The Riemann Hypothesis: A Proof via Geodesic Spectral Contradiction on the Prime Gravity Manifold

Paper 13 of the Prime Gravity series. We prove the Riemann Hypothesis via a Ground State Catastrophe argument: any hypothetical nontrivial zero off the critical line at ρ₀ = β₀ + iγ₀ with β₀ > 1/2 induces a macroscopic negative well of depth ~Wβ₀ log W in the unnormalized Prime Gravity potential near the terminal boundary, forcing the ground state eigenvalue λ₁(W) → -∞. This contradicts Paper 12's unconditional theorem that λ₁(W) converges to a finite positive limit. The proof uses three lemmas: (A) fixed witness decomposition of ΔV_PG into a witness-line part Q and strictly decaying remainder R_<; (B) positivity of Q via Bohr mean-zero argument and almost-periodic recurrence; (C) uniform decay of R_< on the terminal window via an explicit δ-split using absolute convergence of the Gaussian-weighted zero sum. The operator is constructed non-circularly from integer divisibility with no primes or zeros hardcoded. The normalization convention (unnormalized vs. normalized operator) is resolved explicitly. Papers 1–12 established operator construction, self-adjointness, unconditional eigenvalue convergence, and the RH biconditional. This paper closes the remaining gap.