Holographic Saturation I: Fundamental parent action, universal reciprocal geometry, entropic % screening, timelike Madelung branch, and galactic phenomenology.

Abstract

\noindentWe 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:$f=v\alpha_Z^{-13/2}=1.222\times10^{16}\,$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, in galaxies as inclusters, and the scalar modulates how that mass couples. A matched shapeanalysis of $84$ SPARC galaxies identifies the predictedgeometric-versus-dynamical residual signature at $5\sigma$ and shows that acolumn-anchored gate restores the null; we also show that this null belongs tothe plane-parallel idealisation rather than to the field, whose finite-sourcesolution scales with the baryonic potential and reproduces an unmodulated haloto three decimals. The field excursion is $0.5\%$ across an $L^\ast$ disk and$18.9\%$ at Coma, so clusters are the only systems in which the gate acts. Thekinetic-ceiling branchfollows cold dark matter at linear order to $3\times10^{-6}$, whereas applying thegalactic low-acceleration law on linear scales overproduces power by factors$8.8$--$38$, confining the enhanced response to bound systems. Galaxy-pairdata give a first, presently $\sim1.7\sigma$, indication of the finite canonicalCompton length. 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. A joint nonlinearprediction of the condensate profile and weak lensing from one field solutionalso remains outstanding. 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\[ \nabla\!\cdot\!\left[ \frac{|\nabla\Phi|}{\sqrt{a_{\rm sat}^2+|\nabla\Phi|^2}}\nabla\Phi \right]=4\pi G\rho_b, \qquad a_{\rm sat}=\alpha_\star cH_\Lambda =1.2867\times10^{-10}\ {\rm m\,s^{-2}}.\]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 thesame kinetic-ceiling condensate whose cosmological abundance is fixed inPart~III; it is neither a second dark component nor an additional force addedon top of a cold-dark-matter halo. A complementary transport completion addresses ultra-diffuse systems formed afterviolent separation of baryons from the dark component. In the strictBorn--Infeld kinetic-ceiling limit the condensate stress tensor becomes dust;after shell crossing its explicit conservative multistream completion obeysVlasov/Liouville transport. Conditional on the absence of an additionalnon-gravitational condensate collision kernel, and for matched progenitor initialdata and baryonic closures, the resulting Vlasov--Euler--Poisson system isequation-equivalent to the collisionless-dark-matter plus gas system used inpublished mini-bullet simulations. The NGC~1052 trail galaxies DF2,DF4 and DF9 then occupy the condensate-depleted, baryon-only branch rather thanthe relaxed bounded-susceptibility equilibrium branch.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$N_{\rm BR}=60.972260$--$61.079368$ e-folds before the end of acceleration.Formation, survival, and transfer to Mg\,\textsc{ii} absorbers remain open.