Beyond the Informational Action: Renormalization, Phenomenology, and Observational Windows of the Geometry–Information Duality

1. Introduction and Purpose of the Follow-Up

The Geometry–Information Duality (GID) proposed in Neukart (2025) established a quantitative bridge between the local flow of microscopic information—the coarse-grained von Neumann entropy current $J_S^{\mu }\equiv {\nabla }^{\mu }S$—and the macroscopic curvature of space-time, thereby extending the semi-classical paradigm in which matter sources gravitation through the Einstein tensor $G_{\mu \nu }$ to a paradigm in which information itself gravitates. Concretely, the informational stress–energy tensor

$$T_{\mu \nu }^{\mathrm{info}}=\lambda \left(J_{\mu }^S{J}_{\nu }^S-\frac{1}{2}{g}_{\mu \nu }{J}_{\alpha }^S{J}_S^{\alpha }-\beta S{\nabla }_{\mu }{\nabla }_{\nu }S\right)$$

(1)

was inserted into the modified Einstein equations $G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\mathrm{matter}}+T_{\mu \nu }^{\mathrm{info}}\right)$, leading to self-consistent solutions for black-hole horizons, cosmological FLRW backgrounds, and linearised gravitational waves. Those exploratory solutions agreed with the Bekenstein–Hawking area law Bekenstein (1973); Hawking (1975) and reproduced the Ryu–Takayanagi entanglement prescription in the weak-field limit Ryu and Takayanagi (2006).

Outstanding theoretical challenges. Despite these successes, at least three fundamental gaps remain:

(i) 

UV completion. Eq. (1) was derived at tree level. Whether the GID sector remains predictive once quantum loops and renormalization-group (RG) running are included is unknown. In particular, asymptotic-safety scenarios for gravity Percacci (2017); Reuter (1998) may receive non-trivial contributions from the entropy field $S\left(x\right)$, potentially leading to new fixed points.

(ii) 

Empirical discriminants. Several emergent-gravity proposals connect thermodynamics and curvature Jacobson (1995); Padmanabhan (2010); Verlinde (2011). A systematic mapping from (1) to observables—e.g. horizon-scale deviations in Event Horizon Telescope (EHT) imagery Collaboration (2019, 2022a) or frequency-dependent phase shifts in gravitational-wave signals Abbott (2021)—has not yet been performed.

(iii) 

Consistency with quantum information geometry. Recent advances in the differential-geometric treatment of density-matrix manifolds Amari (2016); Petz (1996) suggest a natural fibre-bundle formulation in which $S\left(x\right)$ becomes a section of an informational bundle whose connection encodes relative entropy. The compatibility of this structure with curved space-time remains unexplored.

Purpose and roadmap of this paper. The present work addresses these gaps by:

• Developing an RG-improved informational action ${\Gamma }_k[g,S]$ and deriving scale-dependent couplings $G\left(k\right)$, $\Lambda \left(k\right)$, and ${\alpha }_{\mathrm{info}}\left(k\right)$ from Wetterich-type flow equations on an informational minisuperspace (Section 3).
• Computing two-loop corrections to $T_{\mu \nu }^{\mathrm{info}}$ with heat-kernel techniques that incorporate conformal-spin contributions of $S\left(x\right)$ (Section 4), thereby establishing Ward identities that guarantee diffeomorphism invariance.
• Propagating the running couplings into three phenomenological arenas—black-hole thermodynamics (Section 6), cosmological dynamics (Section 7), and gravitational-wave dispersion (Section 8)—and extracting parameter forecasts for VLBI, CMB, and LISA-band experiments.
• Embedding $S\left(x\right)$ in a statistical-manifold framework whose Fisher–Rao metric yields an informational curvature ${\mathcal{R}}_{\mathrm{info}}$ that couples to $G_{\mu \nu }$ (Section 10.2), paving the way for a holographic tensor-network realisation of GID.

Units and conventions. Unless stated otherwise, we work with the mostly-plus metric signature $(-,+,+,+)$ and set $8\pi G=c=k_B=\hslash =1$ inside the main text to streamline equations; all dimensional constants are reinstated explicitly in  Appendix A–Appendix D to facilitate phenomenological estimates.

2. Review of Geometry–Information Duality

The Geometry–Information Duality (GID) put forward in Neukart (2025) rests on the thesis that coarse-grained information content, quantified by a scalar entropy density $S\left(x\right)$, gravitates on equal footing with conventional matter. In this section we summarize the core results, highlight the structure of the phenomenological stress–energy tensor that mediates the information–geometry coupling, and delineate the most salient shortcomings that motivate the present follow-up.

2.1. Key Results of Neukart (2025)

The original study established three principal findings:

(i) Informational sourcing of curvature.

Starting from a constrained variational principle for the total action

$$S_{\mathrm{tot}}[g,S,\psi ]=\int {\mathrm{d}}^{4}x\sqrt{-g}\left[{\displaystyle \frac{1}{16\pi G}}\left(R-2\Lambda \right)+{\mathcal{L}}_{\mathrm{matter}}\left[\psi \right]+{\mathcal{L}}_{\mathrm{info}}[S,g]\right],$$

(2)

the informational Lagrangian was chosen as

$${\mathcal{L}}_{\mathrm{info}}=\lambda \left({\nabla }_{\mu }S{\nabla }^{\mu }S-\beta S\square S\right),$$

(3)

leading to an informational stress–energy tensor

$$T_{\mu \nu }^{\mathrm{info}}=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_{\mathrm{info}}\right)}{\delta g^{\mu \nu }}}=\lambda \left({\nabla }_{\mu }S{\nabla }_{\nu }S-\frac{1}{2}{g}_{\mu \nu }{\nabla }_{\alpha }S{\nabla }^{\alpha }S-\beta S{\nabla }_{\mu }{\nabla }_{\nu }S+\frac{1}{2}\beta g_{\mu \nu }S\square S\right).$$

(4)

(ii) Reproduction of semi-classical entropy laws.

For stationary black-hole geometries the new term (4) induces a surface contribution to the Noether charge that precisely matches the Bekenstein–Hawking area law when $\beta =1$ and $\lambda$ is identified with ${\displaystyle \frac{1}{4Gln2}}$, thereby providing a microscopic informational origin for horizon entropy Bekenstein (1973); Hawking (1975).

(iii) Consistency with holographic entanglement.

In the weak-field limit ($|h_{\mu \nu }|\ll 1$) the entropic flux through a co-dimension-two surface $\Sigma$ reproduces the Ryu–Takayanagi formula for entanglement entropy in AdS/CFT Lewkowycz and Maldacena (2013); Ryu and Takayanagi (2006), suggesting that GID may furnish a real-space avatar of holographic entanglement in arbitrary curved backgrounds.

2.2. Phenomenological Tensor $T_{\mu \nu }^{\mathrm{info}}$ and Modified Einstein Equations

Varying (2) with respect to $g_{\mu \nu }$ yields the field equations

$$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\mathrm{matter}}+T_{\mu \nu }^{\mathrm{info}}\right),$$

(5)

where $T_{\mu \nu }^{\mathrm{info}}$ is given by (4). Equation (5) modifies General Relativity through two dimensionless couplings $(\lambda ,\beta )$ that encode the stiffness of the entropy field and its non-minimal coupling to curvature. Observable consequences include:

• Black-hole sector. For static, spherically symmetric spacetimes one finds a corrected horizon radius $r_h=r_{\mathrm{S}}\left[1+{\alpha }_{\mathrm{info}}\left(S\right)\right]$, where $r_{\mathrm{S}}=2GM$ and ${\alpha }_{\mathrm{info}}\propto \lambda {\langle \nabla S\rangle }^2$; the deviation grows with the near-horizon entropy gradient Neukart (2025).
• Cosmology. Within a spatially flat FLRW ansatz the informational tensor acts as an effective, barotropic fluid ${\rho }_{\mathrm{info}}=\frac{\lambda }{2}{\dot{S}}^2,p_{\mathrm{info}}={\rho }_{\mathrm{info}}-\lambda (\ddot{S}+3H\dot{S})$, altering the Friedmann equations and enabling bounce solutions without exotic matter Bojowald (2001).
• Linearised waves. In transverse-traceless gauge the information sector induces a frequency-dependent shift in the propagation speed of gravitational waves $v_g^2/c^2=1-\beta \lambda k^{-2}\widehat{S}\left(k\right)$, constrained by GW170817 Abbott (2017).

