The Natural Constant of the Hilbert–Pólya Operator: Arc-Length Oscillation and Asymptotic Convergence in the Prime Gravity Manifold

We identify the natural constant governing the domain geometry of the Prime Gravity Hilbert–Pólya operator Hgeo = −d²/ds² + VPG (u (s) ) introduced in Papers 7–8. The optimal excess arc-length Δs (W) = sₘax (W) − (log W − log 2) converges asymptotically to 3π/2 as W → ∞, with a finite-W correction term that oscillates at frequency ω = π/log (5) — corresponding to a period of exactly 5² = 25× in W-space — and decays as A (W) = B/log (W) ^α. The complete formula is: Δs (W) = 3π/2 − A (W) · cos (π · log (W) / log (5) ) This formula is theoretically motivated and empirically calibrated from two anchor measurements (α = 10. 486, B = 1. 349×10¹²). The asymptotic constant 3π/2 is identified as the Maslov phase of the Dirichlet Schrödinger operator on the Prime Gravity manifold. Applied to 13 prime cutoffs from W = 2×10⁶ to W = 10⁹, the formula achieves coll = 0 — every one of the first 20 nontrivial Riemann zeros resolved as a distinct eigenvalue — at all 13 tested scales with no per-W free parameters. The record alignment error of 0. 290 at W = 10⁹ (50, 847, 534 primes) improves on the Paper 8 record of 0. 349. A complete 10-step reproduction protocol and explicit falsifiability conditions are provided. Two supplementary WebGL visualizations are included: an interactive 3D valley landscape and an animated formula convergence viewer.