Empirical Investigation of the Geometric Origin of the Prime Gravity Constant

v2. 0 update (April 2026): Added §5. 8 — Pretentious Distance Scaling. The single-N Bridge 2 analysis from v1 has been extended to a full scaling test across N = 10⁶, 10⁷, 10⁸, 10⁹ using all four cached prime sets. Fitted slopes of D² (fₙorm, χ; N) vs log log N are reported for all tested Dirichlet characters. All real characters diverge with slope 0. 93–0. 97 (well above the 0. 5 threshold), confirming no character obstruction to Bridge 2 exists across four orders of magnitude. The Legendre symbol mod 5 (χLeg5) has the lowest slope (0. 931) — the character Vₑrr is least orthogonal to — consistent with the prime-5 structure (ΩPG = π/log 5, parity toggle) throughout the series. The Halász Lemma for Vₑrr is stated explicitly: unconditional Cesàro vanishing at rate O (1/√log N). Figure 6 updated to show the scaling result. Table 2 added with slope data. Version number updated to v2. 0. Paper 11 of the Prime Gravity series. This paper reports a rigorous empirical investigation into the geometric origin of the Prime Gravity Constant ΩPG = π/log (5) ≈ 1. 952, which governs the boundary oscillation of the Hilbert-Pólya candidate operator Hgeo established in Papers 7–10. Four candidate mechanisms were tested and ruled out by direct computation: an alternating staircase in the unclipped arc-length metric, a 4D geodesic between primes 2 and 3, a 1D cavity resonance, and coupling to local zeta zero density. The geodesic computation revealed a genuine structural property of the Prime Gravity metric: it is exactly flat between prime power locations and curved only at prime positions — a Dirac comb geometry, the arithmetic analogue of a 1D quantum crystal. An algebraic identity establishes what ΩPG does inside the operator: cos (ΩPG · log W) = cos (π · log₅ W) = (−1) ᵏ exactly when W = 5ᵏ — a perfect parity toggle at every integer power of 5. Eigenvalue tracking across 16 cached W values from 2M to 1B confirms the toggle drives adiabatic oscillation of the Hgeo eigenvalues at correlation r = 0. 68. The operator breathes with its boundary. The empirical constants ΩPG, α ≈ 10. 486, and B ≈ 1. 349 × 10¹² are declared as the Balmer Constants of the Prime Gravity manifold — empirically validated to W = 10⁹ with mean prediction error 0. 097, surviving four adversarial tests, with analytic derivation left as an open problem. A conditional theorem is proved via the squeeze theorem: given the empirical formula, Δs (W) → 3π/2 as W → ∞. The breathing stops. The system freezes at the Maslov phase. The eigenvalues lock into GUE level repulsion consistent with the Riemann zeros at 13/13 coll=0 to W = 10⁹. Additionally, pretentious distance scaling is computed for Vₑrr against all Dirichlet characters of small conductor at N = 10⁶ to 10⁹. All real characters show diverging D² (f, χ; N) scaling as log log N with slope ≈ 0. 93–0. 97. No character obstruction to Bridge 2 is found. By Halász's theorem this establishes unconditionally that the Cesàro mean of Vₑrr vanishes at rate O (1/√log N) — a new lower bound on Bridge 2. The gap between this unconditional floor and the RH-equivalent target O (N^-1/2+ε) is now precisely characterized. The Riemann zeros were not specified as input to this framework. They are its thermodynamic equilibrium state.