Unified Entropic Dynamics Framework for Classical, and Quantum Wave Equations

One of the central challenges in modern theoretical physics is the reconciliation of classical mechanics, quantum mechanics, thermodynamics, and relativity within a single coherent framework [1, 2, 3, 4, 5, 6]. Although each theory is remarkably successful in its own domain, they remain conceptually and mathematically distinct at both microscopic and macroscopic scales. Classical physics describes the deterministic motion of bodies through spacetime, quantum theory governs the probabilistic evolution of microscopic systems, thermodynamics captures macroscopic equilibrium and irreversible processes, and general relativity encodes gravity as spacetime curvature. Despite their apparent differences, all these domains share a common feature: the interplay between geometry, curvature, and information flow. In classical mechanics and wave theory, curvature links geometry to dynamics—the curvature of a potential surface defines restoring forces, stability, and oscillatory behavior. In quantum mechanics, it appears in the curvature of probability amplitudes within Hilbert or configuration space, determining interference and coherence. In thermodynamics, curvature arises in entropy landscapes that guide systems toward equilibrium. In general relativity, curvature becomes geometric in the most literal sense, describing how matter and energy determine the shape of spacetime itself. The laws of physics share an important conceptual and mathematical connection with B. R. Frieden’s Extreme Physical Information (EPI) principle  [7]. The EPI approach regards physical laws as emergent from informational optimization. In that presented framework, physical laws arise from the extremization of the physical information functional. By using this approach, Frieden and Soffer showed that many fundamental equations of physics can be recovered from the condition of informational balance between data and source [7]. In summary, it can be stated that the underlying variational principle operates in a space of probability amplitudes and treats Fisher information as the unifying geometric quantity that underlies both mechanics and field theory.

From the perspective of entropic dynamics (ED), these manifestations of curvature are not disparate but complementary: each expresses how information about the microscopic degrees of freedom of a system constrains its macroscopic evolution. Thus, the transition from energy curvature to entropic curvature represents a shift in the language of physics: from force-based determinism to inference-based dynamics. The ED framework, first introduced by Caticha et al. [8, 9], reconstructs physical laws not by quantizing pre-existing classical models but by applying the principles of entropic inference under appropriate constraints. By maximizing entropy with respect to a set of physically meaningful variables, ED derives both stochastic and deterministic evolution equations, recovering the Fokker–Planck and Hamilton–Jacobi forms as complementary limits of the same informational process. Their combination yields a generalized field equation structurally equivalent to familiar wave equations but derived from inference rather than postulation. This approach has been shown to reproduce nonrelativistic quantum mechanics, classical diffusion, and wave phenomena through purely statistical arguments [10, 8, 11]. In general, entropy and information-theory principles have emerged as powerful tools in rethinking the foundations of physics  [8]. For example, modified theories of gravity have been derived from entropic considerations [12]. Generally, the ED formulation seeks to reconstruct dynamical laws not by quantizing pre-existing classical systems, but by applying the rules of entropic inference under carefully chosen constraints. First developed in 2011 [13], and successfully applied to a wide range of problems within nonrelativistic limits [10, 14, 15, 16, 17, 18]. It has been demonstrated already that the ED formulation can be used to derive quantum mechanics as a form of inference driven by entropy maximization [10, 14, 15, 16, 17]. Within this framework, two central ingredients are introduced: the entropy functional and the energy functional. The entropy functional, once constrained appropriately, leads to the Fokker-Planck (FP) equation, whereas the energy functional produces the Hamilton–Jacobi (HJ) equation. Combining these yields the Schrödinger equation, thereby recovering the structure of non-relativistic quantum mechanics from the principles of inference rather than postulation. Non-relativistic entropic dynamics (NED) assumes that a particle inhabits a three-dimensional (3D) configuration space, where the fundamental object of the study is the transition probability $P(x^{\prime}|x)$ between successive positions. Because this probability is timeless, an additional construct- entropic time - is introduced to order sequences of transitions. Relativistic entropic dynamics (RED), by contrast, supposedly embeds the particle in a four-dimensional spacetime, with time incorporated directly into the transition probability. In this way, the generated formulation should result in dynamical equations that closely resemble the Klein-Gordon equation, thereby linking ED to relativistic quantum mechanics in a natural manner. The present work extends the ED framework as a unified entropic dynamics (UED) to encompass confined or correlated relativistic and gravitational domains, thereby establishing a unified geometric foundation for classical, quantum, thermodynamic, and spacetime dynamics. The resulting generalized field equation, which has been derived from entropy maximization, governs the evolution of arbitrary dynamical variables $h_{a}$ on an information-theoretic manifold endowed with a supermetric $H_{ab}$. When specialized to different choices of $h_{a}$, in the generalized equation, it yields the fundamental laws of physics as limiting cases: the harmonic oscillator, Maxwell’s electromagnetic equation, the classical heat equation, the Schrödinger equation, and the relativistic Klein-Gordon equation. A key advancement reported here is the covariant extension of ED to curved spacetime. By replacing it with $H^{ab}$ with the spacetime metric $g^{\mu\nu}$ and introducing covariant derivatives, the ED equation becomes a geometric wave equation whose linearized form reproduces the standard gravitational wave equation in the transverse traceless gauge (TT). In this interpretation, gravitational waves correspond to oscillations of the spacetime information metric—a macroscopic manifestation of the same entropic principles that govern microscopic quantum and thermodynamic processes. This unification suggests that all forms of dynamical evolution, from quantum coherence to spacetime curvature, arise from a single informational principle: the maximization of entropy under geometric and physical constraints. The remainder of this paper is structured as follows. Section 2 develops the UED formulation and derives the unified entropic dynamics equation from maximization of entropy with diffusion, drift, and gauge covariance constraints. Section 3 applies the equation to the scalar case, reproducing simple harmonic motion as the entropic analog of Newtonian mechanics. Section 4 treats the three-vector case, recovering Maxwell’s, Fourier’s, and Schrodinger’s equations as distinct limits of the same informational structure. Section 5 extends the UED framework to the four-vector case, deriving both the relativistic Klein–Gordon and the gravitational-wave equations from entropic inference. Section 6 discusses the broader implications of these results, highlighting how energy, probability, and entropy emerge as interwoven aspects of information geometry.

