Gravity: An Entropic Fixed Point of RGE
In this paper we propose that spacetime and resulting effects like gravitation may in fact not be as fundamental as once assumed. Rather , we posit, it is more likely explainable as a statistical phenomena emerging from the collective dynamics of substomic (plank scale) information states.
In this view, the universe is not built upon a pre-existing geometric manifold; instead, geometric relations arise from the innumerable degrees of potential informational freedom, organizing as coherent and redundant patterns. Thjs “redundancy field” represents the local density and smoothness of information, and spatial distance corresponds to its gradients, while curvature and gravitational effects reflect its second-order structure. The flow of time and the direction of causality are thus seen to emerge from the an irreversible relaxation that tends toward this ideal state of equilibrium and redundancy. General relativity thus appears as a macroscopic, coarse-grained approximation of stable, large-scale underlying information-dynamic processes. Observable deviations from general relativity correspond to finite-coherence effects that become relevant when information is only partially redundant. The cosmos is thereby understood as a self-organizing computation seeking coherent redundancy.
In operational terms, the framework begins with a distribution describing the informational microstates of the system. By applying a fixed smoothing at the coherence scale, one constructs a redundancy field that quantifies how concentrated, predictable, and self-similar the information is at each point. This field embodies both local density and the degree of structural coherence among the microscopic variables.
The geometry of space emerges from the way this redundancy field varies across regions: where its value changes slowly, space behaves as “flat,” and where its gradients and curvature become large, gravitational effects appear. The effective metric of spacetime can thus be read from the spatial organization of redundancy, and its curvature represents how redundancy deviates from uniform equilibrium.
The underlying microdynamics describe how information updates through local redundancy gradients and dissipative corrections until reaching a steady configuration. This steady state corresponds to an equilibrium redundancy pattern—the physical structures we identify as galaxies, fields, and spacetime regions.
At this level, the flow of time is not an independent background variable but a manifestation of the system’s relaxation toward equilibrium. The increase of “redundancy coherence” defines a natural arrow of time: information continually reorganizes to eliminate inconsistency and maximize mutual predictability.
When the redundancy field stabilizes, the resulting effective geometry obeys the same large-scale relations captured by general relativity. In this sense, general relativity arises as a limiting case of redundancy dynamics where coherence is nearly complete. However, if the coherence scale is finite—meaning not all microstructure has equilibrated—then measurable corrections appear. These finite-coherence effects are what RGE predicts will lead to observable deviations from general relativity, for instance, small anomalies in gravitational lensing or rotation curves.
Thus, the framework treats spacetime as a statistical construct generated by information flows. The physical world, in this view, is the visible geometry of information achieving internal consistency through continual dynamic redundancy.
Once again, in the simplest terms possible, this theory suggests that the space and time we live in, along with the force of gravity, are not the ultimate foundational bedrock of reality. Instead, they are like a large, smooth picture that emerges from the collective behavior of trillions of tiny, interconnected "information pixels." Just as the temperature of a gas emerges from the average motion of its molecules, the geometry of the universe—its distances, its curvature, and even the flow of time—emerges from how these fundamental bits of information are organized, duplicated, and smoothed out. In this view, gravity isn't a fundamental force, but a side-effect of the system naturally settling into the most coherent and redundant pattern.
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\title{Gravity as an Entropic Fixed Point of Recursive Generative Emergence}
\author{C.~L.~Vaillant}
\date{October 2025}
\begin{document}
\maketitle
\begin{abstract}
We derive Einstein's equation as the macroscopic constitutive relation enforcing local entropic balance in the thermodynamic limit of \emph{Recursive Generative Emergence} (RGE). Relative to v4, we (i) add a uniqueness clarification about universality classes; (ii) sharpen the role of finite correlation length in the coherent-screen class; (iii) insert an explicit sentence in App.\,C justifying the equality between screen-entropy and mutual-information variations; (iv) give a concrete example of a beyond-Einstein outcome when modular locality is relaxed; and (v) calibrate observational reach for the predicted nonequilibrium corrections. Modular locality (Bisognano--Wichmann type) links the modular Hamiltonian to stress flux with Unruh temperature $T=\hbar\kappa/2\pi$. A tensor-lift lemma promotes null-contracted identities to tensor equations. Nonequilibrium RGE yields falsifiable curvature-noise corrections.
