Gray-Structured Transition Decomposition and Sector-Resolved Operator Dynamics on Q5

This paper presents a self-contained structural result on the five-dimensional hypercube Q5. The main result establishes a correspondence between Gray code edge decomposition, operator-induced sector transitions, and scalar commutator dynamics on simultaneous eigensectors. The construction is purely mathematical, falling within the scope of discrete and combinatorial structures, with an explicit operator-algebraic realization. This work is independent of the author's concurrent submissions on Q5, each addressing distinct structural results. The present paper focuses specifically on transition decomposition and operator correspondence and does not rely on results from those works.