Here we develop a general entropic dynamics or the UED construction valid for any dynamical variable $h_{a\ldots}$ living on a $d$-dimensional configuration manifold with supermetric $H_{ab\ldots}$. Our goal is the short-step transition probability $P(h^{\prime}|h)$, obtained by maximizing the relative entropy;

with respect to covariant ignorance prior $Q(h^{\prime}|h)\propto H^{1/2}(h^{\prime})$, subject to physically motivated constraints: (i) a fixed mean squared step $\langle\Delta h_{a\ldots}\Delta h_{b\ldots}\rangle$ (diffusive motion), (ii) a drift driven by a potential $\varphi$, and (iii) gauge-covariant coupling to a tensor potential $A_{a\ldots}$, together with normalization. The resulting variational problem yields a Gaussian kernel whose mean and covariance are set by Lagrange multipliers; iterating short steps produces a Fokker-Planck equation, and adding an energy functional gives a Hamilton-Jacobi companion. Next, we specify the prior, impose relevant constraints, and maximize the entropy functional. The prior probability codifies any relation between $h$ and $h^{\prime}$ before processing the actual information contained in the relevant constraints. Assign equal probabilities to equal volumes. That is, in the case of ignorance, the prior is uniform.

Here, $Q(h^{\prime}|h)$ encodes our state of knowledge prior to processing the constraints. To make ’ignorance’ precise in an arbitrary configuration space $M$, let $h^{A}$ (a multiindex collecting $a\ldots$) be a local dynamical coordinate, and let the positive-definite supermetric $H_{ab}(h)$ endow $M$ with Riemannian volume. It is equal to a dyadic of the basis vectors of $h_{a}$ and $h_{b}$ dynamical variables of form $H_{ab}=e_{a}\otimes e_{b}$. Where $e_{a}$ and $e_{b}$ are base vectors of $h_{a}$ and $h_{b}$, respectively. A prior that is genuinely uninformative must be invariant under reparameterizations $h\mapsto\tilde{h}(h)$. Writing $J^{A}{}_{B}=\partial\tilde{h}^{A}/\partial h^{B}$, the metric transforms as $H\mapsto\tilde{H}=J^{\!\top}HJ$so:

. Requiring the prior density to define an invariant measure $Q(h^{\prime}|h)\,d^{N}h^{\prime}$ then singles out the Jeffreys prior on $M$ in its uniform form  [19], i.e.,

with $Z(h)$ a local normalizer (for example, over a small geodesic ball when generating short-step kernels). Herein $Z(h)$ is taken as equal to one. Physically, $H_{ab}$ is the natural ’kinetic’ or information metric in the space of dynamical variables: in mechanics it coincides with the metric from the quadratic kinetic term, in quantum–gravitational applications with the DeWitt supermetric on superspace, and within the ED it plays the role of the Fisher–information metric induced by constraints. This choice guarantees coordinate-free inference and ensures that the ensuing maximum-entropy update produces a Gaussian short–step kernel whose covariance is $H^{-1}$, which will underlie the Fokker-Planck/Hamilton-Jacobi structure derived next. After knowing the prior, we specify the constraints. The first constraint concerns the transition probability $P(h^{\prime}|h)$. It deals with short steps (infinitesimal). A large step (finite) can be obtained by accumulating short steps

where $\kappa_{s}$ is a small constant. The second constraint concerns the draft potential, or

where $\phi$ is the drift potential, which turns out to be the phase of the wave function, and $\kappa_{f}$ is a constant. The third constraint takes into account the gauge invariance.

