Concept of ln 2 as a Curvature Invariant of Classical and Quantum Distinguishability in the Theory of Entropicity (ToE)

1. The Obidi Curvature Invariant and the Role of ln 2 in the Theory of Entropicity (ToE)

Within the Theory of Entropicity (ToE), the quantity \(\ln 2\) is elevated from its conventional role as a statistical constant in classical thermodynamics to a fundamental curvature invariant of the entropic field. This invariant, formally designated as the Obidi Curvature Invariant (OCI), was introduced by John Onimisi Obidi as part of a broader program to reinterpret entropy as a physical, ontic field endowed with geometric structure. The OCI serves as a minimal, irreducible unit of curvature divergence and is used to quantify the smallest physically meaningful degree of distinguishability between configurations of the entropic field.

In this formulation, the entropic field \( S(x) \) is treated as a continuous geometric object whose curvature encodes informational and physical structure. The invariant \(\ln 2\) is interpreted as the minimal non-zero curvature quantum that the universe can resolve. Any curvature divergence smaller than \(\ln 2\) is regarded as physically indistinguishable, implying that the entropic field possesses a built‑in resolution threshold. This threshold plays a central role in the emergence of discrete informational units, the behavior of quantum measurement, and the structure of black‑hole entropy.

2. Formal Definition of the Obidi Curvature Invariant

The Obidi Curvature Invariant is defined by the expression \[ \text{OCI} = \ln 2, \] which is treated not as a thermodynamic artifact but as a geometric invariant of the entropic field. In classical statistical mechanics, \(\ln 2\) appears in the context of binary distinguishability and the entropy of a two‑state system. In ToE, this quantity is reinterpreted as the minimal curvature divergence that can be encoded or resolved by the entropic field. It therefore functions as a quantum of distinguishability, analogous to a pixel in a digital image or a minimal geodesic separation in a discretized manifold.

The significance of this invariant is that it establishes a lower bound on the curvature differences that can be physically meaningful. If two configurations of the entropic field differ by a curvature less than \(\ln 2\), the field cannot distinguish them, and they are treated as effectively identical. This principle introduces a natural discretization into the geometry of the entropic field, even though the field itself remains continuous at the level of its mathematical representation.

3. Minimal Distinguishability and Curvature Resolution

The introduction of \(\ln 2\) as a curvature invariant leads to the concept of minimal distinguishability within the entropic field. In the Theory of Entropicity, the entropic field possesses curvature that encodes the informational structure of physical systems. The OCI establishes the smallest curvature divergence that can be resolved by the universe. This implies that the entropic field has a finite resolution, and that physical reality is constructed from curvature differences that exceed this threshold.

This minimal resolution plays a role analogous to the Planck length in quantum gravity or the quantum of action in quantum mechanics. It defines the smallest meaningful unit of geometric deviation and thereby constrains the granularity of physical information. The OCI thus functions as a geometric quantization rule for the entropic field, determining the smallest unit of curvature that can contribute to physical processes.

4. Curvature, Information, and the Ontic Entropic Field

In ToE, entropy is not a statistical abstraction but a physical field with geometric properties. The curvature of this field encodes information, which is interpreted as localized deviations from entropic uniformity. The OCI provides the minimal curvature unit that can represent a distinguishable informational state. This leads to a geometric interpretation of information in which each bit corresponds to a curvature divergence of at least \(\ln 2\).

This geometric interpretation aligns with the broader ToE program, which seeks to derive spacetime, matter, and quantum behavior from the structure of the entropic field. In this context, the OCI serves as a foundational constant that governs the resolution of the entropic geometry and thereby constrains the emergence of physical structure.

5. Landauer’s Principle as a Geometric Constraint

The reinterpretation of \(\ln 2\) as a curvature invariant leads to a geometric reformulation of Landauer’s principle. Traditionally, Landauer’s principle states that erasing one bit of information requires a minimum energy cost of \( k_{\mathrm{B}} T \ln 2 \). In the Theory of Entropicity, this cost is understood as the energy required to flatten a curvature divergence of \(\ln 2\) in the entropic field. Erasing a bit corresponds to eliminating a localized curvature, and the associated energy cost is a direct consequence of the geometric structure of the entropic field.

This geometric interpretation provides a deeper physical justification for Landauer’s principle, linking it to the curvature dynamics of the entropic field rather than treating it as a purely thermodynamic constraint. It also connects the cost of information erasure to the fundamental resolution limit imposed by the OCI.

6. Applications of the Obidi Curvature Invariant

The OCI plays a central role in several areas of the Theory of Entropicity. In quantum measurement, the collapse of a quantum state is interpreted as a transition between curvature configurations that differ by at least \(\ln 2\). This provides a geometric criterion for when two quantum states become distinguishable. In the context of black‑hole entropy, the OCI contributes to the discretization of horizon entropy, suggesting that the entropy of a black hole arises from a finite number of curvature quanta. In the emergence of spacetime geometry, the OCI constrains the minimal curvature fluctuations that can contribute to the formation of geometric structure.

These applications illustrate how the OCI functions as a unifying constant within the Theory of Entropicity, linking information, geometry, and physical dynamics through a single curvature invariant.

7. Historical and Developmental Context

The concept of the Obidi Curvature Invariant was developed extensively by John Onimisi Obidi in 2026, with detailed expositions published across platforms such as Medium, Substack, and ResearchGate. These works form part of a broader effort to construct a comprehensive entropic field theory in which entropy is treated as the fundamental geometric and dynamical substrate of physical reality. The introduction of \(\ln 2\) as a curvature invariant represents a key step in this program, providing a minimal geometric unit that underlies the emergence of information, quantum behavior, and spacetime structure.