Parity-Optimal Hamiltonian Cycles and a Unique 2-Adic Tier Structure on Q5

A Hamiltonian cycle on the five-dimensional hypercube that maximizes a natural alternating parity functional is forced, up to symmetry, to follow a step-direction sequence governed by the 2-adic valuation of the step index. This determines the flip frequencies uniquely as (16, 8, 4, 2, 2) and produces a canonical three-tier hierarchy with normalized ratio 1:3:12. The structure is not assumed; it is derived entirely from the optimality condition established in the companion result on maximal parity alignment. This theorem establishes the tier structure as a combinatorial consequence of parity optimality on Q5.