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. 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. 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. The linear kinetic-ceiling branch follows CDM to $3\\times10^{-6}$, while therelaxed and collisionless basins differ. The relaxed $L^\\ast$ target is itselfa Vlasov equilibrium ($R^2=0.999763$; Maxwell--Boltzmann/Jeans dispersions$127.65/127.81\\,$km\\,s$^{-1}$). For fixed sources, 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.