Unitary-orbit classification and a refinement theorem for context-independent projective probabilities

Let mathcal H be a finite-dimensional complex Hilbert space, and let w\(P,mathsf M\) be a normalized probability weight assigned to an outcome projection P as it occurs in a projective measurement mathsf M. For fixed P, let G_Pcongmathcal U\(P^perp\) be the group of unitaries acting identically on operatorname{ran}P and arbitrarily on P^perp. We classify the G_P-orbits of projective measurements containing P: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile λ=\(r₁,ldots,r_k\) is a compact homogeneous space of real dimension \(d-operatorname{rank}P\)²-sum_j r_j². On maximal rank-one measurements there is one orbit, so context independence is equivalent to G_P-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome Pneq I coarsens to the unique binary context \{P,I-P\}. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is d-operatorname{rank}P-1 when Pneq I. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form operatorname{Tr}(ρP). The results are structural characterizations, not independent physical derivations of context independence.