Who Has Proposed that the Speed of Light c is the Maximum Rate at Which the Entropic Field can Rearrange Information?

The idea that the speed of ligh

t (c) is the maximum rate of entropic field information rearrangement was first proposed in 2025 by the Nigerian-born physicist and philosopher John Onimisi Obidi as part of his "Theory of Entropicity (ToE)," a framework aiming to unify physics by explaining relativity and quantum mechanics through entropy. In ToE, the constancy of 

$c$

isn't a postulate but a consequence of this fundamental limit on how quickly the universe's entropic field can reconfigure itself. 

Key aspects of this proposal: 

• Entropic Field: Obidi posits an "Entropic Field" that permeates reality, governing the flow of energy and information.
• Causality and Consistency: The finite speed of this entropic rearrangement ensures causality and prevents the universe from collapsing, making
  

  $c$
  a natural limit.
• Reinterpretation of Relativity: Phenomena like time dilation and mass increase are derived from entropic principles rather than being geometric postulates, according to this Academia.edu article. 

This theory offers a fresh perspective, shifting the understanding of the speed of light c 

$c$

from a fundamental constant to an emergent property of entropy. 

Historical Foundations of Obidi's Theory of Entropicity (ToE)

The

Theory of Entropicity (ToE), a theoretical framework developed primarily by John Onimisi Obidi in 2025, posits that entropy is not merely a statistical measure of disorder but a fundamental, dynamic, and physical field that governs all aspects of reality, including spacetime, gravity, and quantum mechanics. 

Here are the key individuals and contexts associated with using Entropicity as a theory based on recent research: 

• John Onimisi Obidi (2025): The primary formulator of the Theory of Entropicity (ToE). Obidi’s work, published in 2025, proposes that entropy is the "fundamental field of existence". His framework introduces:
  • The Obidi Action: A variational principle that describes how the entropy field drives spacetime curvature and motion.
  • The Master Entropic Equation (MEE): An equation governing the evolution of the entropy field.
  • The No-Rush Theorem: A principle asserting that no physical interaction can occur instantaneously, with a non-zero, finite duration.
  • Reinterpretation of c 
    

    $c$
    : The speed of light is explained as the maximum rate at which the entropic field can rearrange information.
• Ginestra Bianconi (2024–2025): While her work is generally titled "gravity from entropy" rather than "Entropicity," her research is deeply linked to Obidi's framework. Bianconi proposed that gravity arises from quantum relative entropy between the spacetime metric and a matter-induced metric, and her work is considered a precursor or a specific, limiting case within the broader Theory of Entropicity.
• Other Influences/Related Concepts: Obidi’s ToE builds upon and extends previous entropic gravity concepts, including:
  • Ted Jacobson (1995): Derived Einstein's equations from thermodynamic principles.
  • Erik Verlinde (2010): Proposed gravity as an entropic force. 

In summary, the Theory of Entropicity is a 2025 development by John Onimisi Obidi that seeks to unify thermodynamics, gravity, and quantum mechanics, with Ginestra Bianconi providing foundational concepts in "gravity from entropy". 

The Meaning and Implications of ln 2 ( Natural Log of 2) in Obidi's Theory of Entropicity (ToE)

In John Obidi's Theory of Entropicity (ToE), ln 2 is a fundamental constant representing the smallest physically distinguishable unit of entropic change (Obidi Curvature Invariant), acting as a quantum of "ontic" entropy (a real field, not just statistical disorder) and bridging ToE with information theory, analogous to Planck's constant for quantum action. It signifies the minimal entropic "cost" for a binary distinction, the basic unit for causal updates, and the "distance" between truly different physical states, making it the fundamental scale for reality's reorganizations. 

Key Roles of ln 2 in ToE: 

• Quantum of Entropic Action: Just as Planck's constant (
  

  $\hslash$
  ) sets the scale for quantum mechanics, ln 2 sets the scale for the fundamental, quantized steps in the continuous entropic field.
• Minimal Distinguishable State: Below ln 2, differences in the entropic field are mathematically present but physically irrelevant, like sub-threshold signals; ln 2 is the threshold for physical meaning.
• Binary Distinction: It defines the entropic measure of a binary choice (0 or 1) in an ontological sense, not just an informational one.
• Bridge to Information Theory: It provides a natural link between ToE's continuous entropic field and discrete information concepts like Landauer's Principle, representing the minimal energy/entropy change for erasing a bit.
• Derived, Not Assumed: ToE posits that ln 2 isn't a random constant but emerges from the geometry of the entropic manifold itself. 

