Cosmological Implications of the Logarithmic Superfluid Vacuum: Dark Energy, the Hubble Tension, and a Vacuum Self-Consistency Equation

The cosmological consequences of modeling the vacuum as a relativistic logarithmic superfluid are derived. The logarithmic equation of state yields w = −1 natively — dark energy is the thermodynamic pressure of the condensate, with w obtained as the exact, coupling-independent ratio P/ε at the extremum of the self-interaction potential, the same background point that fixes cₛ = c and α = −1. Combining the emergent gravitational coupling Gₑff ∼ c²/ (ξ²ρ₀) with the empirical relation Ω_γ ≈ α² gives a vacuum self-consistency equation, α²H₀²ρ₀ξ² ∼ (8π/3) uCMB, linking the fine-structure constant, the Hubble parameter, the vacuum density, and the CMB photon energy density. When the vacuum density varies spatially, the local speed of light, the gravitational constant, and the effective cosmological constant vary together in calculable ways set by the equation of state; the Λ ∝ c⁴ dependence provides a steep lever arm. A modest ~3. 6% vacuum underdensity inside the observed KBC void suffices to raise the locally measured H₀ to ~73 km s⁻¹ Mpc⁻¹ from a global 68. 15 km s⁻¹ Mpc⁻¹, reframing the Hubble tension as an environmental effect rather than a crisis. The CMB acoustic-peak positions are proven exactly invariant under these variations, because both the sound horizon and the comoving distance scale linearly with the local speed of light — so the framework satisfies the most stringent constraint in observational cosmology without tuning. Promoting Ω_γ = α² to a postulate yields a parameter-free prediction TCMB (α, H₀) that matches the FIRAS measurement to ~0. 003%. The CMB temperature is reinterpreted as the present-day equilibrium temperature of the condensate. A first-principles derivation of the Ω_γ = α² relation, the void density profile, and the role of vacuum perturbations in structure formation remain open problems.