Spectral Convergence of the Prime Gravity Hilbert-Polya Operator

We prove unconditionally that the eigenvalue sequence of the Prime Gravity geodesic Schrödinger operator Hgeo (W) = −d²/ds² + VPG (u (s) ; W) converges as the prime cutoff W → ∞. Three results are established: (1) the Prime Gravity potential VPG (u; W) converges in L² to a limiting potential VPG (u; ∞) at rate O ( (log W) ^−1/2) unconditionally, improving to O (W^−1/2+ε) under RH; (2) the operator family Hgeo (W) converges to a limiting operator H_∞ in norm resolvent sense; and (3) by Kato's perturbation theory, each eigenvalue λₖ (W) converges to λₖ (H_∞) unconditionally. We then state the Prime Gravity Spectral Conjecture — that λₖ (H_∞) = γₖ for all k, where γₖ 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