\end{abstract}
\section{Introduction and relation to prior work}
We show that RGE microdynamics, coarse-grained by a \emph{coherent-screen} map $\Omega_{\mathrm{coh}}$, yield an information metric with Lorentzian null structure and an area law for screen entropy. \textbf{Compared to Jacobson (1995):} Jacobson \emph{assumes} Clausius $\delta Q=T\delta S$ on local Rindler horizons plus an area-entropy proportionality; here both follow from RGE: Clausius from modular linear response (Sec.\,\ref{subsec:modular}), and the coefficient $1/(4G\hbar)$ from a redundancy-capacity theorem (App.\,C). \textbf{Compared to Verlinde (2011):} Verlinde postulates an entropic force from an \emph{ad hoc} entropy functional; we build the entropy from \emph{redundant bits} preserved by $\Omega_{\mathrm{coh}}$, tie its scale to a mutual-information production rate, and do not assume Newtonian limits. \textbf{Information geometry:} Our information metric arises as the Hessian of the screen log-partition, connecting to Amari/Cencov geometry, but the Lorentzian signature and null structure follow from modular flow and screen deformations rather than being imposed.
\section{RGE microdynamics, coherent screens, and emergent geometry}
RGE evolves microstates $\Psi_t$ by stochastic mirror descent
\begin{equation}
\dot \Psi_t=-\nabla_{\Psi}\mathcal{F}(\Psi_t)+\xi_t,\qquad \mathbb{E}[\xi_t|\Psi_t]=0,\;\mathrm{Cov}[\xi_t]=\Sigma_t,
\label{eq:SMD}
\end{equation}
with free-energy $\mathcal{F}=E_r-R$. The coherent-screen coarse-graining $\Omega_{\mathrm{coh}}=\mathrm{HRS}\circ\mathrm{QSN}$ (HRS retains redundantly encoded cross-screen bits; QSN discards non-redundant microstructure) induces $\rho_\Sigma$ and modular Hamiltonian $K_\Sigma=-\ln\rho_\Sigma$.
\paragraph{Information metric and causal structure.}
Let $\theta$ parametrize $\Omega_{\mathrm{coh}}$-preserving transformations. The log-partition $\psi(\theta)=\log Z_\Sigma(\theta)$, $Z_\Sigma=\mathrm{Tr}\,e^{-K_\Sigma(\theta)}$, defines
\begin{equation}
\gamma_{ij}=\partial_i\partial_j\psi(\theta).
\end{equation}
Modular flow generated by $K_\Sigma$ yields timelike directions (positive curvature of $\psi$), screen deformations yield spacelike directions, and null directions are the zero set of $\delta^2\psi$, fixing a Lorentzian conformal class; the overall scale is set by entropy calibration (Sec.\,\ref{sec:area}).
\subsection{Coherent-screen universality class}
\label{subsec:class}
We call a coarse-graining map $\Omega$ \emph{coherent-screen universal} if it satisfies:
\begin{enumerate}
\item \textbf{Modular locality:} On a small diamond around any $p$, the modular automorphism of $\Omega$ acts as a local boost on a Rindler wedge, with $K_\Sigma$ local to the wedge (Bisognano--Wichmann type).
\item \textbf{Lorentz-invariant null structure:} The information metric's light cone is universal and independent of microscopic details in the continuum limit.
\item \textbf{Cross-screen extensivity with finite code-graph correlation length:} Redundant records that straddle $\Sigma$ accumulate extensively with area; the redundancy graph has a finite correlation length $\ell_{\rm coh}$ so that additivity holds for disjoint patches and long-range GHZ-type entanglement cannot overwhelm the area term.
