How the Theory of Entropicity (ToE) Recovers and Corrects Einsteinian General Relativity (GR) Curvature at Small Scales: How does the Obidi Curvature Invariant OCI Function in (ToE) Dynamics?

In the dynamics of Obidi's Theory of Entropicity (ToE), the Obidi Curvature Invariant (OCI) $$\ln 2$$ acts as a built‑in “quantum of entropic curvature” that constrains how the entropy field can evolve and how distinguishable configurations can form.[3][9][10]

## Threshold for distinguishable configurations

- OCI arises as the first non‑zero minimum of the distinguishability potential built from a KL‑type functional $$D(\rho_A\Vert\rho_B)$$, evaluated for a binary 2:1 curvature ratio $$\rho_B = 2\rho_A$$, giving a gap of $$\ln 2$$.[9][3]

- Dynamically, this means two configurations of the entropic field only count as physically distinct if their relative curvature surpasses this $$\ln 2$$ threshold; smaller deformations are dynamically treated as indistinguishable fluctuations.[9][10]

## Constraint in the Obidi / Spectral Obidi Action

- In the Spectral Obidi Action, the field dynamics contain a curvature scalar $$R[g]$$, a kinetic term for $$\nabla_\mu S$$, and the distinguishability potential $$D(S,S_0)$$; the OCI value $$\ln 2$$ fixes the first non‑trivial minimum of that potential.[3][10]

- This effectively quantizes curvature response: entropic curvature cannot relax continuously through arbitrarily small distinguishable steps, but does so in increments constrained by the $$\ln 2$$ gap encoded in the potential landscape.[3][9]

## Role in stability and transitions

- Because $$\ln 2$$ corresponds to the smallest stable curvature separation, it sets the **activation barrier** for certain entropic transitions, such as the formation of new informational bits or curvature domains in the entropic field.[9][10]

- Configurations separated by less than this invariant tend to smear into each other under ToE’s entropy‑driven evolution, while those at or above $$\ln 2$$ can persist as robust, dynamically stable structures or “bits” of geometry/information.[9][10]

## Link to emergent geometry and gravity

- Since the curvature scalar $$R[g]$$ in the Obidi Action is induced by the entropy field, the OCI sets a natural curvature scale in the emergent geometry itself—an intrinsic geometric invariant tied directly to entropic distinguishability.[3][10]

- In gravitational regimes, this implies that certain geometric deformations (e.g., small perturbations of the entropic metric) are only dynamically meaningful once their entropic curvature exceeds the $$\ln 2$$ invariant, thus invariably shaping how ToE recovers and corrects Einsteinian General Relativity (GR) curvature at small scales.[3][10]

Citations:

[1] John Onimisi Obidi - Independent Researcher https://independent.academia.edu/JOHNOBIDI

[2] John Onimisi Obidi 1 1Affiliation not available October 15, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/284761/preprint_pdf/0304242fc1b6f7dfc2e1da6d68e30f89.pdf

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

[4] 1 Introduction 2 The Entropic Reformulation of the Unified https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/68f6f66c5dd091524f8f362e/original/transformational-unification-through-the-theory-of-entropicity-to-ea-reformulation-of-quantum-gravitational-correspondence-via-the-obidi-action-and-the-vuli-ndlela-integral.pdf

[5] John Onimisi Obidi 1 1Affiliation not available October 17, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/285164/preprint_pdf/c7acf1b70b62c5ae001365c123d20350.pdf

[6] Curvature-driver d.dynamics on $S^3$: a geometric atlas https://arxiv.org/pdf/2512.14164.pdf

[7] Evolution of curvature invariants and lifting integrability https://www.kent.ac.uk/ims/personal/elm2/liz/papers/elm-kamp.pdf

[8] Curvature invariant characterization of event horizons of four ... https://link.aps.org/doi/10.1103/PhysRevD.96.104022

[9] Entropy as a Physical Field: ToE Theory | John Onimisi ... https://www.linkedin.com/posts/john-onimisi-obidi-a2041911_formal-derivation-of-ln2-as-a-universal-activity-7417781493487374336-buas

[10] (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