The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315

Abstract: The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual $\pm1$ assignment violates. For the four-qubit hyperbolic quadric $\mathcal{Q}<sup>{+}(7,2)$, with $135$ points and $1575$ lines as contexts, Muller et al. conjectured from heuristic searches that the degree is $315$ and that the violated contexts of every optimal assignment form a dual polar space $DW(5,2)$. We prove both conjectures. The lower bound has a short proof by hand: every doily ($GQ(2,2)$) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives $1575/5=315$. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly $240$ optimal violated sets, one orbit of $\mathrm{O}<sup>+(8,2)$, each isomorphic to $DW(5,2)$. The same argument gives the first lower bounds $\lceil L/5\rceil$ for all quadrics with $L$ lines in $N\ge4$ qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.