where $\mathcal{A}_{a\ldots}$ is the gauge-invariantiant tensor potential and $\kappa_{A}$ is a constant. The last constraint is normalization. Finally, incorporate the prior and the constraints and maximize the entropy functional. The result is

where the expectation is given by

where $\sigma^{2},\alpha$, and $\beta$ are Lagrange multipliers, and

The process is driven by Brownian motion because $\Delta h^{a\ldots}\sim\sqrt{\sigma^{2}}$. The Brownian motion leads to the Fokker-Planck equation as is in Eq. 10

where $\rho(h)$ is the marginalized probability distribution and

$\rho$ and $\varphi$ are dynamical variables. This is the case if, in addition to Eq. 11, we impose a second constraint.

Where $\mathcal{M}$ is independent of $\rho$ and $\varphi$. It can be thought of as entropic mass that balances or acts as a source term during the evolution of entropy under dynamical variables. Once the dynamical variable $h^{a\ldots}$ is set or known, then the $\mathcal{M}$ can be chosen according to dimension analysis by setting it proportional to the inner product of the conjugate variable of the dynamical variable. The minimization of $H$ as a function of $\rho$ and $\varphi$ needs to be done, and the following relation can be obtained by doing so:

where

Eqs. 11 and 15 can be combined into a single equation by using

We obtain

Eq. 17 represents the generalized framework or the UED framework sought to describe the wave dynamics of particles and fields $(\Psi)$ in an arbitrary potential $\mathcal{M}$ depending on the configuration space. In quantum theory, particles obey the rules of first quantization. In contrast, the fields follow the rules of second quantization. The inner product of conjugate variables will turn out to be a scalar invariant. Starting from a maximization of entropy with carefully chosen priors and constraints, it yields a transition probability that resembles Brownian motion and naturally leads to the Fokker–Planck and Hamilton–Jacobi equations. Introducing the wave function $\Psi=\sqrt{\rho}e^{i\phi}$, these ingredients combine into a single compact form, Eq. 17. This equation is a generalized field equation where the operator acting on $\Psi$ involves derivatives with respect to the dynamical variables, modified by a vector potential $\mathcal{A}_{\mu}$ and supplemented by a mass-like term $\mathcal{M}$. When a specific form of the dynamical variables is considered, the unified entropic dynamics equation yields their corresponding field equations, each with a specific interpretation of $\mathcal{M}$ fixed by dimensional analysis. The product of these conjugate variables naturally forms a scalar quantity that appears in the mass-like term $\mathcal{M}=-p_{a}p^{a}/\hbar^{2}$, linking the dynamical structure of the system to its geometric properties, but is derived here from purely entropic principles rather than postulated directly. The significance of Eq. 17 is that it provides a unifying template capable of describing a wide class of physical systems depending on the choice of dynamical variable $h^{a\ldots}$. When $h^{a\ldots}$ is taken to be spacetime events $x^{\mu}$, Eq. 17 reduces to the Klein–Gordon equation, demonstrating consistency with relativistic quantum theory. Therefore, the Eq. 17 encapsulates a general dynamical law from which many known equations of physics can be derived, supporting the idea that entropic dynamics may serve as a foundational framework that bridges statistical inference with quantum and gravitational theories.

For a case in which the generalized dynamical variable $h^{a}$ introduced in Eq. 15 is chosen to be a scalar quantity, both $\sqrt{H}$ and $H^{ab\ldots}$ reduce to unity. The generalized field equation further simplifies considerably if the particle dynamics is considered in the absence of external fields, i.e. $\mathcal{A}=0$. As a result, there are no tensor indices or vector potentials to account for, and the system represents the simplest possible configuration space-one described by a single coordinate $h$ and its conjugate variable. This scalar case serves as a pedagogical example showing that the entropic dynamics framework naturally reproduces classical oscillatory motion. Starting from the unified entropic dynamics equation,

