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.