Holographic Saturation I:Universal Reciprocal Geometry, Entropic Screening, and Galactic Phenomenology

We formulate a relativistic bounded complex saturation fieldI=√s\,eiθ, $0<s<1$, from a constrained parent action. TheEinstein--Hilbert and matter sectors share the physical metricgμν; the seed metric is auxiliary and is eliminated on thetimelike phase branch. The reduced action is\[ S=∫\!^4x√- g[ { MPl^2}{2} R+ L_I( g,s,θ)] +S_m[ g,χ],\]and is a two-field first-derivative k-essence system: two scalar modes accompanythe two tensor modes, with the Born phase on the observable cone and theamplitude on the seed cone. Methods.\ A locally uniform (s,u^μ) is removable by a constant tetrad transformation;gradients remain observable. In the static phase-comoving weak field,δΦₛ=δΨₛ=(c²/2)ln B. The standard-kernelnear-zone PPN projection gives\[ γ=β=1, ξ=0, α_1=α_2=α_3=0, ζ_1=ζ_2=ζ_3=ζ_4=0,\]while finite-range/cosmological tidal terms remain separate. Matter sourcesthe scalar sector only through the physical metric, and non-zero finite chargeselects a unique linearly stable timelike phase branch. Results.\ The bounded KL free energy givessvac=αZ³/(4π)=3.80×10⁻⁸; the orbit-channel law givesf=vαZ-13/2=1.220×10¹⁶\,GeV, hencemcan=8.2×10⁻³¹\,eV and λcan=7.8\,Mpc. Thesecharacterise the radial excitation, not a baryon-sourced screening length ordirect fifth-force charge. Galactic phenomenology belongs to the kinetic-ceiling condensate, with boundedsusceptibility\[ ∇\!·\![ {|∇Φ|}{√{aₛₐₜ^2+|∇Φ|^2}}∇Φ] =4π Gρ_b.\]Under the stated physical-horizon response identification,aₛₐₜ=α_ cH_Λ=1.2886×10⁻¹⁰\, m\,s⁻²;the waterbag algebra fixes the response form but not this cross-sector gap.The exact spherical branch gives V_∞⁴=GaₛₐₜMb. Solving the fullaxisymmetric equation for all $165$ usable SPARC systems gives median residual$0.05798$ dex and BTFR point $(m,b)=(3.968,1.7758)$; atΥ_[3.6]=0.60 the orthogonal $123$-galaxy comparison is0.88σ away. The same solution yieldsΔΣ=κ Vflat²/(4GR) withκ=0.959--$1.003$, without an independent lensing normalisation. Conclusions.\ The linear kinetic-ceiling branch follows CDM to 3×10⁻⁶. The relaxedL^ target is itself a Vlasov equilibrium (R²=0.999763;Maxwell--Boltzmann/Jeans dispersions $127.65/127.81\,$km\,s⁻¹). A$2600$-shell preparation audit shows that the previously reported concentrationoffset is not preparation invariant: assigning angular momentum at each shell'sactual turnaround and retuning only the initial overdensity amplitude to the sameM₂₀₀ gives time-averagedCVlasov/Crelaxed=1.31--$1.00$ across the already tested$q=0.12$--$0.19$ interval ($1.23$ at $q=0.15$). The remaining transportuncertainty is therefore the cosmological angular-momentum prior, not apreparation-independent failure to access the relaxed Vlasov basin. For fixedsources, the self-consistent Lyapunovfunctional has positive Hessian and decay rates [Γ,2Γ]. Thethird-moment force proxy gives 6.11±0.69\,Gyr⁻¹ but is not the exactorthogonal Mori kernel. A weak-coupling Mori--register theorem separates thestrict zero-frequency Markov coefficient ΓM= MV(0) fromthe finite-memory single-gap closure Γ= MV(Γ); thelatter follows under explicit functional-CLT and orthogonal-sectorfactorisation hypotheses. Applying the archived proxy givesΓSC=2.833\,Gyr⁻¹; positive-spectrum continuation gives$2.839\,$Gyr⁻¹ and uniqueness under its stated measure assumption. Therequired exact Vlasov FCLT/factorisation remains a microscopic condition to beestablished. The same aₛₐₜ passes theno-fit weak-lensing check; retention separates relaxed and transport regimes,DF2/DF4/DF9 lie on the depleted branch, the galaxy-pair kernel is a sensitivityforecast, and the regular $n=1$ winding solution remains conditional with noidentified observed structure.