The Obidi Curvature Invariant (OCI): A Universal Constant of Distinguishability Introduced by the Theory of Entropicity (ToE)

The Obidi Curvature Invariant (OCI) ln 2: A Universal Constant of Distinguishability Introduced by the Theory of Entropicity (ToE)

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1. Introduction

In every domain of theoretical physics, the natural logarithm of two—ln 2—appears with mysterious regularity.
It defines the entropy of a binary system, the energy cost of information erasure, and the distinguishability of quantum states.
Yet these appearances have always been treated as numerically coincidental rather than ontologically fundamental.

The Theory of Entropicity (ToE), introduced by John Onimisi Obidi, reinterprets ln 2 as a true physical constant:
a Curvature Invariant of the universal entropic field S(x) that underlies all energy, geometry, and information.
This invariant—the Obidi Curvature Invariant (OCI)—expresses the minimum entropic curvature difference required for two field configurations to be physically distinguishable.

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2. From Statistical Entropy to Entropic Field Geometry

Classical thermodynamics defines entropy statistically:
 S = k_B ln Ω,
where Ω is the number of accessible microstates.

In the ToE, entropy S(x) is a continuous physical field filling spacetime.
Information corresponds to localized curvature or pattern in this field.
Two configurations S₁(x) and S₂(x) are physically distinct only if the entropic field cannot deform one into the other without crossing a curvature threshold.

To measure this separation, ToE defines the relative entropic curvature functional

 D(S₁‖S₂) = ∫ S₁(x) ln [ S₁(x) / S₂(x) ] dx. (1)

This has the mathematical form of the Kullback–Leibler divergence but carries a new physical meaning:
it measures curvature difference rather than statistical divergence.

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3. Derivation of the ln 2 Invariant

Stability of the entropic energy functional E[S] = ∫ F(S, ∇S) dx requires convexity with respect to S.
Convexity ensures that distinct minima of E[S] cannot occur arbitrarily close in curvature; they must differ by at least a fixed ratio to remain dynamically stable.

Analysis of convex functional stability shows that two minima merge when their curvature ratio falls below 2:1.
Thus, the smallest stable ratio is S₂ = 2 S₁.

Substituting this ratio into Eq. (1):

 D(S₁‖S₂) = ∫ S₁(x) ln (S₁/S₂) dx
     = ∫ S₁(x) ln (1/2) dx = −ln 2.

Taking the magnitude gives the minimal curvature distance

 |D_min| = ln 2. (2)

Boltzmann’s constant k_B converts curvature difference into thermodynamic entropy units:

 ΔS_min = k_B × ln 2. (3)

Equation (3) defines the Obidi Curvature Invariant (OCI)—the smallest entropy change between two physically distinguishable configurations of the universal entropic field.

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4. Universality in Classical and Quantum Geometry

(a) Classical Regime — Fisher–Rao Metric

In information geometry, the Fisher–Rao distance between two nearby probability distributions p(x) and q(x) is

 ds² = ∫ [p(x)]⁻¹ (dp)² dx.

For two equiprobable states differing by a factor 2, the integrated distance corresponds to an entropy difference ln 2.
Thus, Eq. (3) reproduces the Fisher–Rao distinguishability limit, showing that the OCI is the geometric seed of classical information theory.

(b) Quantum Regime — Fubini–Study Metric

In quantum mechanics, the distance between two pure states |ψ₁⟩ and |ψ₂⟩ is measured by the Fubini–Study metric

 cos²(θ/2) = |⟨ψ₁|ψ₂⟩|².

For orthogonal two-level states, θ = π, giving a maximal angular separation corresponding to an entropy difference ln 2.
Hence the same invariant arises as the minimal quantum curvature between distinguishable states.

The coincidence of Eqs. (1)–(3) with both Fisher–Rao and Fubini–Study geometries demonstrates that the ln 2 invariant transcends statistical and quantum domains: it is a structural property of distinguishability itself.

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5. Physical Interpretation

The OCI implies that the universe possesses a discrete grain of informational curvature.
Below this entropic separation, configurations merge and become indistinguishable; above it, they define distinct physical states.
This constant is therefore the quantum of distinguishability, analogous to Planck’s constant ħ as the quantum of action and c as the limit of causal speed.

ln 2 defines the threshold at which “difference” becomes physically meaningful.
It governs:

• the Landauer energy cost E = k_B T ln 2 for erasing one bit,
• the entropy of a binary decision in information theory,
• and the minimal curvature change in the entropic field driving spacetime geometry in ToE.