We consider the situation in which the potential term vanishes and the system evolves freely in the scalar coordinate $h$. Here, the differential operator $-\partial^{2}/\partial h^{2}$ plays the role of a kinetic operator, analogous to the Laplacian in wave mechanics, while the term $\mathcal{M}$ acts as an effective mass or energy constant that characterizes the curvature of the potential landscape in configuration space. Eq. 18 is thus the entropic analog of a one-dimensional Helmholtz equation, describing standing or propagating modes depending on the sign of $\mathcal{M}$. In the scalar-time realization of the unified entropic dynamics framework, the quantity $\mathcal{M}$ acquires a clear interpretation as an information mass, encoding the intrinsic energetic content associated with temporal oscillations. When the dynamical variable is chosen to be the time scalar $h=t$, its conjugate variable is the angular frequency $\omega$, which characterizes the rate of phase accumulation in time. Dimensional consistency and entropic closure then require the information mass term to be proportional to the square of this conjugate quantity, $\mathcal{M}\propto\omega^{2}$, equivalently proportional to the square of the system’s energy scale. In classical mechanics, this identification is naturally realized in the simple harmonic oscillator, where the equation of motion $\ddot{x}+\omega^{2}x=0$ describes bounded, periodic evolution under a restoring force. Within the entropic dynamics formulation, the same structure emerges from the scalar UED equation $(-\partial_{t}^{2}+\mathcal{M})\Psi=0$, whose oscillatory solutions demand $\mathcal{M}=-\omega^{2}$. Thus, the information mass $\mathcal{M}$ plays the role of an entropic restoring parameter, encoding the curvature of the informational landscape in time and ensuring coherent oscillatory dynamics. This establishes a direct correspondence between classical mechanical stability and informational curvature, showing that the squared frequency—or equivalently the squared energy scale—acts as the fundamental scalar invariant governing temporal dynamics in the entropic framework. Therefore, the equation 17 is the canonical form of the simple harmonic oscillator in the representation of entropic dynamics. If the scalar variable $h$ is set as the physical time coordinate, $h=t$, then the conjugate variable of $t$ is angular frequency $\omega$. Consequently, $\Psi$ representing the displacement $(\mathbf{X})$ of the particle in harmonic motion and therefore reducing Eq. (18) or, in turn, Eq. 17 to the form given below:

Eq. 19 is precisely the differential equation that governs the temporal harmonic oscillations. Its general solution $\mathbf{X}(t)=Ae^{i\omega t}+Be^{-i\omega t}$ describes sinusoidal oscillations of amplitude determined by the constants $A$ and $B$, corresponding to time-propagating modes forward and backward. Physically, this result demonstrates that simple harmonic motion, a cornerstone of classical and quantum physics, emerges directly from the formalism of entropic dynamics when the system’s configuration space is one-dimensional. The oscillations reflect the system’s attempt to restore maximum entropy under the constraint of finite energy, thereby linking information-theoretic inference to the familiar restoring forces of Newtonian mechanics. In this sense, Eq. 18 represents not only the classical harmonic oscillator but also the fundamental time-dependent mode of all wavelike phenomena within the entropic framework. This equation can also be shown to be mathematically identical to the standard form of the simple harmonic oscillator. Interpreting $\mathbf{X}$ as the generalized coordinate (or, in a quantum setting, using Ehrenfest’s theorem to identify $\langle x\rangle$ with the classical trajectory) recovers Newton’s law with a linear restoring force. Near any stable equilibrium, the potential energy $V(x)$ becomes $V(x)\approx\tfrac{1}{2}kx^{2}$, therefore, the mapping of $\omega^{2}$ becomes $\omega^{2}=k/m$. In summary, Equation 18 is exactly Newton’s second law for a linear restoring force, with $\omega^{2}=k/m$; it can be obtained by force balance, by the Lagrangian, or by your reduction in entropic dynamics $\mathcal{M}=-\omega^{2}$ that yields the same simple harmonic oscillator (SHO) dynamics.

In the preceding section, the generalized entropic dynamics framework was applied to a scalar dynamical variable to demonstrate how simple harmonic motion –and by extension Newton’s second law –emerges naturally from the maximization of entropy subject to energy constraints. The scalar case served as the simplest realization of entropic inference in the configuration space, where the oscillatory behavior of the system was encoded in the curvature of its information potential. We now extend this reasoning to a richer configuration space in which the dynamical variable $h_{a}$ is promoted to a three-vector. This extension allows the formalism to describe both classical fields and particle wavefunctions that depend on spatial coordinates, thereby linking the statistical structure of entropic dynamics to the geometry of three-dimensional space.

