Pressure-Mediated Gravity IV (PMG-ENV) Environmental Dependence of a₀, Early-Type Galaxies, and a Joint Kinematic Constraint on the Acceleration Scale

In Pressure-Mediated Gravity (PMG), the global acceleration scale a₀ = cH₀/2π is derived from first principles 7. We conjecture that in inhomogeneous large-scale environments the local superfluid vacuum responds to the local Hubble rate, giving a₀ˡᵒᶜᵃˡ = cHˡᵒᶜᵃˡ/2π. This motivated ansatz predicts Δa₀/a₀ = ΔH/H₀ ~ 20% between void and filament environments. We test this conjecture by stratifying the 175-galaxy SPARC sample from Paper III by environment using CosmicFlows-3 peculiar velocities and the SDSS void catalogue. Void galaxies yield a₀ᵛᵒⁱᵈ = (0.91 ± 0.15) × 10⁻¹⁰ m s⁻², filament galaxies yield a₀ᶠⁱˡ = (1.10 ± 0.12) × 10⁻¹⁰ m s⁻², consistent with the conjecture at 2 for all models the likelihood is misspecified and the AIC/BIC comparison (ΔBIC ≈ −453) cannot be read on the standard Kass we rely primarily on the permutation result. Environmental correction improves median χ²ν from 2.57 to 2.41. We extend to 52 ATLAS3D early-type galaxies via Jeans Anisotropic Modelling, finding median χ²ν = 3.12 and Υe = 0.71 M☉/L☉; the excess χ²ν relative to disk galaxies is quantitatively accounted for by JAM velocity anisotropy. An approximate joint consistency check yields a₀ʲᵒⁱⁿᵗ = (1.09 ± 0.10) × 10⁻¹⁰ m s⁻², consistent with PMG at 0.5σ. Environmental variation a₀ ∝ Hˡᵒᶜᵃˡ is a unique, falsifiable conjecture distinguishing PMG from all fixed-a₀ modified-gravity frameworks.