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

Abstract

Parts~I and II construct a bounded saturation field, one reciprocal physicalmetric, a kinetic-ceiling pressureless branch, a finite chiral register and arank-fifteen determinant-line vacuum operator. The common normalisation is\[ \rho_{\max}=\frac{16}{7}f^4, \qquad \rhoV(\Sc)=\rho_{\max}\Sc^{15}, \qquad \rhoL=\rhoV(\Sv),\]with\[ f=v\alpha_Z^{-13/2},\qquad \Sv=\frac{\alpha_Z^3}{4\pi}, \qquad \rhoL=\frac{16}{7(4\pi)^{15}}v^4\alpha_Z^{19} =2.5168011\times10^{-47}\ {\rm GeV}^4.\]The late vacuum therefore has fourth root $2.2398$ meV, while completesaturation gives $H_{\rm sat}=5.34374\times10^{13}$ GeV. The same late densitydefines $\HL^2=8\pi G\rhoL/(3c^2)$ and the galactic corollary$a_{\rm sat}=\alpha_\star c\HL=1.2867\times10^{-10}$ m s$^{-2}$. The active fifteen-state complement is lifted to a finite fermionic register.Its binomial multiplicities give positive Shannon--Boltzmann entropy, whilethe zero-normalised local Kullback--Leibler functional is exactly the excessHelmholtz free energy of a Bernoulli channel. A permutation-symmetricsingle-flip GKSL generator yields the exact mean law$\dot\Sc=-\Gamma_c(\Sc-\Sv)$ and a Spohn relative-entropy $H$-theorem. Theminimal spectral gap is$\Gamma_c/H_{\rm sat}=(15/8)\alpha_Z^{3/2}$. A charge-conserving rank-one dilation makes the stiff daughter an explicitBorn-phase component. The fifteen active states carry primitive Born charges$6\times1$, $6\times2$, $2\times3$ and $1\times0$, so$\sum q_a=24$ and $\sum q_a^2=48$. Energy-resolved jump operators routeinteger charge either to the compact phase rotor or to the invertible inflowreservoir, commute with the total Born charge, preserve the exact occupationlaw, and remain completely positive over the accelerated branch. An explicitMonte Carlo unravelling of the same integer-charge jumps reproduces the meanoccupation law, preserves $Q_{\rm tot}$ event by event and recovers thefinite-register shot-noise scaling. Defining the exact vacuum-transfer clock$\GammaV:=-\dot{\ln\Sc}=\Gamma_c(1-\Sv/\Sc)$, the sourced system is\[ \dot\rhoV=-15\GammaV\rhoV, \quad \dot\rho_r+4H\rho_r=15b_r\GammaV\rhoV, \quad \dot\rho_\theta+6H\rho_\theta=15b_\theta\GammaV\rhoV.\]Simultaneous scalar-amplitude and exact pivot matching at the minimal gap fixes\[ \Sc_*=0.9302677564,\quad b_r=7.3684942\times10^{-5},\quad b_\theta=0.9999263151,\]\[ n_s=0.9648101406,\quad r=0.0158028531,\quad n_t=-0.0337939326,\quad \alpha_s=-1.57790\times10^{-3}.\]The pivot lies $59.90214$ e-folds before the end of acceleration. The globallymatched red branch obeys the strict boundary $r<0.0280525926$. Reweighting thearchived tensor-transfer kernels with the exact three-fluid primordial shapegives $r_{\rm eff}=0.01655217$ in a flat $n_t=0$ template over$20\leq\ell\leq330$. Adding the separate kinetic-ceiling pressureless daughter and matching$\Omega_ch^2=0.120$ fixes$b_{\rm DM}=8.3246586\times10^{-28}$ without changing the primordialobservables. This is the homogeneous abundance of the same condensate whosenonlinear galactic organisation is described in Part~I, not an additional darkcomponent. Constant branching gives an exact adiabatic separate-universesolution; conservative routing noise gives$\beta_{\rm iso}<3.3\times10^{-22}$. The constant Born-phase shift has zerolinear stress, while the independent momentum mode contributes only$\mathcal P_\zeta^{(\theta)}/A_s=5.09\times10^{-8}$. Extending the declaredBernoulli rate function to its leading Edgeworth cumulants predicts$f_{\rm NL}^{\rm local}=0.0043798$. The remaining open problems are the UVderivation of the bath spectral density, the complete finite-momentumSchwinger--Keldysh influence functional, shape-resolved higher correlators andexperiment-specific likelihoods.