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

We formulate a relativistic theory for a bounded complex saturation field$I=\sqrt{s}\,e^{i\theta}$, $0<s<1$, from an explicit constrained parentaction. The canonical scalar Lagrangian is retained on a seed metric$g_{\mu\nu}$, while the Einstein--Hilbert, matter, electromagnetic, and tensorsectors share one reciprocal physical metric. The auxiliary metric constraintcontains no independent matter cone and can be eliminated exactly on thetimelike branch. The observable geometry is\[ \bar g_{\mu\nu}=\mathcal B^{-1}(g_{\mu\nu}+u_\mu u_\nu) -\mathcal B\,u_\mu u_\nu, \qquad \mathcal B(s)=\frac{1+\alpha_\star s}{1+\alpha_\star s_\infty}, \qquad u_\mu=-\frac{\nabla_\mu\theta}{\sqrt{-(\nabla\theta)^2}},\]and is shared by matter, photons, and tensor modes. In the staticphase-comoving weak-field limit,$\delta\bar\Phi_s=\delta\bar\Psi_s=(c^2/2)\ln\mathcal B$. The same scalarpotential therefore governs non-relativistic dynamics, weak lensing, andShapiro delay; the construction has one physical cone, $c_T=c_\gamma$, andgives $\gamma_{\rm PPN}=1$ at leading Cassini order. The boosted reciprocalmetric further gives an exact partial preferred-frame closure: the disformalamplitude $D=\mathcal B^{-1}-\mathcal B$ vanishes at the asymptoticnormalisation $\mathcal B(s_\infty)=1$, and the directly readable $U$-only$\bar g_{0j}$ coefficient fixes$\alpha_1-2\alpha_2=8\lambda$, with $\lambda\equiv\delta G/G$. The individual$\alpha_1$ and $\alpha_2$ remain open pending the full PPN gauge reduction andthe sourced phase response. A locally uniform pair $(s,u^\mu)$ is removable by a constant tetradtransformation, so dimensionless nongravitational observables depend only onderivatives of the saturation and phase fields. The configurational freeenergy$V=\Lambda_0D_{\rm KL}[\operatorname{Bern}(s)\|\operatorname{Bern}(s_\star)]$produces a Fermi environmentalresponse and a $\cosh^2$ rigidity. The invariant projector on the externalchannel fixes$s_{\rm vac}=\alpha_Z^3/(4\pi)=3.80\times10^{-8}$ and consequently$\mu^2=2\Lambda_0\ln[(1-s_{\rm vac})/s_{\rm vac}]=34.1719\Lambda_0$.The timelike Madelung direction follows from the conserved canonical charge$q=a^3f^2s\dot\theta$: for every $q\neq0$ the radial equation has a uniquesolution $s_q\in(s_{\rm vac},1)$, with positive radial curvature and a stablephase mode satisfying $0<c_\theta^2<1$. The constrained parent action is shown to propagate exactly four degrees offreedom, two tensor and two scalar. With $\bar g_{\mu\nu}$ fundamental themultiplier and the seed metric are eliminated exactly, the reduced actioncontains only first derivatives, and the characteristic determinant factorisesinto two nested cones: matter, photons, tensor modes and the Born phase share$\bar g_{\mu\nu}$ exactly, while the amplitude mode propagates on the seed conewith $c_s^2=\mathcal B^{-2}$, superluminal by at most$2\alpha_\star s_{\rm vac}=1.8\times10^{-8}$. The framework is confronted with galactic and cosmological data. Thekinetic normalisation is predicted, not fitted:$f=v\alpha_Z^{-13/2}=1.220\times10^{16}\,$GeV from the orbit-measure channellaw, whence the matter coupling in closed form,$\beta=\alpha_\star\sqrt{s_{\rm vac}}\,\bar M_{\rm Pl}/f=9.2\times10^{-3}$, and$\delta G/G=2\beta^2q\le1.7\times10^{-4}$. The scalar is therefore not thesource of the galactic amplitude; the condensate is, while the scalar modulateshow that mass couples. A matched shape analysis of $84$ SPARC galaxiesidentifies the predicted geometric-versus-dynamical residual signature at$5\sigma$ and shows that a column-anchored gate restores the null. The fullaxisymmetric bounded-susceptibility equation is then solved for all $165$usable SPARC systems. It predicts the baryonic Tully--Fisher point$(m,b)=(3.968,1.7758)$ in$\log_{10}(M_{\rm bar}/M_\odot)=m\log_{10}(V_{\rm flat}/{\rm km\,s^{-1}})+b$;an orthogonal fit to the $123$ galaxies with measured flat velocity is within$0.88\sigma$ of that point for the plausible stellar mass-to-light ratio$\Upsilon_\star[3.6]=0.60$, with the significance explicitly conditional onthat nuisance choice. Projecting the condensate density from the samenonlinear solution closes the dynamics--lensing calculation and gives$\Delta\Sigma(R)=\kappa V_{\rm flat}^2/(4GR)$ with$\kappa=0.959$--$1.003$ over$3\times10^8\lesssim M_{\rm