What is the Curvature Invariant ln 2 in the Theory of Entropicity (ToE)? From Mathematical Curvature to a Physical Principle and Criterion of Distinguishability in Modern Theoretical Physics 

In the Theory of Entropicity (ToE), the “Curvature Invariant ln 2” is a specific numerical invariant, $$\ln 2$$, that emerges as the first non‑zero minimum of a distinguishability (information‑geometric) potential and is interpreted as an intrinsic curvature scale of the entropic/informational manifold—called the Obidi Curvature Invariant.[1][2]

## Role in the entropic geometry

- The ToE framework defines an information‑geometric potential that measures how distinguishable nearby entropic states are (a kind of “distinguishability curvature” in state space).[2]

- When this potential is analyzed, its first non‑zero minimum occurs at a value equal to $$\ln 2$$, and this value is taken to be a universal curvature invariant of the entropic manifold.[1][2]

## Interpretation as an invariant

- Because it arises as a minimum of a coordinate‑independent potential on the informational manifold, $$\ln 2$$ is treated analogously to a curvature invariant in differential geometry: it characterizes intrinsic geometric structure independent of coordinates or representation.[1][2]

- Within ToE, this Obidi Curvature Invariant $$\ln 2$$ thus sets a natural “quantum” of distinguishability/curvature in the entropic description, playing a role similar to a canonical scalar invariant that helps anchor the geometry of the underlying entropic field.[1][2]

Citations:

[1] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[2] (PDF) Collected Works on the Theory of Entropicity (ToE) Volume I 31 ... https://www.academia.edu/145698037/Collected_Works_on_the_Theory_of_Entropicity_ToE_Volume_I_31_December_2025_V9_S

[3] Curvature Invariants http://kias.dyndns.org/crg/invariants.html

[4] ENTROPIC CURVATURE AND CONVERGENCE TO EQUILIBRIUM https://www.normalesup.org/~mfathi/docs/CW-curvature_revised.pdf

[5] Curvature Invariants for the Alcubierre and Natário Warp Drives https://www.academia.edu/52538550/Curvature_Invariants_for_the_Alcubierre_and_Nat%C3%A1rio_Warp_Drives

[6] The δ(2,2)-Invariant on Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature https://pmc.ncbi.nlm.nih.gov/articles/PMC7516581/

[7] Invariant http://invariant.com

[8] Weighing the curvature invariants http://www.arxiv.org/abs/2502.12242

[9] Lu Wang: Entropy in mean curvature flow https://www.youtube.com/watch?v=JAD-29_mvbU

[10] 1 Introduction https://arxiv.org/html/2502.12242v2