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\title{Dark Energy as Recursive Informational Pressure: A Holographic Model}
\author{C.~L.~Vaillant}
\date{Oct 15 2025}
\begin{document}
\maketitle
\begin{abstract}
We propose that cosmic acceleration arises from a dynamical feedback between quantum vacuum information and spacetime curvature. Dark energy is modeled as \emph{recursive informational pressure}: a metric response to discrepancies between maximal and realized vacuum entanglement. From a covariant informational action,
\[
\Lambda(t)=\alpha_1 H\Delta\mathcal{I}+\alpha_2\dot{\Delta\mathcal{I}},
\]
coupled to an informational deficit field $\Delta\mathcal{I}$ obeying $\dot{\Delta\mathcal{I}}+3H\Delta\mathcal{I}=S(t)$, the model predicts small but measurable departures from $\Lambda$CDM: $w(z)\!\approx\!-1.03$ at $z>2$, $w(z)\!\to\!-0.98$ today, $\sim5\%$ faster expansion in voids, and $\sigma_8\!\approx\!0.78$. We define $\Delta\mathcal{I}$ holographically as a horizon-entropy gap density and relate its source term to structure formation. Embedded in Recursive Generative Emergence (RGE), this formulation connects entanglement, geometry, and cosmology within a testable informational ontology.
\end{abstract}
\section{Introduction}
The observed cosmic acceleration, though consistent with a constant $\Lambda$, lacks microphysical explanation. We reinterpret $\Lambda$ as an emergent property of informational imbalance: a feedback between spacetime geometry and vacuum entanglement deficit. This approach, rooted in \emph{Recursive Generative Emergence} (RGE), views physical order as a product of recursive information–geometry coupling. Extending thermodynamic and holographic gravity \cite{Jacobson1995,Padmanabhan2010,Verlinde2016}, we derive a dynamic $\Lambda(t)$ with concrete, falsifiable predictions.
\section{Recursive Generative Emergence (RGE) Framework}
RGE describes how systems evolve through iterative feedback loops linking informational state and structural configuration. Each iteration generates new constraints that recursively alter subsequent informational flow, driving emergent order. In cosmology, the state variable is the vacuum entanglement network, and the structure is spacetime curvature. When vacuum information deviates from its equilibrium distribution, curvature responds (via Einstein dynamics) to reduce the mismatch, while the resulting expansion modifies the information field—completing the recursive loop. This self-adjusting cycle provides a natural mechanism for inflation and late-time acceleration without introducing new fundamental fields.
\section{Mathematical Origin of the Covariant Informational Action}
A covariant Lagrangian coupling information flux to spacetime expansion must satisfy: (i) invariance under diffeomorphisms, (ii) linear dependence on the expansion scalar $\theta=\nabla_\mu u^\mu$, and (iii) conservation of informational current $J^\mu_{\mathcal{I}}=\Delta\mathcal{I}\,u^\mu$. The unique lowest-order term fulfilling these criteria is
\begin{equation}
S_{\mathcal{I}}=\int d^4x\,\sqrt{-g}\!\left[\frac{\beta_1}{3}\,\theta\,\Delta\mathcal{I}+\frac{\beta_2}{3}\,u^\mu\nabla_\mu \Delta\mathcal{I}\right],
\end{equation}
ensuring local information balance $\nabla_\mu J^\mu_{\mathcal{I}}=S(t)$ in the presence of sources. In an FLRW background ($u^\mu\nabla_\mu=\partial_t$, $\theta=3H$) this reduces to
\begin{equation}
S_{\mathcal{I}}=\int d^4x\,\sqrt{-g}\left[\frac{\alpha_1}{3}H\Delta\mathcal{I}+\frac{\alpha_2}{3}\dot{\Delta\mathcal{I}}\right],
\end{equation}
establishing Eq.~(\ref{eq:Lambda}) after variation and demonstrating that $\beta_{1,2}$ (hence $\alpha_{1,2}$) are the only symmetry-allowed first-order couplings between curvature and information.
\section{Dynamics of the Informational Deficit}
The informational deficit evolves as
\begin{equation}
\dot{\Delta\mathcal{I}}+3H\Delta\mathcal{I}=S(t),
\label{eq:evol}
\end{equation}
with $S(t)$ reflecting structure-driven decoherence. We adopt $S(t)=\xi\,d\sigma_8/dt$ as a phenomenological proxy linking information flow to clustering rate. Integrating Eq.~\eqref{eq:evol} for $S\!\propto\!d\sigma_8/dt\!\propto\!\mathrm{sech}^2[(z-z_c)/\Delta z]$ yields
\begin{equation}
\Delta\mathcal{I}(z)\propto\tanh\!\left[\frac{z-z_c}{\Delta z}\right],
\end{equation}
providing the empirical tanh-parameterization used in observational fits.
The effective equation of state,
\begin{equation}
w(z)=-1-\frac{1}{3H}\frac{d\ln(\eta\,\Delta\mathcal{I})}{dt},
\end{equation}
gives $w(z)\approx-1.03$ at $z>2$ and $w(z)\to-0.98$ today for fiducial parameters $(\alpha_1H_0^{-1},\alpha_2,\xi)=(1,1,1)$.
