Geometric Phase Realization of Q5 Transport Holonomy

This Theorem establishes an explicit geometric-phase realization of the Q5 observable coefficient law μ_obs = 4κ_adj + 2κ_non. A closed loop on the Bloch sphere is constructed from six constant-polar-angle azimuthal arcs of equal azimuthal sweep Δφ, four at polar angle θ_adj and two at θ_non. By the standard solid-angle contribution of constant-polar arcs (Ω = Δφ(1 − cosθ)), the total enclosed solid angle is: Ω = 4Δφ(1 − cosθ_adj) + 2Δφ(1 − cosθ_non) The Berry phase of a spin-½ eigenstate adiabatically transported along this loop is γ_Berry = −½Ω. Defining κ_adj := −½Δφ(1 − cosθ_adj) and κ_non := −½Δφ(1 − cosθ_non) gives γ_Berry = 4κ_adj + 2κ_non = μ_obs. The (4,2) multiplicity structure arises geometrically from the decomposition into four adjacent-class and two non-adjacent-class Bloch segments. Status: Solid angle lemma and Berry phase formula are standard results. This is a realization result: it establishes that μ_obs admits a geometric-phase realization. It does not assert that Gray-constrained transport uniquely induces this specific loop, nor that θ_adj and θ_non are fixed by Q5 structure. Uniqueness and intrinsic derivation are addressed in T41. Dependencies: T39.