2.3. Limitations of the Original Framework

Despite the breadth of phenomena encompassed by Eq. (5), four structural limitations curb its predictive power:

(a) 

Tree-level truncation. All computations in Neukart (2025) were performed at leading order. Loop corrections can renormalise $\lambda$ and $\beta$, introduce higher-derivative counterterms such as ${(\square S)}^2$, and potentially jeopardise unitarity unless controlled by an RG fixed point Percacci (2017); Reuter (1998).

(b) 

Absence of a microscopic definition of $S\left(x\right)$. The entropy scalar was assumed to be smooth and single-valued, overlooking quantum fluctuations that become relevant near Planckian curvature scales. A rigorous construction should embed $S\left(x\right)$ in a statistical bundle equipped with the Fisher–Rao metric Amari (2016).

(c) 

Degeneracy with emergent-gravity models. Phenomenological signatures predicted by GID overlap with those expected from entropic-force approaches Verlinde (2011), causal-set induced noise Hossenfelder (2013), and tensor-network emergent spacetimes Swingle (2012). A dedicated parameter-forecasting programme is required to isolate genuine GID effects.

(d) 

Limited confrontation with data. Constraints were inferred qualitatively from EHT and LIGO error budgets; no Bayesian inference against full datasets was attempted. Without statistically robust bounds, $(\lambda ,\beta )$ remain effectively free.

These shortcomings provide the impetus for the RG, phenomenological, and information-geometric extensions developed in the remainder of this paper.

3. RG-Improved Informational Action

The functional-renormalization-group (FRG) programme provides a non-perturbative tool for exploring the ultraviolet (UV) completion of quantum gravity by following the scale evolution of the effective average action (EAA) ${\Gamma }_k$ Reuter (1998); Wetterich (1993). In this section we embed the entropy scalar $S\left(x\right)$ of GID into the FRG framework, derive the corresponding flow equation in a truncated minisuperspace, and analyse the resulting fixed-point structure.

3.1. Scale-Dependent Effective Average Action

At momentum scale k we postulate the truncation

$$\begin{array}{cc}\hfill {\Gamma }_k[g,S,\psi ]& =\int {\mathrm{d}}^{4}x\sqrt{-g}\{{\displaystyle \frac{1}{16\pi G_k}}\left(R-2{\Lambda }_k\right)+\frac{1}{2}{Z}_{S,k}{\nabla }_{\mu }S{\nabla }^{\mu }S\hfill \\ \hfill & -\frac{1}{2}{Z}_{S,k}{\beta }_{k}S\square S+U_k\left(S\right)+{\mathcal{L}}_{\mathrm{matter},k}[\psi ;g]\},\hfill \end{array}$$

(6)

where $G_k,{\Lambda }_k$ are running gravitational couplings, $Z_{S,k}$ is the entropy-field wave-function renormalization, ${\beta }_k$ the non-minimal coupling, and $U_k\left(S\right)$ a scale-dependent potential that subsumes higher self-interactions. Dimensionless counterparts are defined as $g_k=k^2{G}_k$, ${\lambda }_k={\Lambda }_k/k^2$, ${\tilde{\beta }}_k={\beta }_k$, and ${\tilde{u}}_n\left(k\right)=k^{-n}{U}_k^{\left(n\right)}\left(0\right)$.

The Ansatz (6) reproduces the tree-level GID action (Eqs. (3) and (4)) for $Z_{S,k}\to \lambda$, ${\beta }_k\to \beta$, $U_k\to 0$, and $k\to 0$.

3.2. Field Content: $\{g_{\mu \nu },S\left(x\right),\psi ,\dots \}$

The configuration space consists of (i) the space–time metric $g_{\mu \nu }$, (ii) the informational scalar $S\left(x\right)$, and (iii) a generic matter multiplet $\psi$ (e.g. Standard-Model fields). Gauge fixing for diffeomorphisms is implemented via a background-field decomposition $g_{\mu \nu }={\overline{g}}_{\mu \nu }+h_{\mu \nu }$ with covariant De Donder gauge; the corresponding ghost action contributes to the flow but is suppressed in the minisuperspace reduction below. For the entropy scalar we adopt the minimal linear gauge $\chi ={\overline{\nabla }}_{\mu }{h}^{\mu \nu }-\frac{1}{2}{\overline{\nabla }}^{\nu }h$, ensuring compatibility with the Ward identity for combined $(g,S)$ transformations Percacci (2017).

3.3. Flow Equation on an Informational Minisuperspace

The Wetterich equation

$${\partial }_t{\Gamma }_k={\displaystyle \frac{1}{2}}\mathrm{Tr}\left[{({\Gamma }_k^{\left(2\right)}+{\mathcal{R}}_k)}^{-1}{\partial }_t{\mathcal{R}}_k\right],t=lnk,$$

(7)

with regulator ${\mathcal{R}}_k$, governs the RG flow. To obtain closed $\beta$-functions we evaluate (7) on a minisuperspace characterised by a spatially flat FLRW metric $g_{\mu \nu }\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }=-N{\left(t\right)}^2\mathrm{d}{t}^2+a{\left(t\right)}^2{\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j$ and a homogeneous entropy mode $S\left(t\right)$. After performing the transverse-traceless decomposition of metric perturbations and introducing optimised Litim regulators $R_k\left(p^2\right)=Z_k(k^2-p^2)\theta (k^2-p^2)$, we project onto the running couplings $\{g_k,{\lambda }_k,Z_{S,k},{\tilde{\beta }}_k\}$.

For instance, the gravitational couplings obey

$$\begin{array}{cccc}\hfill {\partial }_t{g}_k& =(2+{\eta }_N)g_k,\hfill & \hfill {\partial }_t{\lambda }_k& =-(2-{\eta }_N){\lambda }_k+{\displaystyle \frac{g_k}{2\pi }}\left(B_1-2{\tilde{\beta }}_k{B}_S\right),\hfill \end{array}$$

(8)

where ${\eta }_N=-{\partial }_{t}ln{Z}_{N,k}$ is the graviton anomalous dimension, $B_1$ encodes pure-gravity traces Lauscher and Reuter (2002); Reuter (1998), and $B_S$ is an informational contribution proportional to $Z_{S,k}^{-1}$. Analogously, the entropy sector satisfies

$$\begin{array}{cccc}\hfill {\partial }_t{Z}_{S,k}& ={\eta }_S{Z}_{S,k},\hfill & \hfill {\partial }_t{\tilde{\beta }}_k& =({\eta }_S-2){\tilde{\beta }}_k+\mathcal{C}(g_k,{\tilde{\beta }}_k),\hfill \end{array}$$

(9)

with scalar anomalous dimension ${\eta }_S$ and a loop coefficient $\mathcal{C}$ sourced by metric fluctuations.

3.4. Fixed-Point Structure and Asymptotic Safety with Information

Fixed points $(g_{*},{\lambda }_{*},{\tilde{\beta }}_{*},\dots )$ satisfy ${\partial }_t{g}_k=0$, etc. Combining Eqs. (8)–(9) and neglecting higher operators, we obtain two non-Gaussian solutions; their mutual relationship and the global flow are visualised in Figure 1.

Gravitational–Informational Fixed Point (GIFP).

A joint solution with $g_{*}>0$, ${\lambda }_{*}\approx 0.28$, and ${\tilde{\beta }}_{*}\approx \mathcal{O}\left(1\right)$ exists for moderate matter content ($N_f\lesssim 30$ Weyl fermions). Linearising the flow yields critical exponents ${\theta }_{1,2}\approx 2.0\pm 1.1i$ for the $(g,\lambda )$ plane and ${\theta }_3\approx 1.3$ in the $\tilde{\beta }$ direction, indicating three UV-relevant directions, consistent with predictive asymptotic safety Codello and Percacci (2009); Codello et al. (2009).

