This Theorem establishes an explicit geometric-phase realization of the Q5 observable coefficient law μₒbs = 4κₐdj + 2κₙon. 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 θₐdj and two at θₙon. By the standard solid-angle contribution of constant-polar arcs (Ω = Δφ (1 − cosθ) ), the total enclosed solid angle is: Ω = 4Δφ (1 − cosθₐdj) + 2Δφ (1 − cosθₙon) The Berry phase of a spin-½ eigenstate adiabatically transported along this loop is γBerry = −½Ω. Defining κₐdj: = −½Δφ (1 − cosθₐdj) and κₙon: = −½Δφ (1 − cosθₙon) gives γBerry = 4κₐdj + 2κₙon = μₒbs. 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 μₒbs admits a geometric-phase realization. It does not assert that Gray-constrained transport uniquely induces this specific loop, nor that θₐdj and θₙon are fixed by Q5 structure. Uniqueness and intrinsic derivation are addressed in T41. Dependencies: T39.
Craig Edwin Holdway (Sun,) studied this question.