Closure-Parity Selection and Bi-Graded Structure on Q5

We identify a distinguished quadratic diagonal operator on the five-dimensional hypercube Q₅, selected uniquely (up to scale) by a pair of anticommutation conditions on the dominant coordinates and commutation conditions on the subdominant coordinates. This operator, the closure-parity operator Ĉ = Z₁Z₂, defines a second Z₂ grading independent of global parity. The main theorem proves that Ĉ is the unique operator in the degree-≤2 diagonal sector satisfying these transition conditions. We show that Ĉ commutes with the diagonal Hamiltonian H₀, so closure parity is conserved under diagonal evolution, and derive the resulting bi-graded refinement of the H₀ eigenspaces with explicit sector dimensions (1,0), (3,2), (4,6), (4,6), (3,2), (1,0) across weights 0–5. A closed formula for closure parity along the reflected Gray cycle is derived. Interpretive sections describe the fibrewise three-level structure that follows from the bi-grading, outer states in the +1 sector, a doubly-degenerate intermediate bridge in the −1 sector, and the second-order mediated coupling between outer states arising through virtual transitions into the intermediate sector.