The abelian scaling bottleneck, and the angular Hurwitz system: where noncommutative arithmetic thermodynamics can and cannot live

Two structural results on the possibility of a genuinely noncommutative arithmetic thermodynamics. First, a universal no-go, the abelian scaling bottleneck: for a right LCM semigroup with scalar one-parameter dynamics driven by a multiplicative weight N, every KMS state is tracial on the gauge-fixed algebra, the classification splits into a scaling measure governed by the abelianization-graded weight and a beta-independent tracial datum on the isotropy, and no nonabelian residual symmetry group can arise on the scaling side - verified concretely on the braid monoids, right LCM and maximally noncommutative, whose entire transition is that of the length series. This retro-explains the abelian essential-value groups, the norm-one reduction, and the hyperfiniteness no-go found earlier in the series. Second, the constructive half: noncommutativity lives transverse to the scaling, as an ordered cocycle into a compact group, and the canonical arithmetic candidate is named - the angular Hurwitz system, whose radial part carries the known thermodynamics with critical exponent 2 and whose angular part a/Nrd(a)^{1/2} in SU(2) is an ordered cocycle whose matrix Kakutani series is exactly computable by the a -> -a symmetry, with abscissa exactly 2: the angle turns critical precisely where the radius does. The programme - transporting the matrix-martingale coboundary theorem to the Hurwitz tail relation, expected verdict KMS simplex Prob(SU(2)) in the entire Gibbs phase, with the Lubotzky-Phillips-Sarnak spectral gap closing the critical edge - is stated with its single hard step named.