Emergent Gravitation from Phase Space Thermodynamics_Letter IV_Cosmological closure, Acoustic horizons and pre registered observational sentinels

Building upon the static phase-space impedance postulates (Letter I), the galactic GinzburgLandau condensate transition (Letter II), and the exact strong-field 2PN/2.5PN relativistic dynamics (Letter III), this fourth letter extends the Thermodynamic Phase-Space Gravity framework to the cosmological regime without invoking particle dark matter. Anchored solely by the Planck CMB expansion rate H0CMB = 67.36 km s-1 Mpc-1 and physical baryon density Ωb,0h2 = 0.02237, the spherical interface hierarchy (1 + 1/4π)n links the fundamental cosmological acceleration a0 = cH0CMB/(2π) = 1.0416 × 10-10 m s-2 (n = 0), the singleinterface local expansion rate H0loc = 72.72 km s-1 Mpc-1 (n = 1), and the galactic rotational transition scale g† = a0(1 + 1/4π)2 = 1.2140 × 10-10 m s-2 (n = 2). We show that the macroscopic susceptibility of the vacuum condensate generates an effective clustering matter density Ωm,eff,0 = 0.31629 (ωcond eff = 0.12115), while two-loop phase-space surface resistance introduces a logarithmic vacuum relaxation with exponent ε = 1/(16π2) ≈ 6.333 × 10-3 and ΩΛ,0 = 0.68361. Restoring the exact comoving redshift scaling yields sound horizons rs(z∗) = 144.14 Mpc at recombination and rdrag(zd) = 146.85 Mpc at the drag epoch. Using the semi-analytical Hu & Sugiyama (1995) formalism with explicitly declared external calibration constants (R32,HS,base = 0.9705, κHS = 0.52) and viscoelastic transfer modules |T (ω2)|2 ≈ 0.0555 and |T (ω3)|2 ≈ 0.0255, the predicted third-to-second CMB peak ratio spans R32,pred ∈ [0.9623, 0.9862] across three relaxation conventions, bracketing the Planck value R32,obs = 0.9708 ± 0.0119. Confrontation with the 13 Baryon Acoustic Oscillation (BAO) measurements of DESI DR2 across 7 tracers refutes a direct local impedance jump applied to BAO shells, whereas the homogeneous phase-space Friedmann background yields χ2 = 20.25/11 (χ2 tot = 21.49/13 including QSOs), traversing the existing ∼ 2.3σ DESI–Planck tension driven by the LRG2 bin (z = 0.71). Exact RK4 integration of linear perturbations establishes a growth factor D(0)/D(100) = 77.90. Finally, we formalize five pre-registered falsifiability sentinels, headed by the Gaia DR4 wide-binary test (γ = 1.00±0.02, due December 2, 2026).