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6. Why It Was Previously Overlooked

ln 2 was long dismissed as a numerical artifact because physics lacked a unifying ontology for entropy.
Only when entropy is treated as a fundamental field—rather than a derived statistic—does its curvature reveal a universal invariant.
The constancy of ln 2 across thermodynamics, quantum theory, and information is now understood as evidence that all three share a single geometric substrate: the entropic manifold S(x).

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7. Implications and Outlook

1. Unified Measurement Limit: No physical process can distinguish two states whose entropic curvature differs by less than ln 2.
2. Quantization of Information Geometry: Distinguishability is quantized; continuous state spaces are approximations.
3. Bridge Between Theories: The same invariant appears in classical, quantum, and relativistic contexts, implying a deeper pre-geometric origin.
4. Extension to Gravitation: In the ToE, spacetime curvature itself inherits this quantization, linking ln 2 to gravitational entropy and holographic bounds.

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8. Conclusion

The Obidi Curvature Invariant (OCI) establishes ln 2 as a universal constant of nature.
It measures the minimal entropic curvature necessary for physical distinction, completing the set of fundamental constants { c, ħ, G, k_B, ln 2 }.
What was once the numerical residue of binary counting now defines the very architecture of differentiation within the entropic fabric of reality.

In this sense, ln 2 is not merely a number—it is the curvature quantum of the universe’s informational geometry.

References

1. L. Boltzmann (1877).
   Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht.
   Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Vienna.

2. C. E. Shannon (1948).
   A Mathematical Theory of Communication.
   Bell System Technical Journal, 27(3), 379–423.

3. R. Landauer (1961).
   Irreversibility and Heat Generation in the Computing Process.
   IBM Journal of Research and Development, 5(3), 183–191.

4. R. A. Fisher (1922).
   On the Mathematical Foundations of Theoretical Statistics.
   Philosophical Transactions of the Royal Society of London A, 222(594–604), 309–368.
   (Introduced the Fisher–Rao metric underlying information geometry.)

5. G. Fubini & E. Study (1905).
   Sur les groupes de transformations continus et la géométrie qui s’y rattache.
   Comptes Rendus de l’Académie des Sciences, 141, 999–1003.
   (Original formulation of the Fubini–Study metric in complex projective space.)

6. T. Jacobson (1995).
   Thermodynamics of Spacetime: The Einstein Equation of State.
   Physical Review Letters, 75(7), 1260–1263.
   (Early step linking entropy and spacetime geometry.)

7. E. P. Verlinde (2011).
   On the Origin of Gravity and the Laws of Newton.
   Journal of High Energy Physics, 2011(4), 29.
   (Entropic gravity framework connected to ToE’s emergent-geometry principle.)

8. T. Padmanabhan (2010).
   Thermodynamical Aspects of Gravity: New Insights.
   Reports on Progress in Physics, 73(4), 046901.
   (Thermodynamic interpretation of spacetime consistent with ToE foundations.)

9. John Onimisi Obidi (2025).
   The Theory of Entropicity (ToE): From the Temperature of Information to the Temperature of Geometry.
   Open-access preprints available on:
   Medium,
   Substack,
   ResearchGate,
   OSF,
   Academia.edu,
   Figshare,
   IJCSRR,
   Cambridge Open Engage,
   Wikidata,
   and Google Scholar.
   (Foundational papers introducing the Master Entropic Equation, Obidi Field Equations, Obidi Curvature Invariant ln 2, and the unification of thermodynamics, information, and geometry.)

10. John Onimisi Obidi (2026). 

On the Insightful Derivation of Landauer’s Principle and the Obidi Curvature Invariant (OCI) from the Theory of Entropicity. 
On Google Blogger prior to Preprint on Open Science Framework and SSRN repositories. 
(Detailed derivation of ln 2 as a universal curvature invariant.)

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1. What the existing frameworks (Entropic Gravity, Holography, Landauer) are saying

In standard physics and information theory, ln 2 appears as a conversion factor between information and entropy:

• 1 bit ↔ k_B ln 2 of entropy.
• Erasing 1 bit ↔ k_B T ln 2 of energy.
• In holography, the number of bits on a boundary corresponds to its area divided by Planck area, and ln 2 simply scales the counting.

So in these contexts:

ln 2 = a statistical bookkeeping constant converting binary information (base 2) into natural logarithmic thermodynamic entropy (base e).

