A Unified Gray-Graded Operator Structure on the Five-Dimensional Hypercube

We establish that four previously independent structural results on the five-dimensional hypercube Q₅ maximal parity alignment in Hamiltonian cycle structure, parity-optimal cycle structure with its induced 2-adic tier organization, the active/complementary transition decomposition, and the graded adjoint operator decomposition under an Ising-type Hamiltonian, are equivalent realizations of a single graded operator system induced by a distinguished coordinate decomposition. Two lemmas establish the core compatibility: the complementary Hamming-weight grading coincides exactly with the spectrum of the grading operator Jᵦ^ (i), and the active/complementary transition partition matches the commutation structure of the transition operators. The unification theorem then identifies the global Hamiltonian cycle structure, the local transition decomposition, and the sector-resolved operator dynamics as consistent manifestations of one underlying finite combinatorial-operator system on Q₅. No new combinatorial or operator identities are introduced. The result makes explicit that the constituent theorems describe the same object.