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. The universal saturationnormalisation is fixed independently by the two-state isolated-horizon count,\begin{equation*} \sum_{j=1/2,1,\ldots}2\, \exp[-2\pi\alpha_\star\sqrt{j(j+1)}]=1, \qquad \alpha_\star=0.2375329579.\end{equation*}With the Part~I physical-horizon completion, the same late density then gives$a_{\rm sat}=\alpha_\star cH_\Lambda=1.2886\times10^{-10}\,{\rm m\,s^{-2}}$.The horizon-counting normalisation is independent of galaxy data; theidentification of that normalisation with the local saturation variable is theexplicit cross-sector step. \textbf{Methods.}\ The spatial causal-patch fluctuation is completed by a singlepermutation-symmetric collective register whose Dicke-basis occupation law isbinomial; this reproduces the declared KL rate without interpreting$15S_H$ as independent classical records. 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 $\Gamma_0/H_{\rm sat}=(15/8)\alpha_Z^{3/2}$. Part~II nowcloses the channel-democratic gauge boundary with the two-loop zero-thresholdPati--Salam chain, $M_I=9.22\times10^{10}$~GeV and$M_U=1.180\times10^{16}$~GeV, and its maximum-entropy two-resolution completion fixes\begin{equation*} \Gamma_c=\kappa_\Gamma^{\rm ME}\Gamma_0, \qquad \kappa_\Gamma^{\rm ME}=1.039276057,\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. \textbf{Results.}\ 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_*=0.9353033134,\quad b_r=5.2703\times10^{-5},\quad n_s=0.967388698,\quad r=0.017066078,\end{equation*}\begin{equation*} n_t=-0.033734887,\qquad \alpha_s=-1.69267\times10^{-3},\qquad N_{*\to\rm end}=60.00600.\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 $b_{\rm DM}=6.31811\times10^{-28}$. Constant branchingis adiabatic; routing noise gives $\beta_{\rm iso}=8.39\times10^{-22}$, thephase fraction is $5.49\times10^{-8}$, and the Edgeworth completion gives$f_{\rm NL}^{\rm local}=0.00368469$. \textbf{Conclusions.}\ Finite-momentum influence, shape-resolvedhigher correlators and experiment-level likelihoods remain open.