Formulation of the Entropic Equivalence Principle (EEP) in the Theory of Entropicity (ToE) Analogous to Einstein's Equivalence Principle of General Relativity - A Foundational Law in Obidi's Theory of Entropicity (ToE)

Abstract

The Theory of Entropicity (ToE) proposes that the fundamental substrate of physical reality is not spacetime, matter, or quantum fields, but a single entropic field $S(x)$ whose curvature generates all observable structure and dynamics. Within this monistic framework, every physical process requires an entropic expenditure, formalized as the Entropic Accounting Principle (EAP). This paper introduces and develops the Entropic Equivalence Principle (EEP), a foundational law that emerges naturally from the EAP and the intrinsic geometry of the entropic field. The EEP states that any two physical processes that produce equivalent reconfigurations of the entropic field must incur equivalent entropic cost, regardless of their classical, relativistic, or quantum description. This principle generalizes and deepens Einstein’s Equivalence Principle by grounding inertia, gravitation, quantum transitions, thermodynamic irreversibility, and informational processes in a single entropic substrate. The paper presents the formal structure of the EEP, derives its consequences for physical law, and demonstrates its role as the unifying mechanism that binds together the emergent phenomena of spacetime, matter, and quantum behaviour within ToE.

1. Introduction

The Theory of Entropicity (ToE) asserts that the universe is fundamentally an entropic system whose dynamics are governed by the curvature and evolution of a single scalar field $S(x)$. In this framework, spacetime geometry, matter distributions, quantum transitions, and thermodynamic processes are emergent manifestations of entropic curvature. The entropic field is the primary ontological entity, and all physical phenomena arise from its structure and reconfiguration.

A central feature of ToE is the Entropic Accounting Principle (EAP), which states that every physical process requires an entropic expenditure. No event, interaction, or transformation occurs without paying an entropic cost. This principle is not an analogy but a literal structural law: the universe maintains an entropic ledger, and all processes must balance their entropic accounts.

From this principle arises the Entropic Equivalence Principle (EEP), which asserts that any two physical processes that produce equivalent changes in the entropic configuration of reality must incur equivalent entropic cost. This principle is deeper and more general than Einstein’s Equivalence Principle, which equates inertial and gravitational mass. The EEP equates the entropic cost of all processes that produce equivalent entropic reconfiguration, regardless of their physical domain. It is the universal law that unifies thermodynamics, relativity, quantum mechanics, and information theory within ToE.

The purpose of this paper is to articulate the EEP in full technical form, to derive its consequences, and to demonstrate its role as the foundational unifying principle of the Theory of Entropicity.

2. The Entropic Field and the Accounting Structure of Reality

In ToE, the entropic field $S(x)$ is defined over a differentiable manifold and possesses curvature determined by its spatial and temporal gradients. The curvature of this field encodes the structure of spacetime, the presence of matter, and the dynamics of physical processes. The Obidi Action governs the evolution of the entropic field and integrates information‑geometric structures such as the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov connections.

The Entropic Accounting Principle (EAP) states that any change in the entropic field requires a quantifiable entropic expenditure. This expenditure is measured by the entropic divergence

$$D(x)=S(x)\ln \left(\frac{S(x)}{S_0(x)}\right)-S(x)+S_0(x),$$

where $S_0(x)$ is the local equilibrium configuration. The divergence quantifies the entropic cost of reconfiguring the field from $S_0(x)$ to $S(x)$. The Obidi Curvature Invariant (OCI), equal to $\ln 2$, represents the smallest distinguishable curvature fold in the entropic field and sets the minimal unit of entropic cost.

The EAP implies that the universe operates as an entropic accounting mechanism. Every physical process corresponds to a reconfiguration of the entropic field, and every reconfiguration incurs a cost measured in units of OCI. This accounting structure is universal and applies equally to gravitational, quantum, thermodynamic, and informational processes.

