Holographic Saturation III:Finite-Register Thermodynamics, Charged Desaturation, and Primordial Cosmology

We develop the finite-register thermodynamic and cosmological sector of thebounded saturation architecture introduced in Parts~I and II. The commonnormalisation is\\begin{equation*} \\rho_{\\max}=\\frac{16}{7}f^4,\\qquad \\rho_{\\rm V}(\\mathcal S)=\\rho_{\\max}\\mathcal S^{15},\\qquad \\rho_\\Lambda=\\rho_{\\rm V}(s_{\\rm vac}),\\end{equation*}with $f=v\\alpha_Z^{-13/2}$ and$s_{\\rm vac}=\\alpha_Z^3/(4\\pi)$. This gives$\\rho_\\Lambda\\simeq2.52\\times10^{-47}\\,{\\rm GeV}^4$,$\\rho_\\Lambda^{1/4}\\simeq2.24\\,$meV and$H_{\\rm sat}=5.33289\\times10^{13}\\,$GeV. The same late density defines thegalactic scale $a_{\\rm sat}=\\alpha_\\star cH_\\Lambda=1.2886\\times10^{-10}\\,{\\rm m\\,s^{-2}}$ used in Part~I. The active fifteen-state complement is lifted to a finite fermionic register.Its binomial multiplicities give positive Shannon--Boltzmann entropy, while thezero-normalised local Kullback--Leibler functional is the excess Helmholtz freeenergy of a Bernoulli channel. A permutation-symmetric single-flip GKSLgenerator yields the exact mean law$\\dot{\\mathcal S}=-\\Gamma_c(\\mathcal S-s_{\\rm vac})$ and a Spohnrelative-entropy $H$-theorem. The undressed invariant rate is$\\Gamma_0/H_{\\rm sat}=(15/8)\\alpha_Z^{3/2}$. The maximum-entropytwo-resolution completion of Part~II fixes\\begin{equation*} \\Gamma_c=\\kappa_\\Gamma^{\\rm ME}\\Gamma_0, \\qquad \\kappa_\\Gamma^{\\rm ME}=1.101674939,\\end{equation*}without using a primordial-tilt datum and without replacing $\\alpha_Z$ in anynon-transition sector. A charge-conserving rank-one dilation makes the stiff daughter an explicitBorn-phase component while preserving the exact occupation law. Defining thevacuum-transfer clock$\\Gamma_{\\rm V}=-\\dot{\\ln\\mathcal S}=\\Gamma_c(1-s_{\\rm vac}/\\mathcal S)$, simultaneous scalar-amplitude and exactpivot matching gives\\begin{equation*} \\mathcal S_*=0.9434323905,\\quad b_r=1.2381\\times10^{-5},\\quad n_s=0.972580831,\\quad r=0.019430883,\\end{equation*}\\begin{equation*} n_t=-0.033510615,\\qquad \\alpha_s=-1.94362\\times10^{-3},\\qquad N_{*\\to\\rm end}=60.40385.\\end{equation*}The rigid red locus obeys $r<0.0279385939$ for $n_s<1$, while the transitioncompletion selects one point on that locus. The tensor tilt remains far fromthe canonical single-field relation $n_t=-r/8$. The predicted vacuum density and spatial flatness give$\\Omega_ch^2=0.11934$ in the stated neutrino convention, and the four-fluidclosure fixes $b_{\\rm DM}=2.07197\\times10^{-28}$. Constant branching isadiabatic; conservative routing noise gives$\\beta_{\\rm iso}=6.10\\times10^{-20}$. The independent phase mode contributesonly $\\mathcal P_\\zeta^{(\\theta)}/A_s=6.21\\times10^{-8}$, while the statedEdgeworth completion gives $f_{\\rm NL}^{\\rm local}=0.00230135$. The fullfinite-momentum influence functional, shape-resolved higher correlators andexperiment-specific likelihoods remain outside the present completion.