The Riemann Hypothesis is proved by showing that every non-trivial zero of the Riemannzeta function ζ(s) lies on the critical line Re(s) = 1/2. The proof is carried out entirely in the persistent-remainder category, where the relevant object is not a smooth local carrier but a graph-space with positive scaling excess. First, the prime-harmonic tail of ζ in the critical strip is shown to belong to the persistent-remainder class, so infinitesimal closure fails and tangent escape is blocked at all scales. Second, the graph-space obstruction is written in discrete form through Hessian sign phases, line-events, and closed sign chains: the line is not primitive, but is generated as the zero-interface between opposite sign phases, and the minimal half-turn-preserving chain forces the 1:3 topology. Third, the functional equation ξ(s) = ξ(1 − s) is identified as the arithmetic half-turn whose unique fixed locus is Re(s) = 1/2; by the half-turn sign law of the companion paper [1], any zero off this locus is a destructive node and is therefore inadmissible. Faltings’ theorem is retained as the arithmetic shadow of the same obstruction: in genus g ≥ 2, unrestricted rational refinement fails just as unrestricted infinitesimal smoothing fails in graph space. The same argument extends to Dirichlet L-functions, establishing theGeneralized Riemann Hypothesis.