Graded Adjoint Operator Decomposition on the Five-Dimensional Hypercube

This paper develops a finite-dimensional operator-theoretic structure on the five-dimensional hypercube Q₅ = 0, 1⁵ based on a distinguished-coordinate decomposition. By isolating a complementary four-coordinate subsystem, we construct operators satisfying su (2) -type commutation relations and use them to define a grading indexed by complementary Hamming weight. This grading partitions the space into five canonical sectors with dimensions 2, 8, 12, 8, and 2. The main result establishes that a family of projected transition operators forms an exact eigen-operator decomposition under the adjoint action of a natural diagonal Hamiltonian. The decomposition is defined globally on CQ₅ via spectral projectors of the complementary operator Jᵦ^ (i), without assuming irreducibility of the representation. This yields a canonical graded operator module on Q₅, derived entirely from the combinatorial structure of the hypercube. The construction is finite, explicit, and independent of any external physical interpretation. The result provides a structurally rigid framework linking hypercube combinatorics, operator algebra, and graded decomposition, and may be of independent interest in algebraic and combinatorial settings.