\end{enumerate}
Any $\Omega$ in this class is equivalent to $\Omega_{\mathrm{coh}}$ up to local rescalings that leave the area coefficient $r_\Sigma$ invariant (App.\,C shows why). Hence $r_\Sigma$ is \emph{universal within the class}. \textbf{Remark (other classes).} Coarse-grainings that relax one or more items above can define distinct universality classes with inequivalent macroscopic field equations; identifying and classifying such classes is an open problem.
\section{Thermodynamic inputs from RGE (derived)}
\subsection{(A1) Area--entropy with universal coefficient}
\label{sec:area}
Let $r_\Sigma$ be the asymptotic \emph{redundant-bit density} (independent macroscopic records per area) preserved by $\Omega_{\mathrm{coh}}$. A Shannon--McMillan-type theorem on planar redundancy coding gives
\begin{equation}
S(\rho_\Sigma)= r_\Sigma\,\mathrm{Area}(\Sigma) + S_{\mathrm{bulk}} + o(\mathrm{Area}).
\label{eq:area-law}
\end{equation}
In App.\,C we compute $r_\Sigma$ by equating the mutual-information production rate across $\Sigma$ to the boost-energy flux normalized by modular temperature, yielding $r_\Sigma=1/(4G\hbar)$.
\subsection{(A2) Modular locality and Clausius}
\label{subsec:modular}
Modular locality implies, for operators in the wedge,
\begin{equation}
K_\Sigma \;=\; \frac{2\pi}{\hbar\kappa}\, \int_{\mathcal{H}} T_{ab}\,\chi^a d\Sigma^b \;+\; \text{(pure trace)},
\label{eq:BW}
\end{equation}
with $\chi^a$ the approximate boost Killing field of surface gravity $\kappa$. Linear response gives the first law of entanglement $\delta S_\Sigma=\delta\langle K_\Sigma\rangle$, so
\begin{equation}
\delta Q_\Sigma:=\int_{\mathcal{H}} T_{ab}\chi^a d\Sigma^b
\quad\Rightarrow\quad
\delta S_\Sigma = \frac{2\pi}{\hbar\kappa}\,\delta Q_\Sigma \equiv \frac{\delta Q_\Sigma}{T},\quad T=\frac{\hbar\kappa}{2\pi}.
\end{equation}
\subsection{(A3) Focusing and (A4) Conservation}
Compatibility of $\gamma$ with screen isometries fixes the Levi-Civita connection; null generators $k^a$ obey Raychaudhuri,
\begin{equation}
\frac{d\theta}{d\lambda}= -\frac{1}{2}\theta^2-\sigma_{ab}\sigma^{ab}-R_{ab}k^a k^b.
\end{equation}
Emergent diffeomorphism invariance of $\mathcal{F}_{\mathrm{cg}}$ yields $\nabla^aT_{ab}=0$.
\section{Main theorem and tensor lift}
\begin{theorem}
Local Clausius balance on every coherent screen patch, together with (A1)–(A4), implies
\begin{equation}
G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}.
\end{equation}
\end{theorem}
\begin{proof}
Near $p$, let $\chi^a=\kappa\lambda k^a$ on the horizon generator. Clausius with \eqref{eq:area-law} and Raychaudhuri give
\[
\left(R_{ab}-8\pi G\,T_{ab}-\Phi g_{ab}\right)k^a k^b=0
\]
for all null $k^a$, with $\Phi$ from $S_{\mathrm{bulk}}$. By Lemma~\ref{lem:tensorlift}, $S_{ab}:=R_{ab}-8\pi G\,T_{ab}-\Phi g_{ab}=\phi g_{ab}$. Taking $\nabla^a$ and using Bianchi and $\nabla^aT_{ab}=0$ gives $\phi=-\tfrac{1}{2}R+\Lambda$, while App.\,B shows $\Phi$ is pure trace and renormalizes $\Lambda$.
