Spectral Convergence of the Prime Gravity Hilbert-Polya Operator

We prove unconditionally that the eigenvalue sequence of the Prime Gravity geodesic Schrödinger operator H_geo(W) = −d²/ds² + V_PG(u(s); W) converges as the prime cutoff W → ∞. Three results are established: (1) the Prime Gravity potential V_PG(u; W) converges in L² to a limiting potential V_PG(u; ∞) at rate O((log W)−1/2) unconditionally, improving to O(W−1/2+ε) under RH; (2) the operator family H_geo(W) converges to a limiting operator H_∞ in norm resolvent sense; and (3) by Kato's perturbation theory, each eigenvalue λ_k(W) converges to λ_k(H_∞) unconditionally. We then state the Prime Gravity Spectral Conjecture — that λ_k(H_∞) = γ_k for all k, where γ_k are the imaginary parts of the nontrivial Riemann zeros — and prove it is equivalent to the Riemann Hypothesis via the Mellin pole structure of Paper 2 and the biconditional of Paper 5. Three routes toward the Spectral Conjecture are identified for Paper 13: inverse scattering (Gel'fand–Levitan–Marchenko), trace formula (Selberg model), and direct eigenfunction construction. Keywords: Prime Gravity, Hilbert-Pólya conjecture, spectral convergence, Kato perturbation theory, norm resolvent convergence, Riemann Hypothesis, limiting operator, Mellin poles, geodesic Schrödinger operator, von Mangoldt function, explicit formula, L² convergence, eigenvalue convergence, Schrödinger operator, self-adjoint operator