Establish a conceptual derivation of the dual polar space structure

Prove directly that the local doily conditions characterizing optimal violated sets of the four-qubit hyperbolic quadric force those sets to be isomorphic to the dual polar space $DW(5,2)$, potentially via the Lagrangian Grassmannian map from planes of $W(5,2)$ to points of $Q(7,2)$.

Background

The paper classifies all optimal violated sets computationally: there are exactly 240, and each is isomorphic to $DW(5,2)$. The preceding local characterization says that an optimal assignment selects a ruling in one grid of every doily. The unresolved issue is whether the global $DW(5,2)$ structure can be derived mathematically from these local constraints without relying on exhaustive SAT enumeration and computational isomorphism checks.