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{equation*} ρₘₐₓ=16/7f^4, ρV( S)=ρₘₐₓ S¹⁵, ρ_Λ=ρV(svac),{equation*}with f=vαZ-13/2 and svac=αZ³/(4π), givingρ_Λ2.52×10⁻⁴⁷\, GeV⁴,ρ_Λ1/42.24\,meV andHₛₐₜ=5.33289×10¹³\,GeV. The universal saturationnormalisation is fixed independently by the two-state isolated-horizon count,{equation*} ∑j=1/2,1,…2\, exp[-2πα_√j(j+1)]=1, α_=0.2375329579.{equation*}With the Part~I physical-horizon completion, the same late density then givesaₛₐₜ=α_ cH_Λ=1.2886×10⁻¹⁰\, 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. 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 interpreting15SH 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 giveṠ=-Γc( S-svac) and a Spohn H-theorem.The undressed rate is Γ₀/Hₛₐₜ=(15/8)αZ3/2. Part~II nowcloses the channel-democratic gauge boundary with the two-loop zero-thresholdPati--Salam chain, MI=9.22×10¹⁰~GeV andMU=1.180×10¹⁶~GeV, and its maximum-entropy two-resolution completion fixes{equation*} Γ_c=κ_ΓMEΓ_0, κ_ΓME=1.039276057,{equation*}without a tilt datum or changes to non-transition uses of αZ. For oneindependent diagonal orientation, the closure projector has an exactnonstationary covariance and a nonnegative stationary spectrum with modesjΓc, 1≤ j≤15; this does not determine a spatial influence kernelor change primordial observables. Results.\ A charge-conserving rank-one dilation supplies the stiff Born-phase daughterwhile preserving the occupation law. WithΓV=-ln Ṡ=Γc(1-svac/ S),simultaneous scalar-amplitude and pivot matching gives{equation*} S_*=0.9353033134, b_r=5.2703×10⁻⁵, n_s=0.967388698, r=0.017066078,{equation*}{equation*} n_t=-0.033734887, α_s=-1.69267×10⁻³, N*→ end=60.00600.{equation*}The rigid red locus has$r<0.0279385939$ for nₛ<1, and nₜ differs strongly from $-r/8$. Vacuum density plus flatness gives Ωch²=0.11934 in the statedneutrino convention and bDM=6.31811×10⁻²⁸. Constant branchingis adiabatic; routing noise gives βᵢₛₒ=8.39×10⁻²², thephase fraction is 5.49×10⁻⁸, and the Edgeworth completion givesfNLˡᵒᶜᵃˡ=0.00368469. Conclusions.\ Finite-momentum influence, shape-resolvedhigher correlators and experiment-level likelihoods remain open.