Linking the BAO Jeans Scale to the Dirac Hierarchy via Static Liouville–Yukawa Screening in a Viscous Vacuum Substrate.

Within the Optico-Hydrodynamic Vacuum (OHV) framework, a single dimensionless viscosity parameter ηcosmo = 0.004182, calibrated exclusively on Pantheon+ Type Ia supernovae (N = 1590, χ2/N = 0.901, RH = 4146.8 Mpc), predicts both nuclear binding energies (96.3% mean accuracy for A ≥ 20) and the stationary cosmological Jeans length λJ = 2π√2 ηcosmoRH ≈ 154.1 Mpc (matching the de-projected physical BAO scale within 4.8% on a background substrate density ρsub = c2 8πGR2 H ≈ 3.27×10-27 kg/m3 = 1 3ρcrit). In this unified communication, we establish a direct cross-scale link between this cosmological Jeans scale and the 39-orders-of-magnitude Dirac hierarchy gap (Πhier = π2 GMp c2a0 = 9.4138×10-40 at proton core radius a0 = 0.84 fm). First, we resolve the wave-speed and tail-energy consistency between the micro- and macro-sectors by distinguishing the massless scalar-field mode □η = 4πG c4 T (propagating at speed c with convergent far-field energy density Etail = c4 8πG|∇η|2 ∝ r-4) from the longitudinal mass-density mode δρ (propagating at acoustic speed cs = ηcosmoc = 1.2537×106 m/s). Second, using a verified 1D boundary-value solution of the nonlinear OHV core equations (Ω = 1.25, core compression s(0) = 4.918, E/Q = 1.599 < 2, virial residual 5.81×10-14), we formulate the 10-39 suppression as static Liouville–Yukawa screening with required optical depth Sscreen = - ln(Πhier) = 89.8612. Crucially, we report a non-trivial parameterfree cross-scale relation: the logarithm of the ratio of the cosmological acoustic scale to the nucleon core radius, Scosmo = ln2ηcosmoRH a0 = ln λJ π√2 a0 = 90.0430, matches Sscreen = 89.8612 to within 0.20% without any adjustable factor, and to within 0.0006% (89.8607, or 0.051% in amplitude via GM pηcosmoRH c2a20= 0.9420 ≈ 3π 10 ) when accounting for the uniform-sphere geometric factor 3 5 . All exact algebraic identities (including the π2 volume-action cancellation and the NIST optical clock redshift closure) are explicitly separated from empirical cross-scale relations.