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

We formulate a relativistic bounded complex saturation field$I=\sqrt{s}\,e^{i\theta}$, $0<s<1$, from a constrained parent action. TheEinstein--Hilbert and matter sectors share the physical metric$\bar g_{\mu\nu}$; the seed metric is auxiliary and is eliminated on thetimelike phase branch. The reduced action is\[ S=\int\!\dd^4x\sqrt{-\bar g}\left[ \frac{\bar M_{\rm Pl}^2}{2}\bar R+\mathcal L_I(\bar g,s,\theta)\right] +S_m[\bar g,\chi],\]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. \textbf{Methods.}\ A locally uniform $(s,u^\mu)$ is removable by a constant tetrad transformation;gradients remain observable. In the static phase-comoving weak field,$\delta\bar\Phi_s=\delta\bar\Psi_s=(c^2/2)\ln\mathcal B$. The standard-kernelnear-zone PPN projection gives\[ \gamma=\beta=1,\qquad \xi=0,\qquad \alpha_1=\alpha_2=\alpha_3=0,\qquad \zeta_1=\zeta_2=\zeta_3=\zeta_4=0,\]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. \textbf{Results.}\ The bounded KL free energy gives$s_{\rm vac}=\alpha_Z^3/(4\pi)=3.80\times10^{-8}$; the orbit-channel law gives$f=v\alpha_Z^{-13/2}=1.220\times10^{16}\,$GeV, hence$m_{\rm can}=8.2\times10^{-31}\,$eV and $\lambda_{\rm can}=7.8\,$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\[ \nabla\!\cdot\!\left[ \frac{|\nabla\Phi|}{\sqrt{a_{\rm sat}^2+|\nabla\Phi|^2}}\nabla\Phi\right] =4\pi G\rho_b.\]Under the stated physical-horizon response identification,$a_{\rm sat}=\alpha_\star cH_\Lambda=1.2886\times10^{-10}\,{\rm m\,s^{-2}}$;the waterbag algebra fixes the response form but not this cross-sector gap.The exact spherical branch gives $V_\infty^4=Ga_{\rm sat}M_b$. Solving the fullaxisymmetric equation for all $165$ usable SPARC systems gives median residual$0.05798$ dex and BTFR point $(m,b)=(3.968,1.7758)$; at$\Upsilon_\star[3.6]=0.60$ the orthogonal $123$-galaxy comparison is$0.88\sigma$ away. The same solution yields$\Delta\Sigma=\kappa V_{\rm flat}^2/(4GR)$ with$\kappa=0.959$--$1.003$, without an independent lensing normalisation. \textbf{Conclusions.}\ The linear kinetic-ceiling branch follows CDM to $3\times10^{-6}$. The relaxed$L^\ast$ target is itself a Vlasov equilibrium ($R^2=0.999763$;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 same$M_{200}$ gives time-averaged$C_{\rm Vlasov}/C_{\rm relaxed}=1.31$--$1.00$ 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 $[\Gamma,2\Gamma]$. Thethird-moment force proxy gives $6.11\pm0.69\,$Gyr$^{-1}$ but is not the exactorthogonal Mori kernel. A weak-coupling Mori--register theorem separates thestrict zero-frequency Markov coefficient $\Gamma_M=\widetilde M_V(0)$ fromthe finite-memory single-gap closure $\Gamma=\widetilde M_V(\Gamma)$; thelatter follows under explicit functional-CLT and orthogonal-sectorfactorisation hypotheses. Applying the archived proxy gives$\Gamma_{\rm SC}=2.833\,$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 $a_{\rm sat}$ 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.