Variational structure of the Obidi Action

The starting point of the ToE derivation is the Obidi Action, a scalar functional of the entropic field and the metric. In its prototypical scalar-field form, the action is written as

\[ \mathcal{A}_\text{Obidi}[S] = \int d^4\lambda \, \sqrt{-g} \left[ \frac{1}{2} (\partial_\mu S)(\partial^\mu S) - V(S) + J(\lambda) S \right], \]

where \( g \) is the determinant of the metric \( g_{\mu\nu} \), \( (\partial_\mu S)(\partial^\mu S) \) is the kinetic term of the entropic field, \( V(S) \) is an effective entropic potential, and \( J(\lambda) \) represents source terms encoding couplings to matter or other degrees of freedom. The kinetic term \( (\nabla S)^2 \) endows the entropic field with genuine dynamical character, analogous to a scalar field in conventional field theory, while the potential and source terms control its equilibrium structure and interactions.

Variation of \( \mathcal{A}_\text{Obidi} \) with respect to the entropic field \( S \) yields the Master Entropic Equation (MEE), a second-order differential equation governing the evolution of \( S(x) \). Variation with respect to the metric \( g_{\mu\nu} \) produces an entropic stress–energy tensor \( T^{(S)}_{\mu\nu} \), which enters the effective gravitational field equations. In this way, the entropic field contributes directly to the curvature of the manifold, and its gradients become the source of gravitational phenomena.

A key structural feature of ToE is the introduction of an entropy-weighted deformation of the metric. The entropic field modifies the effective geometry via an exponential weighting factor,

\[ g^{(S)}_{\mu\nu} = e^{S/k_B} \, g_{\mu\nu}, \]

where \( k_B \) is Boltzmann’s constant. This deformation expresses the idea that the local entropic configuration rescales the effective metric, thereby coupling entropy directly to geometry. The factor \( e^{S/k_B} \) can be interpreted as an entropic weight that modifies distances and intervals in the effective spacetime, making the geometry explicitly dependent on the entropic field.

Entropy-weighted geometry and entropic curvature

The entropic deformation of the metric extends naturally to information-geometric structures such as the Fisher–Rao metric and the Fubini–Study metric, which arise in statistical and quantum contexts. In ToE, these metrics are generalized to entropy-weighted metrics of the form

\[ g^{(S)}_{ij} = e^{S/k_B} \, g^{(\text{FR})}_{ij}, \]

where \( g^{(\text{FR})}_{ij} \) denotes the Fisher–Rao metric on a statistical manifold. The exponential factor introduces an explicit dependence on the entropic field, turning the information manifold into an entropic manifold whose curvature reflects both statistical structure and entropic configuration. This construction provides a bridge between information geometry and spacetime geometry, with entropy acting as the unifying scalar field.

To incorporate irreversibility and directional behavior, ToE employs generalized connections such as the Amari–Čencov \(\alpha\)-connections, which modify the Levi-Civita connection by adding torsion-like terms that encode entropic asymmetry. A representative form is

\[ \Gamma^\lambda_{\mu\nu} = \{^\lambda_{\mu\nu}\} + \frac{\alpha}{2} T^\lambda_{\mu\nu}, \]

where \( \{^\lambda_{\mu\nu}\} \) are the Christoffel symbols of the underlying metric, \( T^\lambda_{\mu\nu} \) captures entropic asymmetry or non-metricity, and \( \alpha \) is a parameter controlling the degree of irreversibility. This modification leads to an entropic connection whose associated curvature tensor depends explicitly on \( \nabla S \), thereby generating an entropic Ricci curvature \( R_{\mu\nu}(S) \).

The resulting effective gravitational field equations take a form analogous to Einstein’s equations but with an entropic source term:

\[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \eta \, T^{(S)}_{\mu\nu}, \]

where \( R_{\mu\nu} \) is the Ricci tensor, \( R \) is the Ricci scalar, \( \eta \) is a coupling constant, and \( T^{(S)}_{\mu\nu} \) is the entropy-stress tensor derived from the Obidi Action. In this formulation, entropy gradients and the dynamics of \( S(x) \) are directly responsible for the curvature of the effective spacetime, and gravitation is interpreted as the geometric manifestation of entropic structure.

Entropic geodesics and the emergence of gravitational motion

The notion of entropic geodesics is central to the ToE interpretation of gravitational motion. In the entropic manifold defined by \( S(x) \) and the entropy-weighted metric, preferred trajectories are those that extremize an entropic action, typically corresponding to paths that maximize entropy flow or minimize entropic disruption. These trajectories are the analogues of geodesics in general relativity, but their origin lies in the variational structure of the entropic field rather than in a purely geometric postulate.

Particles and systems are thus described as following entropic geodesics, which, when projected into the effective spacetime geometry, coincide with null or timelike geodesics of the entropy-weighted metric. The entropy gradients \( \partial_\mu S \) act as effective gravitational potentials, guiding motion in a way that reproduces the familiar phenomenology of gravitational attraction. In regions where \( S \) is uniform, the entropic manifold is effectively flat, and no gravitational effects arise. Where \( \nabla S \neq 0 \), the entropic curvature induces trajectories that correspond to gravitational free fall.

In the appropriate limits, the entropic geodesic equations reduce to Newtonian gravity, with the entropic potential playing the role of the Newtonian gravitational potential. Higher-order corrections, incorporating effects analogous to Unruh and Hawking radiation, yield deviations consistent with general relativity, such as the precession of planetary orbits and the deflection of light by massive bodies. In this way, ToE recovers the empirical successes of general relativity while providing an underlying entropic mechanism.

Phenomenological predictions and correspondence with general relativity

The entropic derivation of gravity in ToE leads to concrete predictions that can be compared with the standard results of general relativity. Key gravitational effects are obtained by analyzing the motion of test bodies and light rays in the entropy-weighted geometry and by solving the entropic field equations in appropriate regimes. The following table summarizes representative effects and their derivation within the ToE framework, together with their correspondence to general relativity.

These correspondences demonstrate that the entropic field formulation of ToE is not merely qualitative but quantitatively compatible with established gravitational tests, while offering a deeper explanatory basis. Gravitation is reinterpreted as the emergent manifestation of entropy gradients and entropic curvature, with the Obidi Action and the Master Entropic Equation providing the unifying mathematical framework. In this way, the Theory of Entropicity derives gravity from entropy gradients in a manner that is both conceptually coherent and empirically anchored.