3. Formal Statement of the Entropic Equivalence Principle (EEP)

The Entropic Equivalence Principle (EEP) states that any two physical processes that produce equivalent reconfigurations of the entropic field must incur equivalent entropic cost. Formally, let ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ be two physical processes that transform the entropic field from an initial configuration $S_i(x)$ to final configurations $S_{f,1}(x)$ and $S_{f,2}(x)$, respectively. If the resulting entropic divergences satisfy

$$D_1(x)=D_2(x)$$

for all points $x$ in the domain of the field, then the entropic cost of the two processes is identical. The EEP therefore asserts that the entropic cost is determined solely by the entropic reconfiguration, not by the physical mechanism through which the reconfiguration is achieved.

This principle implies that gravitational curvature, inertial resistance, quantum transitions, thermodynamic irreversibility, and informational operations are all manifestations of the same entropic expenditure when they produce equivalent changes in the entropic field. The EEP therefore unifies the domains of physics by identifying entropic cost as the universal currency of physical transformation.

4. Consequences of the EEP for Physical Law

The EEP has profound implications for the structure of physical law. In relativity, time dilation and mass increase arise from changes in entropic curvature associated with motion. In quantum mechanics, discrete transitions correspond to the crossing of entropic curvature thresholds determined by the OCI. In thermodynamics, irreversibility arises from the entropic cost of reconfiguring the field away from equilibrium. In information theory, measurement and observation incur entropic cost because they require reconfiguration of the entropic field.

The EEP implies that these phenomena are not separate or domain‑specific but are unified by their entropic cost. A gravitational redshift and a quantum transition may appear different in classical or quantum descriptions, but if they produce equivalent entropic reconfiguration, they incur the same entropic cost. This equivalence is not metaphorical but literal: the entropic field does not distinguish between physical domains; it only registers entropic expenditure.

The EEP therefore generalizes Einstein’s Equivalence Principle. Whereas Einstein equated inertial and gravitational mass, the EEP equates the entropic cost of all processes that produce equivalent entropic reconfiguration. This generalization extends the equivalence principle beyond gravity and inertia to encompass all physical processes.

5. The EEP as the Unifying Principle of ToE

The EEP is the foundational law that unifies the Theory of Entropicity. It ensures that all physical processes, regardless of their classical or quantum description, are governed by the same entropic accounting structure. It explains why the laws of physics exhibit deep structural unity and why phenomena that appear distinct at the classical level share common entropic origins.

The EEP also provides the conceptual and mathematical basis for deriving emergent field equations, including the Einstein Field Equations, from the Obidi Action. It explains the universality of physical constants, the discreteness of quantum transitions, and the invariance of physical laws across reference frames. It is the principle that binds together the emergent phenomena of spacetime, matter, and quantum behaviour within a single entropic substrate.

6. Conclusion

The Entropic Equivalence Principle (EEP) is a natural and necessary consequence of the Entropic Accounting Principle and the monistic entropic ontology of the Theory of Entropicity. It asserts that any two physical processes that produce equivalent reconfigurations of the entropic field must incur equivalent entropic cost. This principle generalizes Einstein’s Equivalence Principle and provides a unified foundation for the laws of physics. It reveals the universe as an entropic accounting mechanism in which existence, motion, interaction, and observation are all governed by the same entropic currency. The EEP is therefore the central unifying law of ToE and the key to understanding the deep structural unity of physical reality.

References

1) The Entropic Equivalence Principle (EEP) in the Theory of Entropicity (ToE): Einstein's Equivalence Principle of General Relativity Finds New Expression in Obidi's Theory of Entropicity (ToE):

https://theoryofentropicity.blogspot.com/2026/01/the-entropic-equivalence-principle-eep.html

2) The Beauty of Obidi's Theory of Entropicity (ToE) - The Universe as an Accounting Mechanism:

https://theoryofentropicity.blogspot.com/2026/01/the-beauty-of-obidis-theory-of.html

3) Achievements of the Theory of Entropicity (ToE): From Formulation to Application:

https://theoryofentropicity.blogspot.com/2026/01/achievements-of-theory-of-entropicity.html

https://theoryofentropicity.blogspot.com/2026/01/the-entropic-equivalence-principle-eep_23.html

 

https://theoryofentropicity.blogspot.com/2026/01/formulation-of-entropic-equivalence.html

https://medium.com/@jonimisiobidi/formulation-of-the-entropic-equivalence-principle-eep-in-the-theory-of-entropicity-toe-a948fe4ed732