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

We develop the finite-register thermodynamic and cosmological sector of thebounded architecture of Parts~I--II. Its common normalisation 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 $s_{\\rm vac}=\\alpha_Z^3/(4\\pi)$, giving$\\rho_\\Lambda\\simeq2.52\\times10^{-47}\\,{\\rm GeV}^4$,$\\rho_\\Lambda^{1/4}\\simeq2.24\\,$meV and$H_{\\rm sat}=5.33289\\times10^{13}\\,$GeV. With the Part~I physical-horizoncompletion, the same late density gives$a_{\\rm sat}=\\alpha_\\star cH_\\Lambda=1.2886\\times10^{-10}\\,{\\rm m\\,s^{-2}}$;this cross-sector identification is not fixed by the local waterbag algebra. The fifteen-state complement is lifted to a finite fermionic register. Itsbinomial entropy, Bernoulli KL free energy and permutation-symmetric single-flipGKSL generator give$\\dot{\\mathcal S}=-\\Gamma_c(\\mathcal S-s_{\\rm vac})$ and a Spohn $H$-theorem.The undressed rate is $\\Gamma_0/H_{\\rm sat}=(15/8)\\alpha_Z^{3/2}$, while thePart~II two-resolution completion fixes\\begin{equation*} \\Gamma_c=\\kappa_\\Gamma^{\\rm ME}\\Gamma_0, \\qquad \\kappa_\\Gamma^{\\rm ME}=1.101674939,\\end{equation*}without a tilt datum or changes to non-transition uses of $\\alpha_Z$. For oneindependent diagonal orientation, the closure projector has an exactnonstationary covariance and a nonnegative stationary spectrum with modes$j\\Gamma_c$, $1\\le j\\le15$; this does not determine a spatial influence kernelor change primordial observables. A charge-conserving rank-one dilation supplies the stiff Born-phase daughterwhile preserving the occupation law. With$\\Gamma_{\\rm V}=-\\dot{\\ln\\mathcal S}=\\Gamma_c(1-s_{\\rm vac}/\\mathcal S)$,simultaneous scalar-amplitude and pivot 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 has$r<0.0279385939$ for $n_s<1$, and $n_t$ differs strongly from $-r/8$. Vacuum density plus flatness gives $\\Omega_ch^2=0.11934$ in the statedneutrino convention and $b_{\\rm DM}=2.07197\\times10^{-28}$. Constant branchingis adiabatic; routing noise gives $\\beta_{\\rm iso}=6.10\\times10^{-20}$, thephase fraction is $6.21\\times10^{-8}$, and the Edgeworth completion gives$f_{\\rm NL}^{\\rm local}=0.00230135$. Finite-momentum influence, shape-resolvedhigher correlators and experiment-level likelihoods remain open.