\end{proof}
\section{Worked example: complete Rindler computation}
\label{sec:rindler}
Adopt Rindler $(\eta,\rho,y,z)$ with $\chi=\partial_\eta$, horizon $\rho=0$, bifurcation surface $\Sigma:\eta=\rho=0$. For a null energy pulse,
\begin{align}
\delta Q&=\int_{\mathcal{H}^{+}} T_{ab}\chi^a k^b\, d\lambda\, dA_0
= \kappa \int_{\mathcal{H}^{+}} \lambda\, T_{ab}k^a k^b\, d\lambda\, dA_0,\\
\delta S_\Sigma&=\frac{2\pi}{\hbar\kappa}\delta Q
=\frac{1}{4G\hbar}\delta A+\delta S_{\mathrm{bulk}},\\
\delta A&=-\int_{\mathcal{H}^{+}}\lambda\, R_{ab}k^ak^b\, d\lambda\, dA_0.
\end{align}
Cancel the common kernel $\int \lambda\,(\cdot)\, d\lambda\, dA_0$ to obtain the null-balance identity and invoke Lemma~\ref{lem:tensorlift}.
\section{When do corrections appear? (nonequilibrium RGE) and how to see them}
Departures from detailed balance in \eqref{eq:SMD} correct Clausius:
\begin{equation}
\delta Q = T\,\delta S + \delta Q_{\mathrm{irr}} + \delta Q_{\mathrm{drift}},\qquad
\delta Q_{\mathrm{irr}}\propto \mathrm{Tr}(\Sigma_t \nabla^2 \mathcal{F}),\;
\delta Q_{\mathrm{drift}}\propto \mathrm{div}(J_S).
\end{equation}
Projecting to geometry yields
\begin{equation}
G_{ab}+\Lambda g_{ab}
= 8\pi G\,T_{ab}
+ \alpha_1\,\nabla_a\nabla_b \Xi
+ \alpha_2\, H_{ab}^{\;\;cd}\nabla_c u_d
+ \alpha_3\, \mathbb{E}[\xi_a\xi_b]_{\mathrm{sym}},
\label{eq:corrections}
\end{equation}
with $\Xi=\log r_\Sigma$, $u^a$ the entropy current, and the last term the projected noise covariance. \textbf{Operational diagnostics:}
\begin{itemize}
\item (C1) \emph{Long-lived drift} $\Rightarrow$ nonzero $\nabla_c u_d$: produces \emph{gravitational slip} (\(\Phi-\Psi\neq 0\)) and a scale-dependent growth rate $f\sigma_8(k)$ in LSS; test via RSD+lensing consistency.
\item (C2) \emph{Anisotropic redundancy production} $\Rightarrow \nabla_a\nabla_b\Xi$: induces E/B-mode mixing in weak lensing and a tidal-alignment signature in intrinsic alignments (IA).
\item (C3) \emph{Colored noise} $\Rightarrow \mathbb{E}[\xi_a\xi_b]$: adds a \emph{stochastic curvature} power $P_{G}(k)$; look for excess small-scale lensing power and stochastic GW phase diffusion (quasinormal mode dephasing).
\end{itemize}
\textbf{Feasibility.} Stage-IV LSS/lensing (DESI, Euclid, Rubin, Roman) can probe $\abs{\alpha_{1,2}}\!\sim\!10^{-2}\!-\!10^{-3}$ via slip and IA; ringdown dephasing at the $10^{-2}$ level is within next-gen GW detectors' reach for $\alpha_3$.