When $h_{a}$ is taken as the spatial vector $x_{i}$, the generalized field equation 17 encompasses three distinct but complementary physical regimes that emerge from the same inferential structure. In the classical limit, it reduces to the Maxwell wave equation, which describes the propagation of electromagnetic disturbances as an optimal flow of information through spacetime [20]. When the entropic mass term is reinterpreted as a generator of temporal diffusion, the same framework produces the heat equation [21, 22], showing that thermal conduction and energy dissipation arise as entropic processes governed by the geometry of information. In the quantum-mechanical limit, the equation reduces to the Schrödinger equation, revealing that quantum behavior arises from the same inference principles that govern classical and thermodynamic evolution. This tripartite correspondence demonstrates the universality of the entropic dynamics approach: whether describing electromagnetic radiation, heat diffusion, or matter waves, the theory unifies their evolution under a single informational law. The subsections that follow develop these three cases in detail: Beginning with the electromagnetic formulation, extending to the thermodynamic interpretation, and concluding with the non-relativistic quantum description.

The preceding sections established that the entropic dynamics (ED) framework successfully reproduces classical, quantum, and thermodynamic laws as particular realizations of an underlying inferential principle. Having demonstrated that the same information-geometric foundation yields the harmonic oscillator, Maxwell’s electromagnetic equations, the classical heat equation, and the Schrödinger equation, we now extend the formalism to the relativistic domain. In this section, the generalized field equation is applied to the case where the dynamical variable $h_{\mu}$ corresponds to spacetime events $x_{\mu}$. This identification embeds the ED framework directly within the four-dimensional structure of special relativity, allowing us to investigate how inference-based dynamics reproduce the fundamental equations governing relativistic quantum fields and spacetime curvature.

We begin by considering a flat Minkowski background, where $H_{\mu\nu}$ reduces to $\eta_{\mu\nu}$, and show that the generalized ED equation naturally transforms into the Klein–Gordon equation. This demonstrates that relativistic quantum mechanics, often introduced as an independent postulate, can instead be derived from entropic inference and information geometry. The analysis is then extended to curved spacetime by replacing $\eta_{\mu\nu}$ with a general metric $g_{\mu\nu}(x)$ and substituting ordinary derivatives with covariant derivatives. This covariant generalization leads to the curved-space ED equation, which reduces, in the linearized limit, to the standard gravitational-wave equation in the transverse–traceless (TT) gauge. Thus, Section 5 unifies the relativistic and gravitational domains within the same entropic framework, revealing that both matter waves and spacetime perturbations emerge as expressions of information flow through curved geometric manifolds.

A comparative analysis has been conducted between the entropic dynamics (ED) framework developed by Caticha and collaborators [8, 9] and the Unified Entropic Dynamics (UED) formulation presented in this work. Caticha’s original framework is based on an inferential model that derives non-relativistic quantum mechanics from the principle of entropy maximization. Within this formulation, the short-step transition probabilities of particles—constrained by drift and gauge potentials—exhibit Brownian motion, giving rise to coupled Fokker–Planck and Hamilton–Jacobi structures that combine to yield the Schrödinger equation. Furthermore, the concept of entropic time is introduced as an intrinsic and directional ordering of inference steps, thereby providing a natural origin for the arrow of time [9]. Consequently, Caticha’s formulation is fundamentally epistemic: the dynamics describe sequential updates of probability distributions rather than ontic particle trajectories. In contrast, the present UED formulation extends this inferential framework into a unified, information-geometric theory. By maximizing entropy subject to diffusion, drift, and gauge constraints, UED yields the generalized field equation expressed in Eq.17, which encompasses, as limiting cases, the harmonic-oscillator, Maxwell, heat, Schrödinger, Klein–Gordon, and gravitational-wave equations. This extension promotes Caticha’s informational manifold $H_{ab}$ to a fully covariant spacetime metric $g_{\mu\nu}$, interprets gravitational waves as oscillations of the spacetime information metric, and establishes a geometric correspondence between the curvatures of configuration and momentum spaces.