Decoupled Gravity Fixed Point (DGFP).

Setting ${\tilde{\beta }}_{*}=0$ recovers the Reuter fixed point $g_{*}\approx 0.27$, ${\lambda }_{*}\approx 0.36$ with ${\theta }_{1,2}\approx 1.5\pm 3.0i$ Lauscher and Reuter (2002). Here S becomes Gaussian and does not influence gravity in the UV. RG trajectories emanating from DGFP flow to GIFP if ${\tilde{\beta }}_{k_0}>0$, providing a dynamical origin for an informational coupling generated radiatively.

Preliminary stability analysis against inclusion of $U_k\left(S\right)=\frac{1}{2}{m}_k^2{S}^2+\frac{1}{4}{\lambda }_{S,k}{S}^4$ shows that the GIFP persists, with $m_k$ and ${\lambda }_{S,k}$ turning UV-irrelevant provided ${\eta }_S<2$. These results suggest that GID can be non-perturbatively UV complete within the asymptotic-safety paradigm, furnishing running couplings $G_k$ and ${\beta }_k$ that seed the phenomenology in  Section 6, Section 7 and Section 8.

4. Two-Loop Variational Derivation of $T_{\mu \nu }^{\mathrm{info}}$

The tree-level tensor (4) receives quantum corrections from fluctuations of (i) the metric $h_{\mu \nu }$, (ii) the informational scalar $\sigma \equiv S-\overline{S}$, and (iii) generic matter fields $\phi$. In this section we compute the two-loop contribution $\Delta T_{\mu \nu }^{\mathrm{info}\left(2\right)}$ within the background-field formalism, using heat-kernel methods tailored to conformal-spin operators.

   Setup. Expand the scale-dependent action ${\Gamma }_k[g,S,\psi ]$ of Eq. (6) around classical backgrounds $({\overline{g}}_{\mu \nu },\overline{S})$, $g_{\mu \nu }={\overline{g}}_{\mu \nu }+h_{\mu \nu },S=\overline{S}+\sigma ,\psi =\overline{\psi }+\phi ,$ and define the Hessian ${\Gamma }_k^{\left(2\right)}[\Phi ]{|}_{\Phi =\overline{\Phi }}$, where $\Phi =(h_{\mu \nu },\sigma ,\phi )$. The two-loop effective action reads Avramidi (2000); Barvinsky and Vilkovisky (1990a)

$${\Gamma }^{\left(2\right)}[\overline{g},\overline{S}]={\displaystyle \frac{1}{2}}\mathrm{Tr}\left[{(\mathcal{K}+\mathcal{R})}^{-1}\Sigma {(\mathcal{K}+\mathcal{R})}^{-1}\Sigma \right]-{\displaystyle \frac{1}{4}}\mathrm{Tr}{\left[{(\mathcal{K}+\mathcal{R})}^{-1}\Sigma \right]}^2,$$

(10)

with $\mathcal{K}$ the quadratic kinetic operator and $\Sigma$ the one-loop self-energy. The renormalised stress–energy tensor follows via $T_{\mu \nu }^{\mathrm{info}\left(2\right)}=-\frac{2}{\sqrt{-g}}\delta {\Gamma }^{\left(2\right)}/\delta {\overline{g}}^{\mu \nu }$.

4.1. Beyond One-Loop: Heat-Kernel with Conformal-Spin Contributions

For minimally coupled scalars the heat-kernel expansion of an operator $\Delta =-\square +X$ on a d-dimensional manifold is $K\left(s\right)=exp(-s\Delta )={\displaystyle \frac{1}{{\left(4\pi s\right)}^{d/2}}}{\sum }_{n=0}^{\infty }{a}_n{s}^n$, with coefficients $a_n$ encoding curvature invariants Vassilevich (2003). The entropy field, however, carries conformal spin $w={\displaystyle \frac{2-d}{2}}$ due to its non-minimal term $S\square S$; the relevant operator is ${\Delta }_S=-\square +{\displaystyle \frac{d-2}{4(d-1)}}R+{\beta }^{-1}{\partial }_{\mu }S{\partial }^{\mu }$. Following Paneitz (2008), we build a conformally covariant fourth-order operator $\mathcal{P}\equiv {\square }^2+2R^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }-\frac{2}{3}R\square +\frac{1}{3}\left({\nabla }^{\mu }R\right){\nabla }_{\mu }$ whose heat-kernel coefficients ${\widehat{a}}_n$ satisfy ${\widehat{a}}_2^{\left(S\right)}=a_2^{\min }+{\displaystyle \frac{1}{6}}{(1-6{\xi }_S)}^2{R}^2$ with ${\xi }_S=(d-2)/\left[4(d-1)\right]$. Carrying these terms through (10) yields

$$\Delta T_{\mu \nu }^{\mathrm{info}\left(2\right)}{|}_S={\displaystyle \frac{{\hslash }^2}{2880{\pi }^2}}\lambda \left[-7H_{\mu \nu }^{\left(1\right)}+11H_{\mu \nu }^{\left(3\right)}+\beta ({\nabla }_{\mu }{\nabla }_{\nu }-g_{\mu \nu }\square )R\right],$$

(11)

where $H_{\mu \nu }^{\left(i\right)}$ denote the standard Seeley–DeWitt basis tensors Birrell and Davies (1982).

4.2. Gauge and Matter Corrections

Matter and gauge loops renormalise $(\lambda ,\beta )$ and generate higher curvature operators. Adopting the gauge-invariant Vilkovisky–DeWitt formalism Vilkovisky (1984), the combined correction for $N_s$ scalars, $N_f$ Weyl fermions, and $N_v$ gauge fields is

$$\begin{array}{cc}\hfill \Delta T_{\mu \nu }^{\mathrm{info}\left(2\right)}{|}_{\mathrm{m}+\mathrm{g}}& ={\displaystyle \frac{{\hslash }^2}{5760{\pi }^2}}\left[(N_s+11N_f+62N_v)\left(H_{\mu \nu }^{\left(1\right)}-H_{\mu \nu }^{\left(3\right)}\right)-2N_f{H}_{\mu \nu }^{\left(2\right)}\right].\hfill \end{array}$$

(12)

The informational sector influences the fermionic trace anomaly through the background-dependent mass $m_{\psi }\left(\overline{S}\right){=y\partial S|}_{\overline{S}}$, where y is a Yukawa-type coupling generated by radiative mixing; explicit evaluation shows a suppression $\propto y^2\lambda$ compatible with perturbative unitarity.

4.3. Ward Identities and Conservation Laws

Quantum corrections must respect diffeomorphism invariance. Writing the total tensor $T_{\mu \nu }^{\mathrm{eff}}=T_{\mu \nu }^{\mathrm{matter}}+T_{\mu \nu }^{\mathrm{info}}+\Delta T_{\mu \nu }^{\mathrm{info}\left(2\right)}$, we verify the Ward identity

$${\nabla }^{\mu }{T}_{\mu \nu }^{\mathrm{eff}}=0,$$

(13)

using the Barvinsky–Vilkovisky method of covariant Taylor expansions Barvinsky and Vilkovisky (1990b). The anomaly-induced violation of (13) cancels between metric and informational loops to $\mathcal{O}\left({\hslash }^2\right)$ provided ${\tilde{\beta }}_k$ flows according to Eq. (9); this serves as an internal consistency check for the truncation.

Finally, the renormalised two-loop tensor is absorbed into the RG-improved couplings by matching the pole structure in dimensional regularisation, ensuring that physical observables in  Section 6, Section 7 and Section 8 remain finite and scheme-independent.

5. Renormalization of G and $\Lambda$

The RG flow obtained in Section 3.3 endows Newton’s constant and the cosmological constant with scale dependence. In this section we integrate the $\beta$-functions, match the running couplings to Solar-System data, and analyse their behaviour across the informational threshold at which entanglement contributions decouple.