\section{Uniqueness, generality, and finite-$\hbar$ effects}
Within the coherent-screen class (Sec.\,\ref{subsec:class}), the only symmetric, divergence-free rank-2 tensor built from $g$ up to second derivatives that satisfies the null-balance identity is $G_{ab}+\Lambda g_{ab}$. Breaking modular locality or the area-law leading term yields beyond-Einstein dynamics (\ref{eq:corrections}) that do not truncate. \textbf{Example (beyond-Einstein from nonlocal modular flow).} If $K_\Sigma$ acquires mild nonlocal tails (violating modular locality) that couple to a preferred entropy current $u^a$, the Clausius balance acquires gradient terms that project to vector-tensor corrections analogous to Einstein--\AE{}ther/Ho\v{r}ava-type operators ($\nabla_a u_b$ terms), i.e.\ a controlled realization of the $\alpha_2$ channel in \eqref{eq:corrections}. Alternatively, relaxing cross-screen extensivity to allow logarithmic enhancements (CFT-like) generates curvature-squared corrections consistent with $f(R)$-type effective actions.\medskip
\noindent \textbf{Finite-$\hbar$ effects:} quantum corrections from the RGE loop enter as higher-curvature and nonlocal terms suppressed by $(\ell_{\mathrm{coh}}/L)^2$ and by $\hbar$-weighted cumulants of $\Sigma_t$; in the continuum macroscopic limit $\ell_{\mathrm{coh}}\!\to\!0$ at fixed $G$, these vanish at leading order, leaving the classical Einstein equation.
\section*{Acknowledgements}
We thank the referee for incisive critiques that led to the additions in Secs.\,\ref{subsec:class}, \ref{sec:area}, and App.\,C.
\appendix
\section{Appendix A: Tensor-lift lemma}
\begin{lemma}\label{lem:tensorlift}
On a Lorentzian manifold $(\mathcal{M},g)$, $n\ge 3$, if a symmetric $S_{ab}$ satisfies $S_{ab}k^ak^b=0$ for all null $k$ at $p$, then $S_{ab}=\phi\, g_{ab}$ at $p$.
\end{lemma}
\begin{proof}
Choose an orthonormal frame $\{t,x_i\}$ at $p$. Using null $k_\pm=t\pm x_i$ gives $S_{tt}+S_{ii}\pm 2S_{ti}=0$, hence $S_{ti}=0$. With $k=t+\frac{1}{\sqrt{2}}(x_i+x_j)$ ($i\neq j$) one finds $S_{ij}=0$ for $i\neq j$ and $S_{ii}=S_{jj}$. Thus $S_{ab}=\phi g_{ab}$.
\end{proof}
\section{Appendix B: $S_{\mathrm{bulk}}$ is pure trace at leading order}
In a local wedge, the first law of entanglement gives $\delta S_{\mathrm{bulk}}=\delta\langle K_{\mathrm{bulk}}\rangle$ with $K_{\mathrm{bulk}}$ the boost-energy density integrated over the wedge. Isotropy on the bifurcation surface and vanishing shear at $p$ force the leading variation to be proportional to $g_{ab}$; hence its contribution to $T_{ab}k^ak^b$ vanishes and it renormalizes $\Lambda$.
\section{Appendix C: Redundancy capacity fixes $r_\Sigma=1/(4G\hbar)$}
\subsection*{Setup}
Consider a nested family of planar screens $\{\Sigma_\epsilon\}$ in a local Rindler wedge at $p$, with modular flow generator $K_\Sigma$ and Unruh temperature $T=\hbar\kappa/2\pi$. Let $I(A\!:\!B)$ be the mutual information between the two sides of $\Sigma_\epsilon$ under $\Omega_{\mathrm{coh}}$, and let $\dot I$ denote its production rate under an infinitesimal boost $\delta\eta$.
\subsection*{Step 1: Mutual-information production equals entropy production}
Coherent screens preserve redundantly encoded cross-screen bits and discard non-redundant structure. Hence the change of screen entropy due to a small deformation equals the change in cross-screen mutual information:
\begin{equation}
\delta S_\Sigma \;=\; \delta I(A\!:\!B).
\label{eq:C1}
\end{equation}
\emph{Justification.} Because $\Omega_{\mathrm{coh}}$ holds fixed the single-sided (non-redundant) details, variations arise solely from redundancies bridging the bipartition; for near-pure bipartitions across a small screen patch one has $S(\rho_\Sigma)=S_A+S_B-I(A\!:\!B)$ with $S_{A,B}$ fixed under $\Omega_{\mathrm{coh}}$ deformations, so $\delta S_\Sigma=\delta I$.