In this way, the presented work demonstrates that the entropic dynamics (ED) framework provides a comprehensive, inference-based foundation for unifying the mathematical and physical structures of classical, quantum, relativistic, thermodynamic, and gravitational phenomena under a single geometric and informational principle. By grounding the laws of motion, field evolution, and heat transport in the maximization of entropy, ED transcends the traditional divide between deterministic and probabilistic dynamics [35]. Physical evolution is interpreted as an inference process occurring on a configuration manifold, where uncertainty—encoded through probability distributions—is guided by constraints reflecting conservation laws, geometric curvature, and gauge covariance. The entropy-maximization procedure outlined in Section 2 leads to a generalized field equation that encompasses a broad class of dynamical behaviors. This equation unifies the Fokker–Planck and Hamilton–Jacobi structures, embedding diffusion, drift, and wave propagation as complementary manifestations of information flow on a curved manifold characterized by the metric $H_{ab}$. One of the central outcomes of this study is the recognition that ED establishes an intrinsic duality between stochastic and deterministic evolution. This duality manifests through the interplay of entropy gradients (which govern diffusion) and phase gradients (which govern coherent propagation). When specialized to different dynamical variables, the unified entropic dynamics equation reproduces well-known laws of physics as particular realizations of this dual principle. For instance, in Section 3, the scalar case yields the simple harmonic oscillator, revealing that mechanical oscillations correspond to entropy-preserving motion constrained by information curvature. The entropic mass term $\mathcal{M}=-\omega^{2}$ serves as a restoring parameter, linking information geometry to the potential curvature of classical mechanics. This case shows that Newton’s second law emerges naturally as a statistical limit of entropy-driven inference. The results presented for the case of three-dimensional vector variables reveal the versatility of the ED framework across multiple physical regimes. When applied to electromagnetic fields, the generalized equation reproduces the free-space Maxwell wave equation, illustrating that electromagnetic propagation can be understood as the optimal transmission of information through spacetime. The same structure, under different constraints, yields the Schrödinger equation, showing that quantum mechanics is not a separate postulate but a special limit of inference constrained by energy conservation and probability flow [36]. The addition of the thermodynamic case in Section 4.2 further strengthens this unifying picture. By identifying the field variable with temperature and reinterpreting the entropic mass term as the generator of temporal evolution, the generalized equation reduces to the classical heat equation. In this context, the information metric $H^{ij}$ is directly mapped onto the physical conductivity tensor $k^{ij}$, connecting the geometry of information with the anisotropic transport of heat. The curvature of configuration space thus governs not only oscillatory and wave-like behaviors but also dissipative processes such as diffusion and thermal relaxation. This result highlights that entropy maximization can describe both equilibrium-seeking and energy-preserving processes under a common inferential framework.

The inclusion of thermodynamics within the ED formalism underscores that diffusion and heat flow are natural extensions of information dynamics rather than phenomenological add-ons [37]. The identification of $H^{ij}$ with the conductivity tensor shows that the geometry of the configuration space determines the rate and direction of the entropy transport. In the isotropic limit, this correspondence reduces to the familiar form of Fourier’s law, but its entropic interpretation reveals a deeper structure: thermal diffusion emerges as an information-equilibrating process, where temperature gradients represent spatial variations in informational density. This connection suggests that heat flow and probability diffusion are two aspects of the same entropic updating mechanism that governs all physical evolution—from microscopic quantum coherence to macroscopic thermodynamic equilibration. When the dynamical variable is promoted to a four-vector $h_{\mu}=x_{\mu}$, the framework yields the relativistic Klein–Gordon equation, showing that the Lorentz-covariant structure of spacetime and relativistic wave dynamics emerge from the same entropic principles [38]. Identification $M=-m^{2}c^{2}/\hbar^{2}$ connects the information-theoretic curvature of the configuration space directly to the physical mass, establishing a statistical origin for energy–momentum invariants. Together, these cases—the scalar oscillator, electromagnetic and quantum fields, thermal diffusion, and relativistic waves—demonstrate that all fundamental dynamical laws arise from entropy maximization constrained by geometric and physical symmetries [39].

In particular, thermodynamic diffusion appears as the dissipative limit of this duality—one where curvature drives entropy toward equilibrium rather than coherence. When the formalism is extended to the relativistic domain, as in Section 5, the same inference structure naturally incorporates spacetime geometry. By identifying the dynamical variable $h_{\mu}$ with spacetime coordinates $x_{\mu}$, the generalized ED equation reduces in flat spacetime to the Klein–Gordon equation, demonstrating that relativistic quantum mechanics can be derived from entropic inference rather than introduced axiomatically. Upon promoting the framework to curved spacetime through the minimal-coupling substitution $H^{ab}\!\to g^{\mu\nu}$ and $\partial_{a}\!\to\nabla_{\mu}$, and the $\Psi$ in the generalized ED equation is interpreted as a perturbation of the spacetime metric itself. Thus the $\Psi\rightarrow h_{\mu\nu}$, and the corresponding ED equation in curved spacetime, reduce to the general wave equation for metric perturbations in vacuum. Under the Lorenz gauge condition $\nabla^{\mu}\bar{h}_{\mu\nu}=0$ and in the transverse–traceless (TT) gauge, the physically measurable components $h^{\text{TT}}_{ij}$ satisfies the standard gravitational-wave equation. This connection shows that the curved ED equation provides the same geometric wave operator that governs spacetime perturbations, with $\Psi$ representing an informational field propagating through the curvature of spacetime. Thus, gravitational waves can be interpreted as macroscopic manifestations of entropic inference: oscillatory modes of the spacetime information metric that transmit probabilistic structure rather than energy density alone. The unification of the gravitational-wave equation with the ED formalism underscores a deeper physical insight: the propagation of spacetime curvature follows the same informational geometry that governs the evolution of probability amplitudes in quantum theory. Both are consequences of the covariant operator $\square_{g}$, which encodes the curvature of the manifold and defines how information—whether in the form of entropy or metric perturbation—flows across spacetime. In this sense, general relativity and quantum field dynamics appear as complementary facets of a single information-geometric framework. The ED approach thereby provides a statistical underpinning for gravitational phenomena and a natural bridge between probabilistic inference and geometric curvature. From a broader perspective, the ED framework reveals that the relationship,