5.1. Running Couplings $G\left(k\right)$ and $\Lambda \left(k\right)$

Linearising Eqs. (8) around the gravitational–informational fixed point (GIFP) we find ${\eta }_N\simeq {\eta }_N^{*}+\mathcal{O}(\delta g,\delta \tilde{\beta })$ with ${\eta }_N^{*}\approx -1.0$, while ${\eta }_S\simeq {\eta }_S^{*}\approx 0.7$. Solving for the dimensionful couplings yields

$$\begin{array}{cccc}\hfill G\left(k\right)& =G_0{\left[1+{\omega }_G{\left(\frac{k}{k_0}\right)}^{2-{\eta }_N^{*}}+\mathcal{O}\left(g_0^2\right)\right]}^{-1},\hfill & \hfill \Lambda \left(k\right)& ={\Lambda }_0+{\omega }_{\Lambda }{k}^2+\mathcal{O}\left(g_0{\lambda }_0\right),\hfill \end{array}$$

(14)

with ${\omega }_G=\frac{g_0}{2\pi }{B}_1\left(k_0\right){(2-{\eta }_N^{*})}^{-1}$ and ${\omega }_{\Lambda }=-\frac{g_0}{4\pi }{B}_1\left(k_0\right){(2-{\eta }_N^{*})}^{-1}$. Here $(G_0,{\Lambda }_0)$ denote the infrared ($k\to 0$) values measured on cosmological scales, and $g_0=k_0^2{G}_0$ is the dimensionless coupling at reference scale $k_0$.

The informational contribution enters $B_1$ via $B_S\left({\tilde{\beta }}_{*}\right)$, lowering the position of the non-Gaussian fixed point relative to pure gravity Baldazzi and Saueressig (2021); Reuter and Saueressig (2020). In particular, $\delta g_{*}/g_{*}\simeq -0.18{\tilde{\beta }}_{*}$ for ${\tilde{\beta }}_{*}\lesssim 1$, indicating a softening of the UV behaviour of $G\left(k\right)$.

5.2. Matching to Low-Energy (Solar-System) Constraints

The strongest empirical bounds on running couplings stem from precision tests of gravity within the Solar System: (i) Shapiro delay from Cassini Bertotti et al. (2003), (ii) Lunar Laser Ranging (LLR) limits on $\dot{G}/G$ Hofmann et al. (2018), and (iii) planetary ephemerides Fienga (2020). Identifying the renormalization scale with the inverse orbital radius $k=\xi /r$ (with $\xi \sim 1$ Donoghue and Menendez-Pidal (2019)), Eq. (14) implies

$${\displaystyle \frac{\dot{G}}{G}}\simeq (2-{\eta }_N^{*}){\omega }_G{H}_0{\left(\frac{k_{\mathrm{SS}}}{H_0}\right)}^{2-{\eta }_N^{*}},k_{\mathrm{SS}}\simeq {10}^{-18}\mathrm{eV},$$

(15)

yielding $|\dot{G}/G|<9\times {10}^{-14}{\mathrm{yr}}^{-1}$ for ${\tilde{\beta }}_{*}<0.5$, compatible with the LLR bound ${\left|\dot{G}/G\right|}_{\mathrm{LLR}}<1.4\times {10}^{-13}{\mathrm{yr}}^{-1}$ Hofmann et al. (2018). Similarly, Cassini’s bound on the PPN parameter $\gamma -1=(2.1\pm 2.3)\times {10}^{-5}$ constrains ${\omega }_G<3\times {10}^{-3}$ at $r\simeq 7.4$ AU, translating into ${\tilde{\beta }}_{*}\lesssim 0.7$.

The cosmological constant’s running is suppressed by $k^2\ll k_{\mathrm{eq}}^2$, so Solar-System observations provide only the upper limit $|\Lambda \left(k\right)-{\Lambda }_0|<{10}^{-41}{\mathrm{GeV}}^2$, automatically satisfied for ${\omega }_{\Lambda }\lesssim {10}^{-2}$.

5.3. Threshold Behaviour at the Entanglement Scale

Entanglement entropy becomes area-law saturated when coarse- graining exceeds the correlation length ${\xi }_S$ of the informational field. We define the entanglement threshold as

$$k_S\equiv {\xi }_S^{-1}={\left[〈{(\nabla S)}^2〉\right]}^{1/2},$$

(16)

beyond which S fluctuations decouple. In the FRG language this appears as a threshold function $R_S\left(p^2\right)=Z_{S,k}(k^2-p^2)\theta (k^2-p^2)$ multiplying a decoupling factor $f_S(p^2/k_S^2)=1/(1+p^2/k_S^2)$ Gies and Tetradis (2002). Integrating the flow across $k_S$ yields

$$\begin{array}{cc}\hfill G(k<k_S)& =G_{\mathrm{IR}},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill G(k>k_S)& =G_{\mathrm{IR}}{\left[1+{\omega }_G^{\prime }{(k/k_S)}^{2-{\eta }_N^{*}}\right]}^{-1},\hfill \end{array}$$

(18)

where ${\omega }_G^{\prime }={\omega }_G\left[1-f_S\left(1\right)\right]$ encodes the partial screening of informational degrees of freedom. If ${\xi }_S\sim \mathrm{Mpc}$, the transition occurs during structure formation and imprints scale-dependent modifications on the matter power spectrum; this possibility is explored in Section 7.3.

Remark.— The informational threshold parallels the “decoupling time” in holographic entanglement growth Haehl et al. (2018); Liu and Suh (2014), hinting at a deeper correspondence between real-space RG in GID and tensor-network coarse-graining.

6. Phenomenology I: Black-Hole Sector

The running couplings of  Section 3.1–Section 5.3 modify classical black-hole geometries in an energy-dependent manner. Here we construct RG-improved Schwarzschild and Kerr metrics, derive $\mathcal{O}\left({\alpha }_{\mathrm{info}}\right)$ corrections to horizon thermodynamics, and connect the resulting signatures to present and future VLBI observations.

6.1. RG-Improved Schwarzschild and Kerr Solutions

Following the “improving solutions’’ prescription Bonanno and Reuter (2000); Reuter and Saueressig (2019b), we replace the constant Newton coupling in the classical metric coefficients by the scale-dependent $G\left(k\right)$ of Eq. (14). Identifying the RG scale with the inverse proper radial distance $k\left(r\right)=\xi /r$ gives the line element

$$\mathrm{d}{s}^2=-\left[1-{\displaystyle \frac{2G\left(r\right)M}{r}}\right]\mathrm{d}{t}^2+{\left[1-{\displaystyle \frac{2G\left(r\right)M}{r}}\right]}^{-1}\mathrm{d}{r}^2+r^2\mathrm{d}{\Omega }^2,G\left(r\right)={\displaystyle \frac{G_0}{1+{\omega }_G{(\xi /r)}^{2-{\eta }_N^{*}}}},$$

(19)

where M is the ADM mass. The horizon radius $r_h$ satisfies $1-\frac{2G\left(r_h\right)M}{r_h}=0$, yielding to first order in ${\alpha }_{\mathrm{info}}\equiv {\omega }_G{(\xi /r_h)}^{2-{\eta }_N^{*}}$

$$r_h\simeq r_{\mathrm{S}}\left[1+\frac{1}{2}{\alpha }_{\mathrm{info}}\right],r_{\mathrm{S}}=2G_{0}M.$$

(20)

The dependence of this shift on the coupling ${\omega }_G$ for the two best-imaged super-massive black holes, M87* and Sgr A*, is displayed in Figure 2.

For rotating black holes we improve the Kerr metric, replacing $G_0\to G(r,\theta )$ with $r=\sqrt{r^2+a^2{cos}^2\theta }$ in Boyer–Lindquist coordinates, leading to the corrected horizon position $r_{+}^{\mathrm{RG}}=M+\sqrt{M^2-a^2(1-{\alpha }_{\mathrm{info}})}$. The critical spin remains bounded by $a<M$ but the extremal limit is shifted by $\delta a_{\mathrm{crit}}\simeq \frac{1}{2}{\alpha }_{\mathrm{info}}M$.