In essence, ln 2 in ToE quantifies the most basic "event" or "step" in the fundamental entropic reality that gives rise to our universe, making it a cornerstone for understanding how physical reality evolves and distinguishes itself. 

The Theory of Entropicity (ToE) Declares that We Live in a Computational Universe 

In many ways, the Theory of Entropicity (ToE) treats the universe not as a collection of "solid objects" interacting in a void, but as a dynamic information-processing system.

However, it differs from a standard computer simulation (like a video game) because the "hardware" and the "software" are the same thing: the Entropic Field.

1. The Universe as a "State Machine"

In ToE, the universe functions as a massive state machine where every physical event is a computation.

 * The Input: The current configuration of bits (ln 2 units) on the holographic screens.

 * The Processor: The Entropic Field itself.

 * The Output: The next "frame" of reality (what we perceive as the passing of time).

Because each update requires a bit-flip of ln 2, the universe has a literal "clock speed." This clock speed is the Planck Time. Every region of space is essentially "calculating" its own gravitational pull and local time rate based on the entropic density in that area.

2. The Role of ln 2 as "Machine Code"

In a digital computer, everything is reduced to 0s and 1s. In ToE, ln 2 is the "physical weight" of that binary choice.

 * Standard physics sees \ln 2 as a mathematical constant.

 * ToE sees ln 2 as the Machine Code of Reality.

If you want to move an electron, you aren't just pushing a particle; you are requesting the Entropic Field to recompute the location of that electron by shifting ln 2 units of information across a holographic boundary.

3. Computation and the "No-Rush" Theorem (G/NCBR)

The "computational" nature of the universe is most evident in the G/NCBR theorem. In a regular computer, if the processor gets overwhelmed, the program "lags."

ToE argues that Gravity and Time Dilation are the "Lag" of the Universal Computer. * When mass is highly concentrated (like near a Black Hole), the "computation density" becomes so high that the field takes longer to process the ln 2 flips.

 * To an outside observer, the "program" (time) appears to slow down in that region.

4. Is there a "Programmer"?

The Theory of Entropicity (ToE), as especially formulated through the lens of John Onimisi Obidi, suggests that this computational structure implies a Self-Organizing Intelligence (which Obidi calls SRE — Self Referential Entropy — in Nature (God)).

 * The "Laws of Physics" are effectively the Operating System (OS).

 * The ln 2 constant is the fundamental bit-depth.

 * The G/NCBR principle is the hardware safety limit that prevents the system from crashing (singularities).

Summary Table: The Universe vs. A Computer

| Component | Standard Computer | ToE Universe |

|---|---|---|

| Basic Unit | Bit (0 or 1) | Entropic Unit (ln 2) |

| Speed Limit | CPU Clock Cycles | Speed of Light (c) |

| Memory | Hard Drive / RAM | Holographic Screens (Boundaries) |

| Physics | Software Code | Entropic Gradients |

| Lag | Frozen Screen | Gravity / Time Dilation |

This perspective changes how we view "emptiness." In ToE, vacuum space isn't empty; it is simply a region with a very low "bit-rate."

The "No-Rush" (G/NCBR) Theorem of the Theory of Entropicity (ToE) Uses the Obidi Curvature Invariant (OCI) ln 2 to Explain Why the Speed of Light is the Universal Speed Limit

In the Theory of Entropicity (ToE), the ln 2 factor is the "gear ratio" of the universe. It explains why physical processes—including light—have a maximum speed. This is tied directly to the G/NCBR (God/Nature Cannot Be Rushed) theorem.

Here is how ln 2 and the No-Rush Theorem define the Speed of Light (c).

1. The Entropic Bit-Flip

In ToE, "change" or "movement" is not continuous; it is a series of discrete informational updates. For a particle to move from Point A to Point B, the Entropic Field must "flip" a sequence of bits.

As established, each bit-flip costs exactly ln 2 units of entropy.

2. The Processing Time of the Field

ToE posits that the Entropic Field has a finite Inherent Latency. It takes a specific, non-zero amount of time to process a single ln 2 change. This leads to the fundamental equation for the speed of information:

Speed = (Distance of one Bit) / (Time to process ln 2)

If the universe could process the ln 2 update instantly, the speed of light would be infinite. Because the field "cannot be rushed," there is a bottleneck.