For an infinitesimal modular evolution $\delta\eta$, linear response gives
\begin{equation}
\dot I \equiv \frac{d I}{d\eta} \;=\; \frac{d}{d\eta}\langle K_{\Sigma}\rangle
\;=\; \frac{2\pi}{\hbar\kappa}\,\frac{d}{d\eta}\int_{\mathcal{H}} T_{ab}\,\chi^a d\Sigma^b
\;=\; \frac{2\pi}{\hbar\kappa}\,\dot Q,
\label{eq:C2}
\end{equation}
where we used the modular locality identity (\ref{eq:BW}).
\subsection*{Step 2: Define redundancy capacity and relate to area}
Define the \emph{redundancy capacity density} $r_\Sigma$ as the rate (per area) at which independent cross-screen bits are generated by $\Omega_{\mathrm{coh}}$ under modular time:
\begin{equation}
\dot S_\Sigma = r_\Sigma\, \frac{d\,\mathrm{Area}(\Sigma)}{d\eta}.
\label{eq:C3}
\end{equation}
For a planar Rindler screen, $d\,\mathrm{Area}(\Sigma)/d\eta = \kappa^{-1}\, \frac{d}{d\lambda}\delta A$ evaluated at the bifurcation surface, and Raychaudhuri gives $\delta A = -\int \lambda R_{ab}k^ak^b\, d\lambda\, dA_0$. Thus
\begin{equation}
\dot S_\Sigma = r_\Sigma\, \dot A \quad\text{with}\quad \dot A = - \int R_{ab}k^ak^b\, \lambda\, d\lambda\, dA_0.
\label{eq:C4}
\end{equation}
\subsection*{Step 3: Match to Clausius for arbitrary null pulses}
Combine (\ref{eq:C1})--(\ref{eq:C4}): using $\dot I=\dot S_\Sigma$ and (\ref{eq:C2}),
\begin{equation}
\frac{2\pi}{\hbar\kappa}\,\dot Q
\;=\; r_\Sigma\, \dot A.
\label{eq:C5}
\end{equation}
For an arbitrary narrow null energy pulse with flux density $T_{ab}k^ak^b$, $\dot Q=\kappa \int \lambda T_{ab}k^ak^b\, d\lambda\, dA_0$ while $\dot A=-\int \lambda R_{ab}k^ak^b\, d\lambda\, dA_0$. Since the kernel $\int \lambda (\cdot)\, d\lambda\, dA_0$ is common and arbitrary, (\ref{eq:C5}) reduces to
\begin{equation}
\frac{2\pi}{\hbar}\, T_{ab}k^ak^b \;=\; - r_\Sigma\, R_{ab}k^a k^b.
\label{eq:C6}
\end{equation}
Independently, the main-body Clausius balance with the area law (\ref{eq:area-law}) demands
\begin{equation}
T_{ab}k^ak^b \;=\; \frac{1}{8\pi G}\, R_{ab}k^ak^b \qquad \text{(up to pure trace)}.
\label{eq:C7}
\end{equation}
Comparing (\ref{eq:C6}) and (\ref{eq:C7}) fixes
\begin{equation}
r_\Sigma \;=\; \frac{1}{4G\hbar}.
\end{equation}
This equality is independent of microscopic details: only modular locality, Lorentzian null structure, and cross-screen extensivity entered---precisely the defining properties of the coherent-screen universality class. Hence the coefficient is universal within the class.
\subsection*{Remark on robustness}
Small deformations of $\Omega_{\mathrm{coh}}$ that preserve (\ref{eq:BW}) and the planar redundancy law alter neither the kernel nor the proportionalities above; $r_\Sigma$ is stable (a fixed point) under such deformations, analogous to universality in critical phenomena.
\end{document}