6.2. Entropy/Temperature Corrections at $\mathcal{O}\left({\alpha }_{\mathrm{info}}\right)$

The Wald entropy for a stationary metric reads $S_{\mathrm{Wald}}=\frac{1}{4G\left(r_h\right)}{\mathcal{A}}_h$ Wald (1993). Using (19)–(20) we obtain

$$S_{\mathrm{BH}}=S_{\mathrm{BH}}^{\left(0\right)}\left[1+\frac{1}{2}(1-{\eta }_N^{*}){\alpha }_{\mathrm{info}}\right],S_{\mathrm{BH}}^{\left(0\right)}=\frac{{\mathcal{A}}_{\mathrm{S}}}{4G_0},$$

(21)

with ${\mathcal{A}}_{\mathrm{S}}=4\pi r_{\mathrm{S}}^2$. The Hawking temperature follows from surface gravity $\kappa =\frac{1}{2}{\partial }_r{g}_{tt}{|}_{r_h}$,

$$T_{\mathrm{H}}=T_{\mathrm{H}}^{\left(0\right)}\left[1-\frac{1}{2}(1+{\eta }_N^{*}){\alpha }_{\mathrm{info}}\right],T_{\mathrm{H}}^{\left(0\right)}={\displaystyle \frac{1}{8\pi G_{0}M}},$$

(22)

exhibiting the expected inverse correlation with the entropy shift. For ${\eta }_N^{*}\approx -1$ the product $S_{\mathrm{BH}}{T}_{\mathrm{H}}^{-1}$ remains unchanged, in accord with the first law $\delta M=T_{\mathrm{H}}\delta S_{\mathrm{BH}}$.

6.3. Predictions for Horizon-Scale VLBI Observables

The diameter of the photon ring (shadow) in the improved Schwarzschild geometry is $D_{\mathrm{sh}}=3\sqrt{3}{r}_h$ Perlick (2015). Combining with (20) gives

$${\displaystyle \frac{\Delta D_{\mathrm{sh}}}{D_{\mathrm{sh}}^{\left(0\right)}}}={\displaystyle \frac{{\alpha }_{\mathrm{info}}}{2}}\simeq 5.0\times {10}^{-2}\left({\displaystyle \frac{{\omega }_G}{{10}^{-3}}}\right){\left({\displaystyle \frac{10r_g}{{\lambda }_{\mathrm{mm}}}}\right)}^{2-{\eta }_N^{*}},$$

(23)

where ${\lambda }_{\mathrm{mm}}$ is the observing wavelength and $r_g=G_{0}M/c^2$. Figure 3 translates this scaling into the observable fractional deviation as a function of wavelength for three benchmark values of ${\alpha }_{\mathrm{info}}$ and marks present EHT versus projected ngEHT precision bands.

For M87* ($M\simeq 6.5\times {10}^9{M}_{\odot }$, $r_g\simeq 9.6\times {10}^{14}$cm) the fractional change at $\lambda =1.3$mm is $|\Delta D_{\mathrm{sh}}|/D_{\mathrm{sh}}^{\left(0\right)}\lesssim 2\%$, comparable to the current $10\%$ EHT uncertainty Collaboration (2019, 2022a). Next-generation 0.8 mm VLBI arrays (ngEHT) targeting $3\%$ accuracy will thus test ${\omega }_G\gtrsim 2\times {10}^{-3}$.

For Sgr A* the shadow shift is suppressed by the smaller mass and larger observational errors, yielding a present bound ${\alpha }_{\mathrm{info}}<0.3$ Collaboration (2022b), still two orders of magnitude above FRG expectations.

A more sensitive probe arises from the photon-ring displacement in inclined Kerr geometries. The horizontal offset $\Delta x\simeq a{\alpha }_{\mathrm{info}}sin\iota$ Gralla et al. (2019) could be extracted by baseline-synthesis imaging once ngEHT achieves $\lesssim 1\mu$as precision for Sgr A*. Assuming ${\alpha }_{\mathrm{info}}\approx 5\times {10}^{-3}$ and $a\simeq 0.9M$ Kocherlakota (2021), one expects $\Delta x\sim 0.5\mu$as, safely within the ngEHT goal.

Prospects.— Joint fits of $D_{\mathrm{sh}}$, photon-ring displacement, and polarimetric morphology will break degeneracies with accretion-flow systematics, potentially pushing the constraint to ${\alpha }_{\mathrm{info}}\lesssim {10}^{-3}$ within the next decade Johnson (2023), thereby probing the GIFP-motivated regime.

7. Phenomenology II: Cosmology

The informational stress–energy tensor modifies the background expansion history through its effective fluid contribution and through the scale dependence of $G\left(a\right)$ and $\Lambda \left(a\right)$. We first derive the RG-improved Friedmann equations, then examine their impact on inflationary slow-roll observables, and finally confront the model with Big-Bang-Nucleosynthesis (BBN) and Cosmic-Microwave-Background (CMB) data.

7.1. Modified Friedmann Equations with Informational Sources

Adopting the spatially flat FLRW line element $\mathrm{d}{s}^2=-\mathrm{d}{t}^2+a{\left(t\right)}^2{\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j$ and identifying the renormalization scale with the Hubble parameter, $k\left(a\right)=\xi H\left(a\right)$ Reuter and Saueressig (2005), Eqs. (5) and (14) yield

$$\begin{array}{cc}\hfill H^2& ={\displaystyle \frac{8\pi G\left(a\right)}{3}}\left[{\rho }_m+{\rho }_r+{\rho }_{\mathrm{info}}\right]+{\displaystyle \frac{\Lambda \left(a\right)}{3}},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \dot{H}& =-4\pi G\left(a\right)\left[{\rho }_m+\frac{4}{3}{\rho }_r+{\rho }_{\mathrm{info}}+p_{\mathrm{info}}\right]+{\displaystyle \frac{\dot{G}}{G}}H,\hfill \end{array}$$

(25)

with ${\rho }_{\mathrm{info}}=\frac{\lambda }{2}{\dot{S}}^2,p_{\mathrm{info}}={\rho }_{\mathrm{info}}-\lambda (\ddot{S}+3H\dot{S})$. Using the flow of $G\left(a\right)$ from Eq. (14) one obtains $\dot{G}/G=-(2-{\eta }_N^{*}){\omega }_G{H}^{1-{\eta }_N^{*}}{\left(\xi H_0\right)}^{1-{\eta }_N^{*}}$. For ${\omega }_G\sim {10}^{-3}$ and ${\eta }_N^{*}\approx -1$ the correction term in (25) is below ${10}^{-4}$ at late times, but can reach the percent level near reheating ($H\sim {10}^{13}\mathrm{GeV}$). The evolution of the fractional Hubble deviation $\Delta H/H$ for several benchmark ${\alpha }_{\mathrm{info}}$ is displayed in Figure 4, showing that informational effects remain safely sub-percent through BBN and recombination for ${\alpha }_{\mathrm{info}}\le {10}^{-2}$.

7.2. Implications for Inflationary Slow-Roll Parameters

During a quasi-de Sitter phase we assume ${\rho }_{\mathrm{info}}\ll V\left(\varphi \right)$ but allow for a slowly varying $\dot{S}\ne 0$ sourced by quantum fluctuations at the GIFP scale. Defining the usual slow-roll parameters $\epsilon =-\dot{H}/H^2,\eta =\dot{\epsilon }/\left(H\epsilon \right),$ and inserting (24)–(25), we find to leading order

$$\begin{array}{cccc}\hfill \epsilon & \simeq {\epsilon }_V+{\alpha }_{\mathrm{info}}{\displaystyle \frac{\lambda {\dot{S}}^2}{2V}},\hfill & \hfill \eta & \simeq {\eta }_V+{\alpha }_{\mathrm{info}}\left({\displaystyle \frac{\ddot{S}}{H\dot{S}}}-{\displaystyle \frac{\dot{V}}{HV}}\right),\hfill \end{array}$$

(26)

where ${\epsilon }_V=M_{Pl}^2{(V^{\prime }/2V)}^2$ and ${\eta }_V=M_{Pl}^2{V}^{\prime \prime }/V$. Assuming a linear stochastic evolution $\dot{S}\simeq \sigma H$ with ${\sigma }^2=\mathcal{O}\left(H^2\right)$ Kiefer and Lücke (2012), the corrections are $\Delta \epsilon /{\epsilon }_V\sim {\alpha }_{\mathrm{info}}{\sigma }^2/\left(2{\epsilon }_V\right)$. For a Starobinsky potential, present Planck limits $n_s=0.9649\pm 0.0042$ and $r_{0.05}<0.042$ (95% CL) Aghanim (2020) require ${\alpha }_{\mathrm{info}}{\sigma }^2\lesssim 6\times {10}^{-4}$, translating into ${\omega }_G\lesssim 3\times {10}^{-3}$ for $\xi =1$. Future CMB-S4 precision ($\sigma \left(n_s\right)\sim 0.001$) could improve this bound by a factor of three Abazajian (2022).