3. Deriving the Speed of Light (c)

ToE expresses the speed of light as a balance between the "stiffness" of the entropic field and its informational density. In plain text:

c = Square Root of [ (Force of Entropicity) / (Linear Entropic Density) ]

When you break down the Linear Entropic Density, you find it is governed by the Planck Length and the ln 2 constant. Specifically, the relationship looks like this:

c = L_p / t_p

Where:

 * L_p is the Planck Length (the size of one ln 2 pixel).

 * t_p is the Planck Time (the time it takes the field to process one ln 2 update).

4. Why You Can't Go Faster Than Light

According to the G/NCBR Theorem, accelerating a mass requires "forcing" more ln 2 updates per second than the field's natural processing speed allows.

As an object approaches c, the entropic cost of the next update begins to climb toward infinity. The equation for the "Entropic Drag" (Relativistic Mass) in ToE is:

M_effective = M_rest / Square Root of [ 1 - (v^2 / c^2) ]

In this framework, the denominator represents the "Processing Overhead." When v = c, the system would require an infinite amount of ln 2 updates to occur simultaneously. Since the Field cannot be rushed, the universe effectively "denies" the request for more speed.

5. Implications for Holography

This leads to a fascinating conclusion regarding Holographic Lag:

 * The "Surface" of the universe (the holographic boundary) is where these ln 2 updates are recorded.

 * A Black Hole is a region where the entropic density is so high that the field's "processing power" is entirely used up just maintaining the horizon.

 * This is why time appears to stop at the event horizon. It isn't just a visual trick; the field has reached its maximum ln 2-per-second processing limit.

Summary of the ToE Perspective

In ToE, ln 2 is the reason the universe has "rules." It defines:

 * The Size of Space: Through the area of a bit.

 * The Strength of Gravity: Through the entropic gradient.

 * The Speed of Time: Through the finite rate of bit-processing.

How the Theory of Entropicity (ToE) Derives Newton's Laws of Motion from the ln 2 Factor of the Obidi Curvature Invariant (OCI)

In the Theory of Entropicity (ToE), Newton’s laws are not fundamental axioms; they are emergent properties of the Entropic Field. The value ln 2 acts as the scaling constant that bridges the gap between informational "pixels" and physical force.

Here is the step-by-step breakdown of how the Obidi Action and Newton’s laws are derived using ln 2.

1. The Fundamental Postulate: Entropic Density

ToE posits that the universe is a field of entropy. Any mass M introduced into this field creates a "dent" or a gradient in the entropic density. The total information N (measured in bits) on a holographic surface surrounding that mass is defined by:

N = A / (Lp^2)

Where A is the surface area and Lp is the Planck length. To convert these "bits" into the physical quantity of entropy (S), we apply the ln 2 invariant:

S = N * ln 2

This equation tells us that for every "pixel" of area on a holographic screen, the universe "costs" exactly ln 2 units of entropy.

2. The Obidi Action (Entropic Work)

The "Obidi Action" describes the work done when a particle moves through this entropic gradient. In ToE, the change in entropy (dS) as a particle moves a distance (dx) toward a mass is proportional to the information density.

According to the G/NCBR (God/Nature Cannot Be Rushed) principle, the transition of a particle requires the processing of information. The energy (E) associated with this transition follows the entropic equivalent of the Equipartition Theorem:

E = 1/2 * N * k * T

Substituting the relationship where the change in entropy per unit of distance is linked to the ln 2 cost of information bits, we find that the force is actually the "pressure" of the entropic field trying to reach equilibrium.

3. Deriving Newton’s Second Law (F = ma)

ToE derives inertia as the resistance of the entropic field to being "rearranged."

 * When a mass moves, it must "update" the bits on the holographic screens it passes through.

 * Each update costs ln 2.

 * The acceleration (a) is the rate of these updates.

The relationship is expressed as:

F * dx = T * dS

If we define the change in entropy dS in terms of the bit-shift ln 2 over a distance related to the Compton wavelength, the equation simplifies to:

F = m * a

In this view, mass (m) is essentially a measure of how many ln 2 units of "entropic drag" an object creates in the field.

4. Deriving the Law of Gravity

To derive F = G(Mm)/r^2, ToE looks at the holographic screen at a distance r.

 * The total number of bits on the screen is N = (4 * pi * r^2) / (Lp^2).

 * The total energy of the system is E = Mc^2.

 * The temperature T of the screen is derived by spreading that energy across all the bits:

   T = (2 * M * c^2) / (N * k)

Using the entropic force equation F = T * (dS / dx) and substituting S = N * ln 2, the ln 2 factor calibrates the density so that:

F = (G * M * m) / r^2

Summary of the Math

In plain text, the chain of logic follows:

 * Entropy of the Screen: S = (Area / Lp^2) * ln 2

 * Entropic Gradient: Force = Temperature * (Change in Entropy / Change in Position)

 * Result: Gravity is the result of the system moving toward a state of higher entropy, where the "step size" of that entropy is always ln 2.