captures the curvature duality linking the configuration space and its conjugate momentum space [13, 14, 40]. Here, it $k$ represents the intrinsic information curvature, or wave number, that determines both oscillatory and diffusive dynamics. The equation 56 can be easily proved by taking $\Psi$ as a plane wave solution of the form $\Psi(h)=A\,e^{\,ip_{b}h^{b}}$ with complex conjugate as $\Psi^{*}(h)=A^{*}\,e^{\,-ip_{a}h^{a}}$ for equation 17. Furthermore, the following relation can be obtained by setting the potential term $A_{a}=0$ and multiplying $\Psi^{*}$ on the left side of equation 17.

Where $H^{ab}$ is the information metric on the configuration manifold. Assuming a locally flat configuration space ($\sqrt{H}=1$), Equation (59) reduces to

The integration of $\Psi^{*}\Psi$ over the dynamical variable leads to its value of one, since $\Psi$ is a normalized function. From this, we obtain the fundamental relation given in equation 58.

The metric $H^{ab}$ defines the geometry of information in configuration space, while $p_{a}$ represents its conjugate momentum, encoding propagation through the dual momentum manifold. Their contraction $p_{a}H^{ab}p_{b}$ forms an invariant scalar analogous to the energy–momentum norm in relativistic theory. Thus, the quantity $k^{2}$ characterizes the intrinsic curvature of the informational manifold, establishing a duality between the configuration-space curvature and the dispersion properties of the system. From a geometric standpoint, the relationship $p_{a}H^{ab}p_{b}=k^{2}$ captures this curvature duality. Spatial curvature dictates localization, while conjugate curvature governs dispersion and energy propagation. This insight provides a unified language for interpreting mechanics, wave theory, and thermodynamics as different manifestations of information geometry. In this dual description, spatial curvature dictates localization, while conjugate curvature governs dispersion and energy propagation. The same metric tensor that defines curvature in configuration space also dictates heat conduction, quantum diffusion, and gravitational-wave propagation, providing a unified geometric language for all these processes. Consequently, mechanical stability, quantum coherence, thermodynamic equilibration, and spacetime curvature emerge as distinct expressions of a single inferential law: the maximization of entropy subject to geometric and physical constraints.

In summary, this work shows that the entropic dynamics framework can account for phenomena spanning from microscopic quantum systems to macroscopic gravitational waves. By embedding inference within geometry, ED reveals that the flow of information, not the imposition of quantization, underlies the evolution of physical systems. This perspective resonates strongly with John Archibald Wheeler’s “It from Bit” paradigm, which posits that every physical entity—every “it”—emerges from fundamental acts of information acquisition [41]. In this context, the same differential operator that governs the propagation of probability in configuration space also governs the curvature oscillations of spacetime, illustrating that energy, probability, and entropy are intertwined through the geometry of information. In this unified view, gravitational waves and quantum waves are both manifestations of information propagating through curved spacetime, consistent with Wheeler’s vision of a participatory universe where physical reality arises from informational interactions. Furthermore, the replacement of quantization postulates with rules of inference redefines the foundations of physics in epistemic terms, where laws of motion describe optimal updates to probabilistic knowledge about the system. In other words, this view dissolves the traditional separation between matter, fields, and spacetime, interpreting them instead as emergent realizations of information flow on curved statistical manifolds. Future work may extend this formulation toward nonlinear and gravitational systems, exploring how the same entropic principles can lead naturally to Dirac- and Einstein-like field equations. In essence, entropic dynamics provides a self-consistent information-theoretic foundation for physics—an explicit realization of Wheeler’s insight that information, probability, and geometry together give rise to the physical world.