\section{Parameter Estimates and Sensitivity}
Nominal priors are:
\begin{itemize}
\item $\alpha_1\!\sim\!H_0^{-1}$, coupling timescale of order cosmic expansion;
\item $\alpha_2\!\sim\!1$, dimensionless relaxation weight;
\item $\eta\!\simeq\!\alpha_1H_0/(8\pi G)$, linking informational to energy density;
\item $\xi\!\sim\!1$, encoding structure–information efficiency.
\end{itemize}
Varying $(\alpha_1,\alpha_2,\xi)$ by $\pm50\%$ changes $|w+1|$ by $\lesssim0.01$ and $\sigma_8$ by $\pm0.02$, remaining within present observational uncertainty. Thus predicted deviations ($\Delta w\!\approx\!0.02$) are robust to moderate parameter shifts.
\section{Holographic Definition of $\Delta\mathcal{I}$}
In a horizon volume $V_H=4\pi/(3H^3)$, maximal entropy is $S_H=A/4G=\pi/(GH^2)$. With $S_V<S_H$ for the actual vacuum entanglement entropy, define
\begin{equation}
\Delta\mathcal{I}=\frac{S_H-S_V}{V_H}.
\end{equation}
Substitution into Eq.~(\ref{eq:Lambda}) reproduces $\Lambda\propto H\Delta\mathcal{I}+\dot{\Delta\mathcal{I}}$, linking the model to holographic entropy bounds \cite{Bousso2002}.
Linear perturbation theory implies $\delta\Delta\mathcal{I}/\Delta\mathcal{I}\!\approx\!-\tfrac{2}{3}\delta_m$, giving $\sim10\%$ higher $\Delta\mathcal{I}$ in voids ($\delta_m\!\approx\!-0.5$) and corresponding $H_{\text{void}}/H_0\!\simeq\!1.05$—consistent with the heuristic estimate used.
\section{Relation to Other Dark-Energy Models}
Unlike quintessence or $k$-essence, which introduce scalar fields and potentials, this framework involves no new fundamental fields. $\Delta\mathcal{I}$ arises from the vacuum’s entanglement structure and evolves via informational continuity, not potential dynamics. The resulting $\Lambda(t)$ represents a geometric backreaction from informational imbalance rather than an additional energy component.
\section{Observational Consequences}
\paragraph{Redshift evolution.} $\Delta w\!\equiv\!w+1\!\simeq\!0.02$ at $z\!\lesssim\!2$; Euclid’s forecasted $\sigma(w)\!\approx\!0.01$ allows $\sim2\sigma$ discrimination from $\Lambda$CDM.
\paragraph{Void expansion.} Regions with $\delta\!<\!-0.5$ have $10\%$ larger $\Delta\mathcal{I}$, implying $H_{\mathrm{void}}/H_0\simeq1.05$, testable with void SN\,Ia data.
\paragraph{Structure growth.} Time-varying $\rho_{\mathcal{I}}$ suppresses clustering, lowering $\sigma_8$ from $0.83$ to $\sim0.78$, consistent with Planck and DESI \cite{Planck2018,DESI2024}.
\section{Conceptual Synthesis}
Informational pressure provides a unified explanation for both inflation and late acceleration: early-time large $\Delta\mathcal{I}$ decays as entanglement builds, while present acceleration arises from renewed deficit as structure forms. Spacetime functions as an adaptive medium approaching informational equilibrium—an attractor dynamic consistent with RGE recursion.
\section{Conclusion and Outlook}
We present a covariant, holographic model of dark energy as recursive informational pressure. The framework unifies entanglement, geometry, and cosmic dynamics; introduces no new fields; and yields quantitative, falsifiable predictions. Future work will:
(i) derive $S(t)$ from QFT entanglement dynamics;
(ii) compute $\eta$ from horizon thermodynamics;
(iii) constrain $(\alpha_1,\alpha_2,\xi)$ with joint Planck–DESI–Euclid fits;
(iv) extend the RGE recursion formalism to early-universe inflation.
\appendix
\section{Derivation of Eq.~(\ref{eq:Lambda})}
Starting from
\[
S=\int dt\,a^3\!\left[-\frac{3}{8\pi G}H^2+\frac{\alpha_1}{3}H\Delta\mathcal{I}+\frac{\alpha_2}{3}\dot{\Delta\mathcal{I}}+\mathcal{L}_m\right],
\]
variation of the lapse (Hamiltonian constraint) gives
\[
H^2=\frac{8\pi G}{3}\!\left(\rho_m+\frac{1}{8\pi G}\left[\alpha_1H\Delta\mathcal{I}+\alpha_2\dot{\Delta\mathcal{I}}\right]\right),
\]
identifying $\Lambda(t)=\alpha_1H\Delta\mathcal{I}+\alpha_2\dot{\Delta\mathcal{I}}$.
The same expression follows from projecting the covariant action along $u^\mu$ for homogeneous backgrounds.
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