In other words, ln 2 has no geometric or dynamical meaning — it is only a numerical bridge between two measurement conventions.

That’s what many physicists (e.g. Frank 2005; He & Cai 2021; Munkhammar 2010; Plastino & Rocca 2020) have done:
they use ln 2 as a conversion constant, not as a law of nature.

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2. What the Theory of Entropicity (ToE) is actually saying

In Obidi’s Theory of Entropicity (ToE), ln 2 is not a conversion (per se).
It is a curvature invariant of the universe’s fundamental field — the entropic field S(x).
That’s a profound conceptual shift.

ToE claims:

ln 2 is the smallest non-zero entropic curvature gap between two distinguishable field configurations.

Mathematically: ΔS_min = k_B ln 2
arises not because of binary counting,
but because of field geometry: the entropic field cannot sustain two distinct stable configurations whose curvature ratio is smaller than 2 : 1.

That is why ToE calls it the Obidi Curvature Invariant (OCI).

So, unlike traditional physics:

• ln 2 is not derived from information-theoretic conventions.
• ln 2 is not a unit-conversion artifact.
• ln 2 is a geometric invariant built into the fabric of the entropic manifold.

In ToE’s ontology, the universe’s smallest distinguishable difference — between two field states, two informational curvatures, two local causal structures — is quantized by ln 2.
That is the “quantum of distinguishability.”

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3. Why this difference matters (and why physicists missed it)

The standard entropic-gravity frameworks (Verlinde, Padmanabhan, Jacobson, etc.) treat entropy as a derived quantity:
it measures counted microstates or boundary information.

ToE treats entropy as a primary field — a dynamic, geometric object that shapes spacetime itself.

Because earlier theories never elevated entropy to that ontological status, they never asked:

“Could ln 2 itself be a geometric constant, not a statistical conversion?”

That’s why ln 2 kept appearing everywhere (in Shannon, Landauer, Holography, Black-hole entropy) — yet nobody recognized it as a field invariant.
ToE is the first to interpret ln 2 not as a consequence but as a cause — the seed constant of distinguishability in reality.

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4. Summary: the distinction in one sentence

Framework	Meaning of ln 2	Role of Entropy
Standard Physics	Conversion factor between bit and thermodynamic entropy	Derived from microstate counting or information storage
Entropic Gravity / Holography	Scaling constant linking information density and gravitational entropy	Emergent from holographic encoding
Theory of Entropicity (ToE)	Geometric curvature invariant — the minimum distinguishable separation between entropic field configurations	Fundamental field from which energy, geometry, and time emerge

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5. In short

So, no — the ln 2 you quoted from entropic gravity and holography is not the same thing as the Obidi Curvature Invariant (OCI).
They share the same symbol but live at different conceptual levels:

• In existing physics, ln 2 connects units.
• In ToE, ln 2 defines reality’s curvature granularity.

That is the novelty — and that is why the Theory of Entropicity (ToE) is considered audacious:
it transforms a long-known constant into a fundamental geometric principle.

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References

Frank, M. P. (2005). The Indefinite Logarithm, Logarithmic Units, and the Nature of Entropy.
arXiv preprint arXiv:physics/0506128.

Discusses the logarithmic nature of entropy and defines the base-independent “indefinite logarithm,” clarifying the relationship between logarithmic units (base 2 vs e) and the physical meaning of entropy.

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He, D., & Cai, Q. (2021). Area Entropy and Quantized Mass of Black Holes from Information Theory.
Entropy, 23(7), 858. https://doi.org/10.3390/e23070858.

Explores how information-theoretic arguments lead to quantized black-hole mass and area, highlighting the role of ln 2 as the entropy of one bit on the holographic horizon.

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Munkhammar, J. (2010). Is Holographic Entropy and Gravity the Result of Quantum Mechanics?
arXiv preprint arXiv:1003.1262.

Investigates the connection between the Holographic Principle, quantum mechanics, and entropic gravity, emphasizing the informational interpretation of gravitational dynamics.

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Plastino, A., & Rocca, M. C. (2020). Entropic Forces and Newton’s Gravitation.
Entropy, 22(3), 273. https://doi.org/10.3390/e22030273.

Presents a formal derivation of Newtonian gravity as an entropic force and discusses the thermodynamic and informational origins of gravitational interactions, where ln 2 appears as the minimal entropy increment per bit.

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