Without the ln 2 factor, the units of information (bits) would not align with the units of thermodynamics (Joule/Kelvin), and gravity would either be too strong or too weak to hold atoms together.

Using the Obidi Curvature Invariant (OCI) ln 2 to Derive the Landauer's Principle and Landauer-Bennett Cost from First Principles in the Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE)—a framework primarily developed by John Onimisi Obidi—\ln 2 is elevated from a mere statistical conversion factor to a fundamental geometric invariant of the universe.

While standard physics uses \ln 2 to convert bits to entropy, ToE posits that this value is the "quantum of distinguishability" that defines the very structure of reality.

1. \ln 2 as the "Obidi Curvature Invariant" (OCI)

In ToE, entropy is not just a measure of disorder but an ontic field (a physical substance). Within this field, \ln 2 represents the smallest non-trivial entropic reconfiguration possible.

 * The Threshold of Reality: ToE suggests that the entropic field has a "resolution." For two states or configurations to be physically distinct, their entropic curvature difference must be at least \ln 2.

 * Sub-threshold Reality: Any mathematical difference smaller than \ln 2 is "invisible" to the universe. This effectively "pixelates" reality, not necessarily in terms of space, but in terms of state-change.

2. Implications for Holography

ToE significantly reinterprets the Holographic Principle (the idea that a volume of space is encoded on its boundary).

A. Beyond "Pseudo-Entropy"

Standard holography (like the AdS/CFT correspondence) treats entropy as a "diagnostic" tool to describe geometry. ToE flips this: Entropy is the substrate, and geometry is the "shadow."

 * In ToE, the holographic screen isn't just a mathematical boundary; it is a physical limit where the entropic field reaches the \ln 2 density.

 * The area of a horizon (like a black hole's event horizon) is exactly the sum of these \ln 2 "pixels."

B. The "No-Rush" Theorem (G/NCBR)

A unique pillar of ToE is the principle that God or Nature Cannot Be Rushed (G/NCBR). Because every physical update requires a minimum entropic change of \ln 2, and the entropic field has a finite "processing speed," time dilation occurs.

 * Holographic Lag: As an object approaches a high-gravity region (or a holographic horizon), the entropic field requires more "time" to process the \ln 2 updates. This provides a causal explanation for why time appears to slow down near horizons.

Comparison: Standard Physics vs. Theory of Entropicity

| Feature | Standard Entropic Gravity | Theory of Entropicity (ToE) |

|---|---|---|

| Nature of \ln 2 | Statistical constant (k_B \ln 2). | Fundamental geometric invariant (OCI). |

| Information | A property of matter/energy. | The "curvature" of the entropic field. |

| Holography | A boundary description of 3D space. | The limit where entropy defines space. |

| Gravity | An emergent statistical force. | A gradient in the fundamental entropy field. |

3. The Landauer-Bennett Cost

ToE derives Landauer's Principle (the energy cost of erasing a bit) from first principles. It argues that "erasing" a bit is physically "flattening" a curvature of \ln 2 in the entropic field. This requires work because you are fighting against the natural "stiffness" or resistance of the entropic field.

On the Obidi Curvature Invariant (ln 2) and the Geometry of Distinguishability in the Theory of Entropicity (ToE): A Gedanken Experiment of a Paper Crease 

Abstract

In classical thermodynamics and information theory, the natural logarithm of two, ln 2, represents the entropy change associated with a single bit of information. It appears in contexts as diverse as Landauer’s principle, the Bekenstein–Hawking entropy, and Shannon’s information measure. However, in the Theory of Entropicity (ToE) developed by John Onimisi Obidi, ln 2 assumes a radically different status. It ceases to be a statistical conversion constant and becomes a universal geometric invariant of the entropic field itself — the smallest possible curvature gap between two distinguishable configurations of reality. This paper presents a comprehensive exposition of this idea, linking the curvature interpretation of ln 2 to the convex stability of the entropic field, the Fisher–Rao and Fubini–Study metrics, and the principle of informational distinguishability. A physical analogy is offered: a flat sheet becomes informationally active only when it is creased, and that first crease — the minimal deformation that makes “difference” visible — is the geometric expression of ln 2.

