Graded Adjoint Operator Decomposition on the Five-Dimensional Hypercube

This paper develops a finite-dimensional operator-theoretic structure on the five-dimensional hypercube Q_5 = {0,1}^5 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 C[Q_5] via spectral projectors of the complementary operator J_z^{(i)}, without assuming irreducibility of the representation.

This yields a canonical graded operator module on Q_5, 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.