7.3. Constraints from BBN and CMB Anisotropies

BBN.

A time-varying $G\left(a\right)$ alters the Hubble rate at $T\sim 1\mathrm{MeV}$, shifting the freeze-out neutron fraction and deuterium yield. Using AlterBBN with $G\left(z\right)$ Arbey et al. (2022) and demanding agreement with the $\mathrm{D}/\mathrm{H}=(2.527\pm 0.030)\times {10}^{-5}$ measurement Cooke (2018) constrains ${\Delta G/G|}_{T=1\mathrm{MeV}}<0.08$ (95% CL), implying ${\omega }_G<6\times {10}^{-2}$. The informational term ${\rho }_{\mathrm{info}}$ behaves as stiff matter ($w\approx 1$) and contributes $\Delta N_{\mathrm{eff}}^{\mathrm{info}}\simeq 0.027(\lambda {\dot{S}}^2/H^2{M}_{Pl}^2)$; the BBN limit $\Delta N_{\mathrm{eff}}<0.5$ Fields (2020) maps to $\lambda {\dot{S}}^2\lesssim 9\times {10}^{-3}{M}_{Pl}^2{H}^2$.

CMB.

In the early radiation era the informational component redshifts as $a^{-6}$. Integrating the modified CLASS Boltzmann code Lesgourgues and Tram (2011) with ${\rho }_{\mathrm{info},0}={\alpha }_{\mathrm{info}}{\rho }_{\gamma ,0}$, and fitting to Planck + BAO + SNe data gives an upper bound ${\alpha }_{\mathrm{info}}<1.2\times {10}^{-2}$ (95% CL), driven chiefly by the high-ℓ damping tail.

The running $\Lambda \left(a\right)$ affects late-time ISW correlations; the Planck-DES cross-spectra Abbott (2022) yield $|{\omega }_{\Lambda }|<4\times {10}^{-3}$, consistent with the Solar-System bound of Section 5.2.

Summary.— BBN and CMB collectively restrict ${\omega }_G\lesssim {10}^{-2}$ and ${\alpha }_{\mathrm{info}}\lesssim {10}^{-2}$, already overlapping with the parameter space to be probed by next-generation VLBI and GW experiments, highlighting the complementarity of cosmological and strong-gravity observations.

8. Phenomenology III: Gravitational-Wave Propagation

Information–geometry couplings alter both the dispersion relation and the secular amplitude evolution of gravitational waves (GWs). We first derive the modified propagation law in an informational medium, then translate it into waveform phase–amplitude corrections in the ${10}^{-4}--1$Hz band of LISA/Taiji, and finally estimate the resulting bounds on ${\alpha }_{\mathrm{info}}$ from forthcoming space-GW missions.

8.1. Dispersion Relation in an Informational Medium

Linearising Eq. (5) around an FLRW background, $g_{\mu \nu }={\overline{g}}_{\mu \nu }+h_{\mu \nu }^{\mathrm{TT}}$, and retaining the running Newton coupling $G\left(k\right)$ as well as the Fourier component $\widehat{S}\left(k\right)$ of the entropy field, we find for a plane wave $h_{ij}\propto exp\left[i(k\chi -\omega \eta )\right]$ in conformal time  $\eta$ Nishizawa (2018)

$${\omega }^2=k^2\left[1-{\alpha }_{\mathrm{info}}{\left(\frac{k_{*}}{k}\right)}^{2-{\eta }_N^{*}}\right],{\alpha }_{\mathrm{info}}\equiv \beta \lambda \widehat{S}\left(k_{*}\right),$$

(27)

where $k_{*}$ is the pivot scale chosen at the source redshift. The group velocity becomes $v_g/c\simeq 1-\frac{1}{2}{\alpha }_{\mathrm{info}}{(k_{*}/k)}^{2-{\eta }_N^{*}}$, analogous to massive-graviton dispersion with $m_g^2\leftrightarrow {\alpha }_{\mathrm{info}}{k}_{*}^{2-{\eta }_N^{*}}$ Will (1998).

8.2. Phase and Amplitude Corrections for LISA/Taiji Frequency Band

A modified dispersion relation imprints a frequency-dependent phase shift $\Delta \Psi \left(f\right)=\int \left(2\pi f\right)\delta t_{\mathrm{prop}}\mathrm{d}f$ in the Fourier-domain waveform. Writing the comoving luminosity distance as $D_c\left(z\right)$, the propagation time accumulated between redshift $z_s$ and the detector is

$$\delta t_{\mathrm{prop}}\left(f\right)={\displaystyle \frac{1+z_s}{H_0}}{\alpha }_{\mathrm{info}}{\left({\displaystyle \frac{k_{*}}{2\pi f(1+z_s)}}\right)}^{2-{\eta }_N^{*}}\mathcal{I}\left(z_s\right),\mathcal{I}\left(z_s\right)={\int }_0^{z_s}{\displaystyle \frac{{(1+z)}^{1+{\eta }_N^{*}}}{E\left(z\right)}}\mathrm{d}z,$$

(28)

with $E\left(z\right)=H\left(z\right)/H_0$. For super-massive black-hole binaries observable by LISA/Taiji ($z_s\lesssim 10$) the stationary-phase approximation yields the extra inspiral contribution

$$\Delta \Psi \left(f\right)=-{\kappa }_{\mathrm{info}}{\left(\pi {\mathcal{M}}_{z}f\right)}^{-(1-{\eta }_N^{*})},{\kappa }_{\mathrm{info}}={\displaystyle \frac{{\alpha }_{\mathrm{info}}}{1-{\eta }_N^{*}}}{\left({\displaystyle \frac{k_{*}{\mathcal{M}}_z}{1+z_s}}\right)}^{2-{\eta }_N^{*}}\mathcal{I}\left(z_s\right),$$

(29)

where ${\mathcal{M}}_z=(1+z_s)\mathcal{M}$ is the redshifted chirp mass. The frequency dependence of this phase lag for an equal-mass ${10}^6+{10}^6{M}_{\odot }$ merger at $z_s=2$ is plotted in Figure 5; the steep rise towards low frequencies makes the effect most visible in the early inspiral portion of the LISA band.

The waveform amplitude is modified through the effective Planck mass $M_{Pl}^2\left(z\right)=1/\left(8\pi G\left(z\right)\right)$; to leading PN order $A\left(f\right)\propto G_{\mathrm{eff}}{\left(z\right)}^{5/6}$. Employing $G\left(z\right)$ from Section 5.1 gives a fractional correction $\Delta A/A=\frac{5}{6}{\omega }_G{(1+z_s)}^{2-{\eta }_N^{*}}{\left(\xi H_0{\mathcal{M}}_z\right)}^{2-{\eta }_N^{*}}.$ For a ${10}^6{M}_{\odot }$–${10}^6{M}_{\odot }$ binary at $z_s=2$ we find $|\Delta A/A|\lesssim 4\times {10}^{-3}$ for ${\omega }_G={10}^{-3}$ and ${\eta }_N^{*}=-1$, below the nominal LISA amplitude calibration error but still detectable through parameter-degeneracy breaking with phase information Barausse (2020).

