Phase Emergence from Ordered Transport on Q5

We establish that a scalar phase-like invariant arises from ordered transport on the five-dimensional hypercube Q5 through a purely algebraic mechanism. The commutator of the defect and dressed transport generators is the unique source of order sensitivity in the branch difference. Under a symmetry-constrained reduction pipeline, all symmetric contributions are eliminated by mirror-orbit cancellation, and the surviving operator is confined to a one-dimensional antisymmetric defect sector spanned by the skew-Hermitian generator Ed = iσᵧ, derived algebraically from the Gray code closure defect. A canonical linear extraction functional then yields a unique scalar invariant μ with no free parameters. The extraction functional is formally identical to the Berry connection integrated over a closed cycle, and μ admits an explicit realization as the Berry phase of a six-segment loop on the Bloch sphere with multiplicity structure (4, 2) mirroring the Q5 adjacency structure.