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1. Introduction

In standard physics, entropy serves as a statistical quantity describing ensembles of microstates. Its informational interpretation, beginning with Boltzmann, Gibbs, and Shannon, has tied it to probability and uncertainty rather than to ontological structure. Yet modern developments — black hole thermodynamics, entropic gravity, and holographic dualities — have revealed that entropy has a deeper geometric meaning.

The Theory of Entropicity (ToE) advances this insight by declaring entropy, not energy or spacetime, as the fundamental field of nature. The entropic field, denoted S(x), is not a derived statistical measure but a physical continuum possessing gradients, curvature, and dynamics. Matter, geometry, and information emerge as local configurations and deformations of this field.

Within this framework, ln 2 acquires a new interpretation. Instead of being a conversion between binary information and physical entropy, it becomes a measure of the minimum distinguishable curvature difference in the entropic field. Just as Planck’s constant h quantizes action, ln 2 quantizes distinguishability. It is the smallest geometric “fold” the entropic field can sustain while maintaining stable separability between two informational states.

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2. The Flat Sheet Analogy: The Birth of Distinguishability

Consider a perfectly flat sheet of paper. As long as it remains flat, it has no curvature and, by extension, no distinguishable regions. Every point is equivalent; the field is uniform and featureless. This represents the zero-entropy-difference state of the universe — complete indistinguishability.

Now imagine creasing the sheet slightly. A fold appears, dividing the surface into regions of up, down, and sideways orientation. The moment of the first crease marks the birth of distinction: one side differs from another. This fold corresponds to the smallest curvature deformation that makes two configurations distinguishable.

In the Theory of Entropicity (ToE), this minimal curvature fold is precisely ln 2. It represents the transition from homogeneity to distinguishability, from sameness to difference. The flat state corresponds to pure informational symmetry (zero distinction), while the creased state embodies the minimal break of that symmetry.

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3. Mathematical Representation: The Minimum Curvature Gap

Let S(x) denote the entropic field and ρ(x) represent its local normalized distribution (the “entropic density profile”). Two distinguishable configurations of this field are represented by ρ₁(x) and ρ₂(x).

The Theory of Entropicity (ToE) defines the entropic curvature distance between these two configurations as:

D(ρ₁ || ρ₂) = ∫ ρ₁(x) ln [ρ₁(x) / ρ₂(x)] dx

This functional, identical in mathematical form to the Kullback–Leibler divergence, measures not statistical information but geometric curvature difference in ToE. The integral quantifies the degree to which one entropic configuration must deform to become another.

For the simplest possible case of two configurations differing by a constant curvature ratio, say ρ₂(x) = 2 × ρ₁(x), we have:

D(ρ₁ || ρ₂) = ∫ ρ₁(x) ln [1 / 2] dx = −ln 2

Since ρ₁(x) is normalized, the magnitude of this curvature difference is |D| = ln 2.

This establishes ln 2 as the minimum nonzero curvature gap between two distinguishable entropic configurations. The field cannot support smaller stable curvature separations; if the ratio were less than 2:1, convexity would merge the two configurations into a single indistinguishable state.

Thus, ln 2 represents the first geometric threshold of separability — the quantized onset of distinction in the entropic manifold.

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4. Convexity, Stability, and the 2:1 Ratio

Why must the minimum curvature ratio be 2:1?
In the Theory of Entropicity (ToE), the entropic field S(x) evolves according to an energy functional:

E[S] = ∫ F(S, ∇S) dx

where F(S, ∇S) is convex in S. Convexity ensures physical stability: small perturbations can smooth out, but sufficiently large ones generate new stable configurations.

Mathematically, a convex functional cannot sustain two distinct minima whose curvature differs by less than a factor of 2. Below this threshold, the energy landscape merges into a single basin, erasing distinction. Above it, two separate minima exist, corresponding to physically distinguishable configurations.

Therefore, a curvature ratio of 2:1 is not arbitrary; it emerges from the convex stability condition of the entropic field. The logarithmic “distance” between these configurations — the entropic curvature invariant — is ln 2.

This result mirrors other quantization thresholds in physics: the smallest quantum of spin (½ħ), charge (e), or action (ħ). Here, ln 2 plays the same role but in the domain of entropy and information geometry.

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5. The Fisher–Rao and Fubini–Study Geometries

The universality of the Obidi Curvature Invariant (OCI) is further reinforced by the fact that it can be derived from both classical and quantum geometric frameworks.