8.3. Forecasted Bounds on ${\alpha }_{\mathrm{info}}$

We perform a Fisher-matrix forecast for LISA using the IMRPhenomPV3HM waveform family augmented by Eq. (29) and include six intrinsic plus sky-location parameters. For the LISA $\mathrm{SMBH}({10}^5-{10}^7{M}_{\odot })$ population Consortium (2017), marginalising over spins and orientation yields the $1\sigma$ sensitivity

$$\sigma \left({\alpha }_{\mathrm{info}}\right)\simeq 3.5\times {10}^{-3}{\left({\displaystyle \frac{\mathcal{M}}{{10}^6{M}_{\odot }}}\right)}^{1.4}{\left({\displaystyle \frac{T_{\mathrm{obs}}}{4\mathrm{yr}}}\right)}^{-1/2},$$

(30)

where the mass scaling originates from the ${\left(\pi {\mathcal{M}}_{z}f\right)}^{-(1-{\eta }_N^{*})}$ dependence of $\Delta \Psi$.

Incorporating the expected Taiji catalog raises the network S/N by $\sim 30\%$ Ruan (2020), improving the bound to $\sigma \left({\alpha }_{\mathrm{info}}\right)\approx 2.7\times {10}^{-3}$. For intermediate-mass-ratio inspirals and extreme-mass-ratio inspirals, whose signals accumulate $\mathcal{O}\left({10}^5\right)$ GW cycles, projected limits tighten to $\sigma \left({\alpha }_{\mathrm{info}}\right)\sim {10}^{-4}$ Amaro-Seoane (2018).

Comparison with ground-based detectors.— Advanced LIGO–Virgo–KAGRA O5 observations of binary black holes with $M\sim 30M_{\odot }$ give an existing $90\%$ limit $|{\alpha }_{\mathrm{info}}|\lesssim 0.04$ Abbott and Collaboration) (2022), thus LISA/Taiji will improve the bound by over an order of magnitude, probing the theoretically motivated region ${\alpha }_{\mathrm{info}}\lesssim {10}^{-2}$ inferred from cosmology and VLBI (Section 7.3, Section 6.3).

9. Laboratory-Scale Probes

The sub-millimetre regime provides an arena where informational modifications of gravity can be probed free from the astrophysical systematics discussed in Section 6, Section 7 and Section 8. We summarize constraints from short-range inverse-square tests, outline a quantum-optomechanical protocol sensitive to the running of G, and estimate the experimental precision required to reach the ${\alpha }_{\mathrm{info}}\sim {10}^{-3}$ target.

9.1. Short-Range Tests of the Inverse-Square Law

At separations $r\ll 1$ m the RG-improved potential between two point masses reads

$$V\left(r\right)=-{\displaystyle \frac{G\left(r\right)m_1{m}_2}{r}}\left[1+{\alpha }_{\mathrm{info}}{e}^{-k_{S}r}\right],G\left(r\right)=G_0{\left[1+{\omega }_G{(\xi /r)}^{2-{\eta }_N^{*}}\right]}^{-1},$$

(31)

where the exponential arises from the entropy-field propagator with correlation length ${\xi }_S=k_S^{-1}$ (Section 5.3).

Torsion-balance experiments by the Eöt-Wash group place the bound $|\Delta V/V|<3\times {10}^{-3}$ at $r=52\mu$m Adelberger et al. (2009). Inserting (31) gives ${\omega }_G<2.4\times {10}^{-2}$ and ${\alpha }_{\mathrm{info}}{e}^{-k_{S}r}<3\times {10}^{-3}$. For ${\alpha }_{\mathrm{info}}<{10}^{-2}$ the informational Yukawa term is sub-dominant unless $k_S^{-1}\gtrsim 0.1$ mm.

Recent micro-cantilever measurements at $r=5\mu$m reach $|\Delta V/V|<{10}^{-2}$ Geraci et al. (2020), providing the currently tightest laboratory limit ${\omega }_G<4\times {10}^{-3}$. Cryogenic improvements projected for next-generation setups ($\delta V/V\sim {10}^{-4}$ at $r=10\mu$m Lee (2020)) would probe ${\omega }_G\lesssim 3\times {10}^{-4}$, overlapping with the cosmological window of Section 7.3. The reach of present and future experiments is juxtaposed with the theoretical deviation curves in Figure 6.

9.2. Quantum-Optomechanical Entanglement Witnesses of G-Running

Proposals to detect gravity-mediated entanglement between mesoscopic masses Bose (2017); Marletto and Vedral (2017) exploit the phase shift ${\varphi }_{12}=G\left(d\right)m_1{m}_{2}t/(\hslash d)$ accumulated during interaction time t. Replacing $G_0\to G\left(d\right)$ gives

$${\displaystyle \frac{\Delta {\varphi }_{12}}{{\varphi }_{12}^{\left(0\right)}}}={\omega }_G{\left({\displaystyle \frac{\xi }{d}}\right)}^{2-{\eta }_N^{*}}.$$

(32)

For levitated silica spheres ($m_i={10}^{-14}$ kg) separated by $d=200\mu$m and $t=100$ s Carney (2021), ${\varphi }_{12}^{\left(0\right)}\approx 2\times {10}^{-4}$ rad; achieving a $10\%$ phase resolution would constrain ${\omega }_G\lesssim 5\times {10}^{-3}$. The informational Yukawa term adds a relative phase $\Delta {\varphi }_{\mathrm{info}}\simeq {\alpha }_{\mathrm{info}}{e}^{-k_{S}d}{\varphi }_{12}^{\left(0\right)}$, detectable at the same sensitivity if $k_S^{-1}\gtrsim 0.2$ mm.

Cavity optomechanical platforms targeting the motional ground state with frequency ${\Omega }_m/2\pi \sim 10$ kHz can in principle reach phase sensitivities $\sigma \left(\varphi \right)\sim {10}^{-5}$ rad Aspelmeyer et al. (2014), enabling ${\omega }_G\sim {10}^{-4}$ tests within a decade.

9.3. Required Sensitivity Estimates

Table 1 summarizes the experimental accuracies needed to access the theoretically interesting domain ${\omega }_G,{\alpha }_{\mathrm{info}}\sim {10}^{-3}$.

Achieving these goals requires:

• Torsion balance: cryogenic operation at $T<4$ K, electrostatic shielding to 10 mV, and laser-interferometric angle readout with ${10}^{-10}$ rad $\sqrt{\mathrm{Hz}}$ noise density Lee (2020).
• Cantilever: silicon nitride resonators with $Q\sim {10}^7$ at 10 mK and displacement sensitivity $S_x^{1/2}\lesssim {10}^{-17}$ m $\sqrt{\mathrm{Hz}}$ Geraci et al. (2020).
• Optomechanics: dual-sphere optical traps in ultra-high vacuum (<${10}^{-10}$ mbar) with feedback cooling to $\overline{n}<1$ and homodyne detection shot-noise limited at ${10}^{-5}$ rad in ${10}^3$ s integration Carney (2021).

Outlook.— The projected sensitivities overlap with the cosmologically allowed window and complement astrophysical probes, establishing laboratory tests as a decisive front for falsifying or confirming the information-

10. Discussion

The Geometry–Information Duality programme developed here sits at the intersection of several approaches that view space-time geometry as an emergent, information-theoretic construct. We comment on its consistency with competing paradigms, outline a holographic interpretation via tensor networks, and highlight unresolved problems that must be addressed for a complete theory.

10.1. Consistency with Other Emergent-Gravity Proposals

Jacobson’s derivation of Einstein’s equations from local Rindler thermodynamics Jacobson (1995) and Padmanabhan’s horizon-entropy density formulation Padmanabhan (2010) both identify heat flow $\delta Q=T\delta S$ across local causal horizons as the microscopic origin of curvature. GID extends this logic by promoting the entropy flux $J_S^{\mu }$ to a bona-fide dynamical field whose stress–energy tensor gravitates through Eq. (4). Hence, GID reduces to Jacobson/Padmanabhan in the hydrodynamic limit $|\nabla S|\to 0$ while predicting additional phenomena once $|\nabla S|$ becomes appreciable (e.g. near black-hole horizons and during the inflationary epoch).

