Holographic Saturation III: Boltzmann Entropy, Primordial Saturation, and the Cosmological Arrow of Time

We investigate the statistical and primordial-cosmology consequences of the fifteen-channel vacuum closure developed in Parts I and II. The active module is lifted to the fermionic register Λ∙A\Lambda^\bullet\mathcal AΛ∙A, where the top-exterior projector represents complete occupation. Within a permutation-symmetric single-flip Lindblad completion, the collective occupation variable obeys the exact relaxation law S˙=−Γc(S−svac)\dot{\mathcal S}=-\Gamma_c(\mathcal S-s_{\rm vac})S˙=−Γc(S−svac). Quantum relative entropy contracts according to the Spohn inequality, producing a Kullback-Leibler HHH-theorem and non-negative entropy production. The collective vacuum density scales as ρV∝S15\rho_V\propto\mathcal S^{15}ρV∝S15, while the apparent-horizon entropy scales as S−15\mathcal S^{-15}S−15, yielding an exact closure-horizon reciprocity and a generalised second law during desaturation. The background evolution is analytically solvable and exits acceleration automatically. A conditional invariant spectral-gap completion fixes the relaxation rate without introducing a new continuous parameter and gives approximately 102 total e-folds. Primordial fluctuations are closed through a holographic large-deviation postulate that respects the horizon Hilbert-space capacity. Combined with the single-clock attractor, this produces a one-dimensional consistency curve for AsA_sAs, nsn_sns, and rrr. The observed scalar amplitude selects a unique red-tilted solution with N∗=59.17N_*=59.17N∗=59.17, ns=0.96502n_s=0.96502ns=0.96502, and r=0.01588r=0.01588r=0.01588. These predictions depend explicitly on the large-deviation, spectral-gap, and single-clock completions.