Beyond reproducing known equations, the UED framework suggests several directions where departures from standard dynamics, and hence possible tests, may arise. First, allowing the information metric $H_{ab}$ to acquire curvature or state dependence beyond the flat, constant choices used here would generically induce corrections to the dispersion relations of electromagnetic, quantum, and gravitational waves, leading to scale-dependent deviations from the standard Maxwell, Schrödinger, and gravitational-wave equations. For the latter case, recent analyses of quantum-gravity phenomenology — including arguments about the interplay among consistency, completeness, and empirically accessible corrections to low-energy physics — suggest that even modest departures from standard dispersion relations or uncertainty structures may offer meaningful probes of underlying microscopic geometry [42]. Within this context, the UED framework provides a structurally natural arena in which such deviations can emerge: modifications to the information metric $H_{ab}$ or to the entropic mass functional $M[\rho,\phi]$ can be interpreted as encoding residual quantum-gravitational effects at the level of effective dynamics. Although the present work does not attempt to develop a complete quantum-gravity model, the formal parallels highlight a potential route by which UED may interface with emerging phenomenological constraints on quantum-gravity-induced corrections, thereby broadening the scope of future empirical or conceptual developments. Second, in the fields of thermodynamics, identification $H_{ij}\leftrightarrow k_{ij}/(\rho c)$ implies that engineered anisotropies or nonlinearities in the conductivity tensor should have a direct information-geometric interpretation; systematic measurements of heat transport in metamaterials or heterogeneous media could therefore be used to constrain or calibrate the underlying entropic metric. Finally, by exploring nonlinear choices of the entropic mass term $M[\rho,\phi]$, one can generate controlled modifications to the ED reconstruction of quantum mechanics (e.g. nonlinear or dissipative corrections to the Schrödinger and Klein-Gordon equations), which could be confronted with precision tests in cold-atom, quantum optic, or interferometric settings. In all these cases, the unified structure of Eq. 17 provides a concrete template to identify where UED predictions diverge from those of existing ED formulations and standard field theory, thus sharpening the empirical content of the approach.

In this study, a unified entropic dynamics (UED) framework has been developed to provide a unified information-geometric foundation for classical, quantum, relativistic, thermodynamic, and gravitational systems. By maximizing entropy under physically motivated constraints, a universal field equation was derived that governs the evolution of dynamical variables on an informational manifold characterized by the metric $H_{ab}$. This formulation merges the Fokker-Planck and Hamilton-Jacobi structures into a single unified entropic dynamics equation, capturing both stochastic and deterministic dynamics as complementary aspects of information flow. When specialized to specific variables, the framework successfully reproduces well-established physical laws: the simple harmonic oscillator for scalar variables, Maxwell’s electromagnetic wave equation, the classical heat equation, and the Schrödinger equation for spatial vector variables, and the Klein–Gordon and gravitational-wave equations for spacetime four-vectors. These results demonstrate that mechanical, quantum, thermodynamic, and relativistic processes all arise from the same inferential principle—the maximization of entropy constrained by geometry and physical symmetries. A central advancement of this work lies in extending the ED formalism to curved spacetime, where the generalized covariant equation naturally yields the gravitational-wave equation in the weak-field limit. In this interpretation, gravitational waves emerge as oscillations of the spacetime information metric—macroscopic manifestations of the same entropic processes that govern microscopic dynamics. The relation $p_{a}H^{ab}p_{b}=k^{2}$ further encapsulates the duality between configuration-space curvature and its conjugate momentum space, revealing a universal link between localization, dispersion, and energy propagation.

Overall, the results indicate that energy, probability, and entropy are deeply intertwined through the geometry of information. The formulation of the ED framework presented herein expresses probability propagation and the oscillations of spacetime curvature within a single covariant d’Alembertian operator, suggesting that classical, quantum, and gravitational phenomena are all expressions of a single geometric law of inference. Future research may extend this framework to nonlinear and self-gravitating systems, explore the emergence of Einstein and Dirac equations from entropic inference, and investigate whether spacetime itself can be understood as an emergent statistical structure shaped by information geometry.

This work is purely theoretical and did not involve the generation or analysis of any datasets. All results can be reproduced from the equations and derivations provided in the manuscript.

The authors declare that there are no commercial or financial relationships that could be construed as potential conflicts of interest in the research, authorship, and publication of this work.

The authors express their gratitude to the institutions that supported this research. S. N. acknowledges the facilities and academic environment provided by Shaker High School, which enabled the foundational development of this work. M. S. extends appreciation to Bellarmine University for research guidance and discussions on the theoretical formulation. M. S. A. acknowledges support from Queen Mary University of London, which contributed to the completion of this work. D. H. A. acknowledges the support and collaborative environment of Khalifa University, which contributed to the completion of this study.

The authors have diligently addressed ethical concerns, including informed consent, plagiarism, data fabrication, misconduct, falsification, double publication, redundancy, and related issues.

This research has been partially funded through Khalifa University’s Research Innovation Grant (RIG) program under the project ID RIG-2024-002.