The angular transport: tree-level filtrations, predictable matrix martingales, and the two regimes of the angular Hurwitz system

The matrix-martingale coboundary theorem for noncommutative cocycles, previously proved for independent geometric coordinates, is transported to the Hurwitz tail relation, whose coordinates decompose along (p+1)-regular Bruhat-Tits trees. The Gibbs conditional law on tree paths is computed exactly from the scaling axiom: geometric depth with ratio p^{1-beta} and uniform non-backtracking labels. The martingale survives with a predictable correction - a reversed product of inverse conditional means - and for beta > 2 it is L2-bounded with invertible limit, so the angular cocycle a -> a/Nrd(a)^{1/2} in SU(2) is a coboundary and the KMS simplex of the angular Hurwitz system is Prob(SU(2)) throughout the Gibbs phase, modulo two named verification points. The role of the Lubotzky-Phillips-Sarnak/Deligne bound is corrected: coboundary and mixing are mutually exclusive regimes, so the Ramanujan contraction belongs to the critical edge, where it supplies the mixing hypothesis and forces full restoration of the angular symmetry. One boundary at beta = 2, two regimes: a genuinely nonabelian simplex above, uniqueness at the edge.