Understanding Reality in a new way by treating entropy as a field, not just a statistical or probabilistic measure of disorder

The central question is whether the curvature of an information‑geometric manifold can give rise to the physical curvature of spacetime. The answer, developed rigorously within the Theory of Entropicity (ToE), is that information curvature does not transform into physical curvature; rather, physical curvature emerges from information curvature once entropic dynamics are imposed. This section presents the conceptual and mathematical structure of that emergence. Any statistical manifold endowed with distinguishable states possesses a natural metric: the Fisher information metric The crucial result is that the evolution of the information metric cannot be arbitrary. The dynamics must satisfy internal consistency conditions: the evolution must preserve the probabilistic structure, the metric compatibility, and the entropic updating rules. When these consistency conditions are imposed, the resulting 4‑geometry satisfies the vacuum Einstein field equations: This is the Obidi Action.

The present Letter — Letter ID in the Theory of Entropicity (ToE) Living Review Letters Series — introduces and fully formalizes the Entropic Seesaw Model (ESSM) as a self-contained, mathematically complete entropic theory of quantum entanglement. ESSM is developed within the broader framework of the Theory of Entropicity, an entropy-first program that posits the entropic field as the ontological ground of physical reality. The model is constructed in two conceptually distinct but mathematically unified stages. First, a formation stage, in which two previously independent entropic sectors — each described by a local entropic field configuration on its own manifold — undergo a local, finite-time, topological merger into a single shared entropic manifold. This merger is not an instantaneous kinematic fact but a genuine dynamical process requiring finite entropic resources and finite time, governed by a formation drive equation with a well-defined threshold-crossing time.

The Theory of Entropicity (ToE) establishes entropy not as a statistical byproduct of disorder but as the fundamental field and causal substrate of physical reality. Central to this formulation is the Obidi Action, a variational principle. By integrating the Fisher–Rao and Fubini–Study metrics through the Amari–Čencov alpha-connection formalism, ToE provides a rigorous information-geometric foundation for entropy-driven dynamics. The Obidi Action comprises the Local and Spectral Obidi Actions. The passage above is unmistakably about the Theory of Entropicity (ToE), and in fact it captures several of the theory’s defining conceptual moves. Let us break down why each component is directly tied to ToE and not to any pre‑existing framework in physics or information geometry. No existing theory unifies these into a single entropic action. This is one of the most distinctive mathematical signatures of ToE.

Some readers have asked whether the Theory of Entropicity (ToE) is simply a restatement of Maxwell’s electromagnetism. This is understandable — both theories involve a field, both involve waves, and both involve the constant \(c\). But the resemblance ends there. Maxwell’s theory describes electric and magnetic fields in spacetime. ToE describes the Entropic Field, which is the foundation of spacetime. Maxwell’s \(c\) is the speed of electromagnetic waves. ToE’s \(c\) is the maximum rate at which entropy can rearrange in the universe. Maxwell’s field is one of many classical fields. ToE’s field is the single, fundamental field from which all others emerge. So ToE is not replacing Maxwell — it is explaining why Maxwell’s equations work at all, and why the speed of light is universal. Maxwell discovered the wave; ToE explains the medium.

ToE: 1. From Statistical Distance to Physical Curvature To understand why the Obidi Curvature Invariant ln 2 is not a decorative reinterpretation of existing mathematics, one must carefully distinguish between formal distance and physical curvature. Before the Theory of Entropicity (ToE), measures such as Kullback–Leibler divergence, Fisher–Rao distance, and quantum relative entropy were understood as tools for comparing probability distributions or quantum states. They quantified distinguishability, but only at the level of description. What ToE does—quietly but decisively—is reinterpret these structures as measures of deformation of a physical field, namely the entropic field S(x). This shift is not cosmetic. It transforms relative entropy from a bookkeeping device into a curvature functional. In standard information theory, when one writes a relative entropy of the form D(p‖q) = ∫ p(x) ln[p(x)/q(x)] dx, one is comparing two probability distributions over an abstract sample space.

2.3 ln 2 in Landauer’s Principle Landauer’s principle famously states that erasing one bit of information dissipates an energy of at least kB T ln 2. This result is often described as “deep,” yet ln 2 is still treated as the entropy of a bit—an input, not something derived from field dynamics or geometry. Landauer’s principle tells us the cost of erasing a distinction, but not why the distinction exists in the first place. 2.4 ln 2 in Black-Hole Physics and Holography In black-hole thermodynamics, entropy is proportional to horizon area, and ln 2 frequently appears when entropy is expressed per bit of area. Holographic theories speak of “pixels” on a boundary, each storing one bit. Once again, ln 2 appears—but as a scaling factor. It sets the size of informational units, not the nature of geometry itself. ToE Insight: These interpretations treat ln 2 as a passive quantity. ToE treats it as an active geometric invariant that governs the formation of physical distinctions.

Kolmogorov’s Axioms: Formulated the rigorous mathematical foundation for probability theory, defining probability as an axiomatic system over σ-algebras, independent of thermodynamic or cosmological context. Information-Theoretic Progression: Shannon entropy, Bekenstein-Hawking gravitational thermodynamics, and Jacobson's and Verlinde’s work on emergent spacetime extended these principles into physics. Obidi Action: Introduced as the central variational principle in the Theory of Entropicity, unifying discrete algorithmic measures (Kolmogorov complexity) with continuous entropic field dynamics. 2. Core Concepts of KOL KOL serves as a bridge between classical information-theoretic quantities and entropic physics:Obidi Action as Limiting Principle:Every standard information-theoretic quantity (e.g., Shannon entropy, Kolmogorov complexity K(x), Kolmogorov–Sinai entropy, Solomonoff–Levin probability measures) is derivable as a limiting case of the Obidi Action.

The Theory of Entropicity (ToE) enters this landscape with a foundational claim: entropy is not a statistical or probabilistic or book-keeping summary of underlying mechanical degrees of freedom but a real, fundamental, dynamical field — the primary ontological entity from which all physical structure emerges. The entropic field S(x), defined on an entropic manifold M_S, generates gravitational geometry, quantum behavior, and thermodynamic law as emergent consequences of its dynamics, governed by the Obidi Action [1, 3, 6]. The ToE program has been developed across a series of Letters and papers: Letter I [1] establishes the ontological primacy of entropy; Letter IA [2] identifies the deep correspondence between the ToE framework and John Haller's action-as-entropy formulation [19]; Letter IB [3] formalizes the Haller-Obidi Action and Lagrangian; and Letter IC [4] presents the Alemoh-Obidi Correspondence, a monograph-scale examination of the mathematical and conceptual foundations.

The Kolmogorov–Obidi Lineage (KOL) historical and structural summary in Subsection 19.7 traces the intellectual genealogy from Kolmogorov's foundational axioms through Shannon, Bekenstein, Hawking, Jacobson, Verlinde, Padmanabhan, and Bianconi to the Obidi Action, establishing the Theory of Entropicity as the natural culmination of a century-long convergence between probability, information, and gravitation. Subsection 19.8 presents the rigorous derivation of the Obidi Curvature Invariant (OCI), proved by seven independent methods: the geodesic maximum on the Binary Entropic Manifold, the regularized relative entropy, the Landauer–Obidi derivation via the Entropic Description Theorem, the Holevo bound, quantum hypothesis testing via the Chernoff–Stein exponent, the channel capacity of the fundamental binary entropic channel, and the direct derivation from the Minimum Difference Principle (the open methodology). These seven derivations establish that OCI = ln 2 is a geometric constant.