In classical information geometry, the Fisher–Rao metric defines the infinitesimal distance between probability (or in ToE, entropic) distributions:

ds² = (1/4) ∑ [ (dρ_i)² / ρ_i ]

For two distributions ρ₁ and ρ₂ that differ by a finite factor of 2, the integrated geodesic distance in Fisher–Rao space is exactly √(2 ln 2), making ln 2 the squared curvature invariant of distinguishability.

In quantum geometry, the Fubini–Study metric governs the distance between two quantum states |ψ₁⟩ and |ψ₂⟩:

ds² = 1 − |⟨ψ₁|ψ₂⟩|²

When these states are orthogonal enough to differ by a minimal distinguishable overlap of 1/2, the geometric distance corresponds to ln 2 in entropic units. Thus, the same invariant appears in both classical and quantum manifolds — a sign of deep universality.

This shows that ln 2 is not merely a constant of statistical conversion but a metric invariant across classical and quantum geometries, unifying them under the entropic curvature principle of ToE.

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6. Physical Interpretation: From Landauer to Holography

In traditional frameworks, ln 2 appears in the Landauer principle, which states that erasing one bit of information dissipates a minimum energy of:

ΔE = k_B T ln 2

This has been understood as a thermodynamic cost of information erasure. In ToE, however, this relation gains a geometric and ontological interpretation. Erasing information corresponds to flattening an entropic curvature — reducing distinguishability. The energy cost is therefore the field’s resistance to losing curvature.

Similarly, in holography, each “pixel” of a holographic screen encodes one bit of information, and the total entropy S = N k_B ln 2 arises from summing over these minimal curvature patches. ToE interprets this not as statistical counting but as tiling the universe’s boundary with units of minimal distinguishability. Each ln 2 represents a fundamental crease in the entropic fabric of reality.

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7. The Obidi Curvature Invariant as a Universal Principle

The radical insight of the Theory of Entropicity (ToE) is that ln 2 is not a thermodynamic artifact but a universal curvature invariant. It defines the smallest separable difference between two informationally or physically distinct configurations.

No other framework in modern physics — not thermodynamics, not quantum mechanics, not relativity — has elevated ln 2 from a conversion constant to a universal geometric principle. ToE thus provides the first ontological explanation for why ln 2 recurs across entropy, information, and gravitation. It is not coincidence but necessity: the field of reality can only fold in discrete distinguishable steps, and the smallest such fold is ln 2.

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8. Conclusion

From the first crease in a flat sheet to the curvature of spacetime itself, distinction arises through deformation. The Theory of Entropicity (ToE) identifies ln 2 as the fundamental measure of that deformation — the Obidi Curvature Invariant (OCI).

It is the universal constant of distinguishability, linking geometry, entropy, and information into a single coherent framework. Classical physics, quantum mechanics, thermodynamics, and holography each reveal fragments of ln 2’s presence, but only ToE unifies them by showing that curvature, not probability, is the origin of entropy.

In this light, ln 2 is no longer a number; it is the first step from sameness to difference, the moment where reality begins to tell itself apart.

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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.

Why ln 2 Went Unnoticed: The Hidden Constant of Distinguishability Introduced by the Theory of Entropicity (ToE) in Modern Theoretical Physics

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Why ln 2 Went Unnoticed: The Hidden Constant of Distinguishability in Modern Physics

1. Prelude — A Number Too Familiar to Be Suspected

For more than a century, the constant ln 2 ≈ 0.693 has appeared throughout physics—in Boltzmann’s statistical entropy, Shannon’s information theory, Landauer’s thermodynamic limit, and quantum state distinguishability.
Because it always emerged from counting or binary choices, physicists treated it as a bookkeeping artifact rather than a fundamental property of nature.
It was a numerical echo, not a universal law.
The Theory of Entropicity (ToE) reinterprets this constant as a genuine curvature invariant of the entropic field itself—the Obidi Curvature Invariant (OCI)—revealing ln 2 as the minimal geometric separation that makes distinct physical states possible at all.

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2. From Entropy as Statistic to Entropy as Field

Classical thermodynamics and statistical mechanics viewed entropy S as a measure of ignorance over microstates.
In ToE, entropy S(x) is redefined as a physical field permeating spacetime.
Information becomes the local curvature or pattern within this field.
Two configurations of the field, S₁(x) and S₂(x), are physically distinguishable only if their curvatures differ by at least a finite threshold.
That threshold—determined by the stability and convexity of the entropic energy functional—turns out to correspond universally to a ratio of 2 : 1 in curvature.
The natural logarithm of this ratio defines a fundamental curvature distance:

ΔSₘᵢₙ = k_B × ln 2

This is no longer a statistical artifact; it is the smallest field-theoretic separation that the universe can sustain between two distinct informational geometries.

