Grading-Admissibility Forces Gray: A Dual Characterization of Hamiltonian Cycles on Q5

We establish a dual characterization of Hamiltonian cycles on the five-dimensional hypercube Q₅ through two independent mechanisms. The first is variational: the alternating parity functional H(C) is uniquely maximized by the reflected Gray code, achieving ‖H(C*)‖² = 344. The second is structural: a grading-admissibility condition derived from a sector decomposition of Q₅ relative to a distinguished coordinate eliminates all combinatorial freedom in the construction of a Hamiltonian cycle. The main theorem proves these two characterizations are equivalent: a Hamiltonian cycle on Q₅ is grading-admissible if and only if it is parity-optimal, and both conditions uniquely determine the reflected Gray cycle up to cube automorphism and reversal. The structural proof proceeds through five successive eliminations: bypass exclusion, middle transmission, support contiguity, noncrossing transmission, and reflection block identification. The reflected Gray cycle is thereby identified as the unique fixed point of two independent constraint systems (one global and variational, one local and structural), neither of which references the other.