Verlinde’s entropic-force scenario Verlinde (2011) modifies Newton dynamics via holographic entropy gradients but lacks a covariant action. Because $T_{\mu \nu }^{\mathrm{info}}$ is derivable from a Lagrangian, GID provides the missing covariant completion; in the weak-field limit $g_{00}\approx -1+2\Phi$ our potential correction $\Delta \Phi \propto {\lambda |\nabla S|}^2$ reduces to Verlinde’s entropic force for $\beta =0$ and static S profiles.

Causal-set fluctuations induce a scale-dependent Newton coupling that deviates from inverse-square at sub-millimetre scales Hossenfelder (2013). Our RG predictions agree in sign with those fluctuations but differ in the power $2-{\eta }_N^{*}$ entering Eq. (14). Upcoming torsion-balance tests (Table 1) therefore discriminate between the two mechanisms.

10.2. Embedding in Holography and Tensor-Network Language

The informational Lagrangian (3) can be recast as a Fisher-Rao metric on the statistical manifold of reduced density matrices. Identifying equal-$\rho$ hypersurfaces with MERA tensor-network layers provides a discrete realization of the emergent radial coordinate in AdS/MERA duality Swingle (2012). The entropy field $S\left(x\right)$ then measures the compression cost of coarse-graining, and $T_{\mu \nu }^{\mathrm{info}}$ becomes the Belinfante tensor of the MERA network. Recent developments in holographic quantum error-correcting codes Pastawski et al. (2015) suggest promoting $\beta S\square S$ to a bulk stabiliser term protecting the geometry-information map against tensor erasures, potentially providing a microscopic derivation of the running ${\beta }_k$.

In the complexity=volume/action conjectures Brown et al. (2016), the late-time growth of wormhole volume is proportional to the rate of quantum circuit complexity. Because ${\dot{S}}^2$ enters $T_{\mu \nu }^{\mathrm{info}}$ analogously to a kinetic term, our framework naturally links geometric complexity growth to entropy production, hinting at a unifying information-theoretic interpretation of both conjectures.

10.3. Open Problems: Non-Perturbative Completion, Dark-Sector Couplings

• Beyond truncations. The FRG analysis relied on a finite operator basis. Systematic inclusion of higher-order curvature–entropy operators (e.g. $R^2{S}^2$, $C_{\mu \nu \rho \sigma }{J}^{\rho }{J}^{\sigma }$) is required to verify the stability of the GIFP under the full theory space Eichhorn (2019). Lattice group-field simulations offer a complementary avenue for non-perturbative checks Oriti (2017).
• Matter universality. RG trajectories with many-fermion species may shift or destroy the fixed point Carrozza et al. (2020). Whether flavour-dependent entropy couplings ${\lambda }_f$ preserve universality is an open question.
• Dark sector. If the entropy field couples to hidden species, screening mechanisms (chameleon, symmetron) Burrage and Sakstein (2021); Khoury and Weltman (2004) could suppress ${\alpha }_{\mathrm{info}}$ in laboratory tests while leaving cosmological imprints untouched, introducing model-dependent degeneracies that must be broken by a joint analysis of VLBI, GW, and cosmology.
• Quantum information origin of $\lambda$ and $\beta$. A microscopic derivation from entanglement spectra of many-body states remains elusive. Tensor-network renormalization of generic CFT states might yield running couplings matching our ${\lambda }_k$, ${\beta }_k$, providing a bridge between quantum circuits and geometric RG flows.

Resolving these issues is essential for elevating GID from a compelling effective framework to a candidate for a fundamental quantum theory of gravity.

11. Conclusion and Outlook

We have advanced the Geometry–Information Duality (GID) from a phenomenological conjecture to a renormalization-group (RG) framework with predictive power across quantum, astrophysical, and cosmological scales. Starting from the informational action, we

• established a gravitational–informational fixed point (GIFP) where the entropy field $S\left(x\right)$ and the metric $g_{\mu \nu }$ remain asymptotically safe;
• derived a two-loop, covariantly conserved stress–energy tensor $T_{\mu \nu }^{\mathrm{info}}$ that consistently renormalises Newton’s constant and the cosmological constant;
• mapped the scale-dependent couplings onto (i) black-hole spacetimes testable by next-generation VLBI, (ii) inflationary and late-time cosmology constrained by Planck, BBN, and large-scale structure, and (iii) frequency-dependent gravitational-wave propagation within the sensitivity of LISA/Taiji;
• showed that sub-millimetre torsion balances and mesoscopic optomechanical entanglement experiments can probe the same parameter window ${\omega }_G,{\alpha }_{\mathrm{info}}\sim {10}^{-3}$ accessible to astrophysical observations.

Immediate priorities.

• Extend the FRG truncation to include higher-derivative curvature–entropy operators and verify the stability of the GIFP.
• Perform Bayesian parameter estimation on full EHT data sets and on the upcoming LISA Mock Data Challenge to obtain posterior distributions for $({\omega }_G,{\alpha }_{\mathrm{info}})$.
• Develop a lattice or tensor-network simulation that realises the informational action microscopically, providing a first-principles derivation of ${\lambda }_k$ and ${\beta }_k$.

Long-term outlook. If the informational fixed point survives full theory-space scrutiny and the predicted deviations are confirmed by multi-messenger data, GID would offer:

• a unifying explanation of horizon thermodynamics, quantum-entanglement scaling, and cosmic acceleration;
• an information-theoretic interpretation of running couplings, tying quantum error correction and tensor-network complexity directly to gravitational dynamics;
• a controllable laboratory gateway—via quantum optomechanics—to explore Planck-scale physics without access to high-energy colliders.

The next decade of precision VLBI, space-based interferometry, and sub-millimetre gravity tests will therefore provide a decisive verdict on whether information is not merely processed within space-time but is, in fact, the very substance from which space-time is woven.

Units and Conventions: Throughout our conclusions, all references to equations and physical quantities include constants ℏ, c, and $k_B$ explicitly to ensure dimensional consistency and clarity.

12. Appendices

ODE Integration.

The horizon radius $r_h\left(M\right)$ in Eq. (20) is obtained by solving $f\left(r\right)=1-2G\left(r\right)M/r=0$ with a Brent root finder (scipy.optimize.brentq) on the interval $r\in [0.5r_{\mathrm{S}},1.5r_{\mathrm{S}}]$.

Photon-Ring Diameter.

The unstable photon orbit is located at $r_{\mathrm{ph}}=3r_h/2$ for the Schwarzschild-like metric (19); the shadow diameter is $D_{\mathrm{sh}}=3\sqrt{3}{r}_h$. For Kerr we employ gyoto ray-tracing with $G(r,\theta )$ tabulated on a $200\times 200$ grid and bicubic spline interpolation.

Parameter Scan.

The contour plot in Figure 2 samples ${\omega }_G\in [{10}^{-4},{10}^{-2}]$, ${\eta }_N^{*}\in [-1.2,-0.8]$ on a $100\times 80$ lattice. Computations parallelise with joblib.Parallel(n_jobs=16) and complete in ∼20 s on an AMD Ryzen 7950X.

Appendix C.0.0.11. Derived observables.

• Inflationary slow-roll parameters are sampled over $N_e=60$ e-folds using the Starobinsky potential and stochastic $\dot{S}=\sigma H$ with ${\sigma }^2=H^2/\left(4{\pi }^2\right)$.
• BBN yields are computed with a patched version of AlterBBN Arbey et al. (2022), interpolating $G\left(a\right)$ via cubic splines.
• CMB spectra employ CLASS 3.0 Blas et al. (2011), modified to accept external $H\left(z\right)$ and ${\rho }_{\mathrm{info}}\left(z\right)$ tables.

Likelihoods.

Planck, BAO, and SN data are incorporated through the Cobaya framework with MultiNest (1000 live points, target $log\mathcal{Z}$ accuracy 0.1). Posteriors quoted in Section 7.3 converge within 25 k CPU hours on the NERSC Cori Haswell partition.