance theorem remains open. These passes prove that every word of weight <625 has chamber leader at least 33; leaders ≥33 and the weight-625 equality shell remain open.
Pass5182 closes leader 31: the weakest adjacent-pair layer has weight lower bound 673. Pass5183 adds the exact geodesic coupling P5≤2N4, raising the sole failing leader-32 layer to 631. Strict q=5 counterexamples therefore require chamber leader ≥33.
The chamber distance scheme now has the exact quadratic-form bridge
(xᵀAPx+yᵀALy)/2 = N1+2N2+N3.
Pass5185 also promotes full cut-coset minimality: for every Levi vertex set S, 2|Y∩δ(S)|≤|δ(S)|. At q=5 this forces every host incidence between selected-degree-three vertices to be selected and constrains all 3–2–3 host paths.
The P-component tensor decomposition reconstructs the original W(3,q) point geometry. Every P component is the union H∪H⊥ of a hyperbolic projective line and its polar, inducing Kq+1,q+1. Its minimum tensor atoms are exactly the edges of this dual grid.
If B is point-by-P-component incidence and N is ordinary point-line incidence, then
BBᵀ=(q²−1)I+(q−1)A+J=(q−1)NNᵀ+J.
Every W-line meets every dual grid in 0 or 2 points, hence NᵀB=0 over F₂.
At q=5, rank₂(N)=91 and rank₂(B)=65, so the 325 dual grids span the entire binary incidence dual:
C(W(3,5))⊥ = [156,65,12]₂.
Its 325 minimum words are exactly the 325 P-component dual grids. More generally, for every q the binary W-line incidence dual has minimum distance 2(q+1), and its complete minimum shell consists exactly of the q²(q²+1)/2 dual grids.
The P-heavy-free weight-625 sector is no longer an unconstrained apartment search: it is a 25-atom gluing problem over these canonical dual grids. The missing step is the connected L-side compatibility, not the P-side local code.