What question did this study set out to answer?

This research aims to identify the natural constant related to the geometry of the Prime Gravity Hilbert–Pólya operator.

April 13, 2026Open Access

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

Key Points

This research aims to identify the natural constant related to the geometry of the Prime Gravity Hilbert–Pólya operator.
Mathematical derivation of the arc-length formula Δs(W) from the Hilbert–Pólya operator.
Empirical calibration using two anchor measurements for constants α and B.
Numerical application of the formula across 13 prime cutoff values, measuring alignment with Riemann zeros.
The optimal excess arc-length Δs(W) converges to 3π/2 as W approaches infinity.
Every one of the first 20 nontrivial Riemann zeros aligns as a distinct eigenvalue at all tested scales.
Achieved a record alignment error of 0.290 at W = 10⁹, an improvement over previous findings.

Abstract

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.

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