From Cosmological Viscosity to the Dirac Hierarchy: Static Liouville–Yukawa Screening in a Gradient Vacuum Substrate

The Optico-Hydrodynamic Vacuum (OHV) framework connects gigaparsec-scale cosmology to femtometer-scale nuclear physics via a single dimensionless vacuum viscosity parameter η =0.4182%, calibrated exclusively on Pantheon+ Type Ia supernovae (N = 1590, χ2/N = 0.901,H0 = 72.30 km s-1Mpc-1) and predicting nuclear binding energies across stable isotopes (A ≥ 20) with 96.3% mean accuracy using zero nuclear-fitted parameters. In this communication, we examine the extension of this viscoelastic substrate model to Newtonian gravitation and the 39-orders-of-magnitude Dirac hierarchy problem. We demonstrate analytically that in a strictly homogeneous barotropic background (ρbg = const), simultaneously matching the nucleon restmass deficit (Mp at core radius a0 = 0.84 fm) and the far-field gravitational constant G requires a dimensionless core-to-tail amplitude ratio of Πhier = 2π GM p c2a0 = 9.4152 × 10-40, which is strictly independent of the polytropic exponent γ. By introducing a radially inhomogeneous compression shell sbg(r) = ρeff(r)/ρ∞ supported by internal phase stresses—whose existence we verify numerically via a 1D boundary-value solution of the OHV field equations (Ω = 1.25,core compression s(0) = 4.918, binding ratio E/Q = 1.599 < 2, virial residual 5.81 × 10-14)—the static (ω = 0) radial perturbation equation transforms via a spherical Liouville substitution into a screened Yukawa–Schrödinger equation. Absorbing the 10-39 hierarchy factor reduces to an inverse boundary-value requirement for the dimensionless optical screening depth .Sscreen = - ln(Πhier) = 89.8611 across the femtometer shell, outside which an unscreened 1/r potential tail is recovered. Finally, we calibrate the terrestrial density gradient against the NIST 27Al+ optical clock redshift experiment (∆h = 0.33 m; Chou et al., 2010), explicitly demonstrating how the macroscopic parameters R⊕ and Np cancel algebraically to establish exact cross-scale self-consistency, and formulate falsifiable predictions at sub-femtometer and high-redshift scales.