bar}/M_\odot\lesssim6\times10^{11}$ and$50\,{\rm kpc}\le R\le1\,$Mpc, with no independent lensing normalisation.The kinetic-ceiling branch follows cold dark matter at linear order to$3\times10^{-6}$, whereas applying the galactic low-acceleration law on linearscales overproduces power by factors $8.8$--$38$. Galaxy-pair data give afirst, presently $\sim1.7\sigma$, indication of the finite canonical Comptonlength. The preferred-frame sector is partially closed by the exactboosted-metric relation $\alpha_1-2\alpha_2=8\lambda$; the remaining decisivepost-Newtonian step is the full gauge reduction separating the individualcoefficients together with the sourced phase response. The final galactic closure is supplied by a bounded-susceptibility actionrather than by an imported transition function. Its constitutive potential andnormalisation admit a conditional holographic microfoundation: a symmetricone-dimensional square-root waterbag has pressure$\mathcal P(K)=\sqrt K\sqrt{a_{\rm sat}^2+K}-a_{\rm sat}^2\operatorname{arsinh}(\sqrt K/a_{\rm sat})$, whose Legendre conjugate is$\eta=\partial_K\mathcal P=\sqrt{K/(a_{\rm sat}^2+K)}$ and whose convex dual isexactly $a_{\rm sat}^2(\operatorname{artanh}\eta-\eta)$. Interpreting onenatural-unit quantum of Bekenstein--Hawking entropy-area density as onelongitudinal response channel fixes the prefactor$(4G)^{-1}(2\pi)^{-1}=\bar M_{\rm Pl}^2$; the remaining microscopic question iswhether the DBI parent dynamics uniquely populate this saturated channelensemble. Its Euler equation is\[ \nabla\!\cdot\!\left[ \frac{|\nabla\Phi|}{\sqrt{a_{\rm sat}^2+|\nabla\Phi|^2}}\nabla\Phi \right]=4\pi G\rho_b, \qquad a_{\rm sat}=\alpha_\star cH_\Lambda =1.2886\times10^{-10}\ {\rm m\,s^{-2}}.\]The spherical branch gives the exact deep-regime limit$g\sim\sqrt{a_{\rm sat}g_b}$ and hence$V_\infty^4=Ga_{\rm sat}M_b$. The full axisymmetric equation has been solvedfor all $165$ usable SPARC galaxies: the median logarithmic residual is$0.05798$ dex, against $0.05962$ dex for the algebraic reconstruction, with amedian geometric velocity change of $1.05\%$. This equation organises therelaxed galactic and poor-group response of the same kinetic-ceiling condensatewhose cosmological abundance is fixed in Part~III; it is neither a second darkcomponent nor an additional force added on top of a cold-dark-matter halo. Itsretention consistency supplies a calculable crossover scale,$M_{\rm bar}^{\rm cross}=4.34\times10^{13}M_\odot$ in closed form and$5.03\times10^{13}M_\odot$ for the full-profile solve. Above that scale therelaxed closure cannot be extrapolated automatically to rich clusters becauseit would require a baryon fraction exceeding the cosmic share. A complementary transport completion treats the same condensate on unrelaxedbranches. In the strict Born--Infeld kinetic-ceiling limit the condensatestress tensor becomes dust; after shell crossing its explicit conservativemultistream completion obeys Vlasov/Liouville transport. A direct shellcalculation shows that collisionless transport does not by itself reproduce therelaxed bounded-susceptibility core: the condensate mass fraction inside$0.1r_{200}$ differs by factors $1.78$ at galaxy scale and $7.39$ at clusterscale after normalisation at $r_{200}$. This leaves the microscopiccoarse-graining that prepares the relaxed constitutive branch as an open task,while the unrelaxed Vlasov branch is the appropriate starting point for richclusters and for systems formed after violent baryon--condensate separation.Conditional on the absence of an additional non-gravitational condensatecollision kernel, and for matched progenitor initial data and baryonicclosures, the Vlasov--Euler--Poisson system is equation-equivalent to thecollisionless-dark-matter plus gas system used in published mini-bulletsimulations. The NGC~1052 trail galaxies DF2, DF4 and DF9 then occupy thecondensate-depleted, baryon-only branch rather than the relaxed equilibriumbranch.Finally, the Big Ring is retained as a deliberately conditional two-scale test,not as a claimed origin calculation: a regular Born-phase winding BVP gives$r_{90}=27.8025\,$Mpc, while the final sourced Part~III background maps a$360$--$400\,$Mpc present diameter to$N_{\rm BR}=61.473827$--$61.580920$ e-folds before the end of acceleration.Formation, survival, and transfer to Mg\,\textsc{ii} absorbers remain open.