The Binary, Diffeomorphism‑Invariant Measure of Curvature of the Entropic Field in the Theory of Entropicity (ToE): The Obidi Curvature Invariant OCI of ln 2—Its Interpretation, Meaning, Universal Applicability, as well as Physical and Practical Use-cases

The Obidi Curvature Invariant (OCI) is introduced in the Theory of Entropicity (ToE) as a **binary, diffeomorphism‑invariant measure of curvature of the entropic field**, encoding the minimal curvature contrast needed to register one bit of information in spacetime.[1][2]

## What the OCI is (core definition)

In the ToE formulation, entropy $$S(x)$$ is promoted to a continuous physical field over spacetime, with an associated entropic energy‑density or curvature density $$\rho(x)$$.[1][2] Information then corresponds to localized deformations of this entropic field, rather than to an abstract bookkeeping over microstates.[1][2] Within this setup, the Obidi Curvature Invariant is defined via a binary curvature ratio: a minimally distinct pair of entropic configurations $$A$$ and $$B$$ obeys

$$

\rho_B(x) = 2\,\rho_A(x),

$$

and this fixed ratio $$2:1$$ defines the invariant “curvature gap” associated with one bit (ln 2) of distinguishable information in the entropic field.[1] The OCI is constructed to be non‑negative and invariant under smooth coordinate transformations, so it functions analogously to a scalar curvature invariant, but now tied to the entropy field rather than purely to the metric.[1][3]

## Intrinsic physical meaning

1. **Entropic field curvature**

   In ToE, curvature is not only a property of the spacetime metric $$g_{\mu\nu}$$, but also of the entropic field $$S(x)$$ and its associated density $$\rho(x)$$.[1][2] The OCI captures the smallest stable “bifurcation” in this field that can support two distinct, physically meaningful entropic states, interpreted as a binary informational distinction.[1] Thus, an OCI‑quantized curvature difference is the field‑theoretic analogue of a single bit encoded in spacetime, assigning geometric meaning to ln 2 as a universal activity increment of the entropic field.[1]

2. **Causal and informational structure**

   Because ToE treats entropy flow as the fundamental driver of dynamics and time’s arrow, gradients and curvatures of $$S(x)$$ determine both effective forces and the directionality of causal evolution.[4][2] Within this view, regions where the entropic curvature crosses the OCI threshold represent loci where new causal/informational distinctions can emerge—e.g., when a quantum state transitions between macroscopically distinguishable outcomes under an entropy‑driven collapse criterion.[4] This links the causal structure (what can influence what) to discrete jumps in the entropic curvature spectrum, with the OCI acting as the elementary step in that spectrum.[1][4]

3. **Relation to GR curvature scalars**

   Classical curvature invariants in GR—such as the Ricci scalar $$R$$ or Kretschmann scalar—are diffeomorphism‑invariant scalars built from the metric and its derivatives, used to characterize the geometric and tidal structure of spacetime.[3] ToE extends this paradigm by introducing an “entropic curvature scalar,” where metric curvature and entropy production are linked, and in suitable limits the entropic framework reproduces Einstein’s field equations as an effective description.[4][5] The OCI then plays the role of a fundamental unit of this entropic curvature, tying the geometric content of curvature scalars to discrete information‑bearing deformations of the entropy field, in a way that resonates with thermodynamic and holographic interpretations of curvature as a driver of irreversible information flow.[1][4][5]

## Role in the ToE action and field equations

Recent ToE work formulates a Spectral Obidi Action of the schematic form

$$

A_{\text{ToE}}[S] = \int d^4x\,\sqrt{-g}\,\big[\alpha^2 R[g] - \beta^2 g^{\mu\nu}\nabla_\mu S\nabla_\nu S - \lambda\,D(S,S_0)\big],

$$

where $$R[g]$$ is a curvature scalar “induced by the entropic field”, the gradient term encodes kinetic dynamics of entropy, and $$D(S,S_0)$$ is a distinguishability potential between the local entropy and a reference configuration.[6] In this setting, the OCI shows up as the minimal spectral gap in $$D(S,S_0)$$ corresponding to a binary entropic distinction, effectively quantizing the distinguishability structure entering the action.[1][6] Varying this action yields coupled equations for $$g_{\mu\nu}$$ and $$S$$, in which classical GR emerges as a limiting case when entropic curvature effects reduce to an effective Ricci scalar with appropriate source terms.[4][6]