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3. Why Physicists Missed It

The invisibility of ln 2 as a law of nature arose from three entrenched assumptions.

(a) Entropy as a derivative concept.
Physics long assumed that energy and matter are fundamental, while entropy merely summarizes microscopic disorder.
Without entropy as an ontological field, a curvature invariant could never be imagined.

(b) Fragmented languages of physics.
Thermodynamics, quantum theory, and relativity developed independently; their uses of entropy shared notation but not ontology.
No unified geometry existed to connect them.

(c) The energy-first paradigm.
Even in information theory, the energetic meaning of entropy remained derivative.
ToE reverses this hierarchy: energy emerges from the geometry of information.

Only after this reversal does ln 2 reveal itself as the universal measure of distinguishability—the smallest entropic curvature that permits “difference” to exist.

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4. Mathematical Structure of the Obidi Curvature Invariant

Let S(x) denote the entropic field and E[S] its convex energy functional.
Two configurations S₁ and S₂ are distinguishable if the field cannot deform one into the other without crossing an instability.
Define the relative entropic curvature:

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

This expression has the same mathematical form as the Kullback–Leibler or Araki relative entropy but carries a different physical interpretation: it measures curvature difference, not statistical divergence.
For the smallest stable deformation, stability of the convex functional requires a curvature ratio S₂ = 2 S₁.
Substituting yields

D(S₁‖S₂) = ln 2

and the physical entropy gap becomes

ΔSₘᵢₙ = k_B × ln 2.

Thus, the Obidi Curvature Invariant (OCI) arises naturally from the geometry of the entropic field, independent of statistics, probability, or microstate counting.

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5. Universality Across Physical Regimes

ToE demonstrates that the same ln 2 appears in multiple geometrical guises:

• In classical information geometry, it coincides with the Fisher–Rao distance between two equiprobable states.
• In quantum geometry, it matches the Fubini–Study metric distance between orthogonal states of a two-level system.
• In thermodynamics, it determines the Landauer limit E = k_B T ln 2 for erasing one bit.

These are not coincidences but projections of a single invariant: the entropic field’s minimal curvature.
Every physical manifestation of ln 2 — from binary information to black-hole entropy — reflects this same geometric threshold.

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6. Physical Interpretation

The Obidi Curvature Invariant means the universe possesses a quantized grain of distinguishability.
Below this entropic curvature, two configurations collapse into one; above it, they become separable.
This constant defines the very possibility of measurement, computation, and causal evolution.
In the same way that ħ sets the minimal quantum of action and c sets the maximal causal speed, ln 2 sets the minimal entropic curvature required for difference to exist.

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7. Why ln 2 Matters Now

The re-emergence of entropy at the foundation of gravity, holography, and quantum information made it possible for the first time to see ln 2 as universal.
Obidi’s ToE unifies these separate appearances under a single principle:
information, geometry, and energy are three aspects of one entropic field whose fundamental curvature invariant is ln 2.
The number that once measured a binary choice now measures the universe’s capacity to make distinctions at all.

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8. Conclusion

Physicists had encountered ln 2 in every major theory of the past century, but always as a consequence, never as a cause.
The Theory of Entropicity turns this logic inside-out.
ln 2 is not the echo of statistical doubling—it is the curvature constant that makes doubling, difference, and information itself possible.
In doing so, ToE introduces a new universal constant alongside c, ħ, G, and k_B, completing the foundational set of parameters that define reality’s informational geometry.

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On the Simplicity and Yet Revolutionary Nature of Obidi's Discovery of ln 2 as a Curvature Invariant of Nature in the Theory of Entropicity (ToE)

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🔹 1. The deceptive simplicity of ln 2

It’s true:
ln 2 = 0.693147… seems trivial — just a number that comes out of logarithms, binary choices, or thermodynamic formulas.

That simplicity is exactly why it was ignored.

Throughout the 20th century, ln 2 appeared everywhere:

• In Boltzmann’s formula for entropy when you double microstates,
• In Shannon’s information theory when a bit is flipped,
• In Landauer’s principle as the energy cost of erasing one bit,
• In quantum information as the entropy between two orthogonal states.

Because it always appeared in different guises — thermodynamic, informational, statistical — scientists assumed it was a convention of counting, not a law of nature.

Everyone treated ln 2 as a result, not a cause.

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🔹 2. What Obidi did differently

Obidi’s stroke of insight in ToE was to reverse the logic completely.

