Finite Capacity Gravitational Vacuum: Simultaneous Resolution of Galactic Kinematics, Wide-Binary Anomalies and Cosmological Tensions

We present a relativistic scalar-tensor framework of gravity founded upon a non-linear capacity saturation principle of the spacetime vacuum: the Finite Capacity Dynamic Vacuum. The theory is governed by a single universal acceleration threshold, the Sykes Saturation Scale g_S ≈ 1.20 × 10⁻¹⁰ m/s². In high-acceleration environments (g ≫ g_S), the Sykes Screening Mechanism exponentially suppresses scalar gradient deviations (S_S ~ exp(-√(g/g_S)) ~ 10⁻¹²⁴,¹⁷³ at terrestrial scales), preserving strict adherence to Solar System ephemerides and laboratory equivalence principle tests. In lowacceleration regimes (g ≲ g_S), the vacuum transitions to an unsaturated state, recovering the baryonic Tully-Fisher relation (v⁴ = G M g_S) without cold dark matter halos. Incorporating the 3D Galactic External Field Effect (EFE), we demonstrate that the Milky Way background field (g_ext ≈ 1.80 × 10⁻¹⁰ m/s²) imposes an asymptotic velocity boost plateau capped at +14.5% to +14.9% for wide stellar binaries (s > 5,000 AU), providing an exact statistical fit (χ²/dof ≈ 0.95 vs. Newtonian χ²/dof ≈ 6.42) to high-purity binned Gaia DR3 kinematics. Integrated into the CLASS Boltzmann engine as a dynamical vacuum fluid (w₀ = -1.216, w_a = 0.184, with cold dark matter density ω_cdm = 0.1030 and baryon density ω_b = 0.0224), the model simultaneously resolves the Hubble tension (H₀ = 72.18 km/s/Mpc) and the cosmic shear matter clustering tension (σ₈ = 0.7888). Furthermore, the finite capacity ceiling prevents central black hole curvature singularities.