## Applicability and use‑cases

1. **Unification and emergent gravity**

   ToE posits that gravity is an emergent phenomenon arising from the dynamics of the entropic field, rather than a fundamental interaction coded solely in the metric.[4][2][7] The OCI provides a natural scale at which entropic curvature becomes informationally nontrivial, furnishing a bridge between microscopic information registers and macroscopic curvature, similar in spirit to frameworks where entropy production and Ricci curvature are directly related.[1][5][7] In principle, this supports unification efforts in which spacetime geometry, quantum information, and thermodynamics are all different manifestations of the same entropic substrate.

2. **Mathematical framework and invariants**

   Mathematically, the OCI motivates constructing curvature invariants not only from $$g_{\mu\nu}$$ and the Riemann tensor, but also from $$S(x)$$, its gradients, and their couplings.[1][3][6] It suggests a hierarchy where conventional curvature invariants (e.g., $$R$$, $$R_{\mu\nu}R^{\mu\nu}$$) are seen as coarse‑grained summaries of a more primitive entropic curvature spectrum, whose elementary gap is fixed by the OCI.[1][4] This provides a systematic way to derive effective laws: start from entropic invariants, then recover metric‑based invariants as emergent quantities in appropriate limits.[4][2]

3. **Complex systems and cosmology**

   In cosmological applications where entropy gradients and production drive large‑scale structure, an entropic curvature invariant can serve as a diagnostic for when new macroscopic structures or phase transitions become possible.[4][5][2] For example, a coarse‑grained entropy field in an FLRW background can be coupled to curvature so that the divergence of an informational flux is sourced by the Ricci scalar, linking cumulative entropy generation to integrated curvature over a causal region.[5] Embedding an OCI‑type quantization into such a framework would allow one to identify discrete entropic curvature thresholds associated with horizon entropy, black hole microstructure, or the onset of new cosmological regimes.[1][5][2]

4. **Phenomenology: black holes and quantum systems**

   Since ToE is designed to reproduce horizon thermodynamics and Bekenstein–Hawking entropy in appropriate limits, an entropic curvature gap like the OCI naturally suggests discrete steps in horizon information capacity or in black hole microstate structure.[4][5] In quantum contexts, where ToE ties wave‑function collapse to entropic thresholds governed by the Vuli–Ndlela integral and related inequalities, the OCI offers a geometric interpretation of the “one‑bit” threshold at which a quantum superposition becomes macroscopically distinguishable and undergoes an entropy‑driven phase transition.[4] This links phenomenological predictions about measurement, decoherence, and black hole evaporation to a single, geometrically defined entropic invariant.[1][4][5]

Overall, the Obidi Curvature Invariant can be read as the **minimal, coordinate‑invariant curvature contrast of the entropic field corresponding to one bit of physical information**, from which ToE builds a unified account of spacetime geometry, gravity, and quantum phenomena as emergent aspects of entropy dynamics.[1][4][2]

Citations:

[1] 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

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

[5] Entropy, Information, and the Curvature of Spacetime ... - Sciety https://sciety.org/articles/activity/10.20944/preprints202601.0455.v1

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

[7] Here is a Hypothesis: Spacetime Curvature as a Dual-Gradient Entropy Effect—AMA https://www.reddit.com/r/HypotheticalPhysics/comments/1lj0x5x/here_is_a_hypothesis_spacetime_curvature_as_a/

[8] The Theory of Entropicity (ToE) Goes Beyond Holographic ... https://www.authorea.com/users/896400/articles/1360831-the-theory-of-entropicity-toe-goes-beyond-holographic-pseudo-entropy

[9] Reconstructing Quantum Field Theory in Curved Space-time https://arxiv.org/abs/1803.07493

[10] Jose Gracia's Post - RealClock Quantum Mechanics https://www.linkedin.com/posts/jose-gracia-0686398_realclock-quantum-mechanics-preface-chapter-activity-7420149947049603072-sURJ