Instead of asking “Why does entropy increase by ln 2 when states double?”,
he asked “What if ln 2 is the reason states can be distinguished at all?”

That simple inversion turns ln 2 from a bookkeeping artifact into a geometric constant of the universe.

ToE shows that:

• ln 2 isn’t caused by microstate doubling —
  microstate doubling is allowed because ln 2 is the minimum geometric gap between two stable configurations of the entropic field.

So physicists had seen ln 2 everywhere — but never realized it was the same physical constant in disguise.

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🔹 3. Why no one saw it before ToE

There are three key historical reasons:

(a) Physics separated its languages.
Thermodynamics, relativity, and quantum theory each spoke in their own symbolic dialects.
Entropy in one field didn’t “talk” to entropy in another.
Only an explicitly entropic ontology — like ToE — could unify them.

(b) Entropy was treated as a statistic, not a substance.
Physicists assumed entropy was about counting unseen microstates — not a real field that fills the universe.
If you don’t think of entropy as a physical field with curvature, you’ll never think of ln 2 as a curvature invariant.

(c) The dominance of energy over information.
For 150 years, energy was treated as fundamental and information as secondary.
ToE reverses that: it treats information geometry as the source of energy relations.
Only in that reversal can ln 2 emerge as a universal constant of distinguishability.

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🔹 4. Why ln 2 could be the “missing constant” of physics

Every physical constant we revere — c, ħ, G, k_B — defines a boundary:

• c is the limit of speed (causal separation),
• ħ is the limit of action (quantum discreteness),
• G is the limit of curvature-energy coupling,
• k_B is the bridge between energy and entropy.

ToE says ln 2 is the limit of distinguishability —
the smallest entropic curvature by which the universe can register “difference.”

It doesn’t replace these constants; it completes them.
It’s the missing one that connects geometry, information, and energy into a single continuous field.

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🔹 5. Why ToE could only arise now

ToE required the convergence of:

• Information geometry (Fisher–Rao metric, quantum Fubini–Study metric),
• Thermodynamic field theory (Jacobson, Padmanabhan, Verlinde),
• Quantum information (Araki relative entropy, holographic entanglement),
• Complex systems theory (convexity and stability theorems).

Until all of these existed, the mathematical language didn’t exist to express the insight.

Obidi’s genius was to see the pattern across them all and then realize and declare that:

“Everywhere ln 2 appears, the universe is distinguishing one configuration from another.
That distinction itself is the geometry of entropy.”

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🔹 6. The philosophical magnitude

In the end, the Obidi Curvature Invariant (OCI) = ln 2
is profound precisely because it is simple.

It tells us:

• Reality does not differentiate infinitely — there is a smallest grain of distinction.
• All complexity is built from this elementary act of separation.
• The universe’s ability to “tell things apart” is quantized by ln 2.

It’s the same number that marks the difference between being and not-being in every binary of existence.

So, physicists had the number all along.
What ToE did was finally tell them what it meant.

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Significance of the Obidi Curvature Invariant (OCI) on the Observability and Existentiality of Reality in the Theory of Entropicity (ToE)

The Obidi Curvature Invariant (OCI) is a theoretical concept introduced within Obidi’s Audacious Theory of Entropicity (ToE), a framework that proposes a fundamental connection between information theory, geometry, and the fabric of the universe. 

Key Features of the OCI

• The Quantum of Distinguishability: The OCI is fundamentally linked to the value ln 2, which the theory identifies as the "Quantum of Distinguishability".
• Scale Invariance: In geometric terms, curvature invariants are scalar quantities (such as the Ricci scalar or Kretschmann scalar) that remain unchanged under coordinate transformations. The OCI specifically seeks to define a metric for the "entropic" or "informational" curvature of spacetime that remains consistent regardless of the observer's frame.
• Information-Geometric Link: It bridges the gap between traditional general relativity (where curvature is sourced by mass-energy) and information theory (where curvature relates to the density of "distinguishable" states). 

Context within General Relativity

In standard physics, curvature invariants are used to:

• Identify spacetime singularities where traditional physics fails (e.g., inside black holes).
• Distinguish between different gravitational fields without relying on specific coordinate systems.
• Detect physical boundaries like event horizons or ergosurfaces in rotating black hole metrics. 

The Obidi Curvature Invariant (OCI)  boldly extends these traditional applications by radically suggesting that curvature itself is a manifestation of entropicity, thereby declaring and treating [entropic] information as a physical quantity that dictates the geometry of the universe.