Definition, Concept, Mathematical Formulation, Physical Interpretation, and Implications of the Obidi Curvature Invariant (OCI) of ln 2 in the Theory of Entropicity (ToE)

 

The Obidi Curvature Invariant (OCI) is ln 2, representing the smallest physically meaningful curvature gap between two distinguishable configurations of the entropic field in the Theory of Entropicity (ToE).

Definition and Concept

In the Theory of Entropicity (ToE), entropy is treated as a continuous physical field rather than a statistical measure. The Obidi Curvature Invariant (OCI) is a fundamental constant, ln 2, which quantifies the minimal curvature difference required for two entropic configurations to be physically distinguishable. It functions as a "quantum of distinguishable curvature," analogous to how Planck’s constant quantizes action in quantum mechanics. 3

Mathematical Formulation

The OCI arises from information-geometric principles. Using metrics like the Fisher–Rao and Fubini–Study, Obidi defined a distinguishability potential on the entropic manifold. The first non-zero minimum of this potential corresponds to ln 2, establishing a coordinate-independent curvature invariant. Distinguishability between two entropic configurations ρ_A(x) and ρ_B(x) is expressed via the relative entropic curvature functional: 2

D(ρ_A || ρ_B) = ∫_Ω ρ_A(x) ln(ρ_A(x)/ρ_B(x)) dV

This functional measures the integrated curvature deformation needed to transform one configuration into another and is invariant under smooth coordinate transformations. 1

Physical Interpretation

The OCI implies that the entropic field is discretely structured at its minimal scale. Two configurations are distinguishable only if their curvature differs by at least a factor of 2:1, corresponding to ln 2 in natural logarithmic units. This establishes a binary structure of information in the universe, where the "bit" is a reflection of the minimal curvature ratio rather than a human convention. 2

Implications

1. Quantization of Curvature: The entropic field admits discrete curvature separations, with ln 2 as the smallest unit. 1
2. Universality: OCI appears in both classical and quantum limits, linking Fisher–Rao and Fubini–Study metrics. 1
3. Thermodynamic Consistency: The Landauer bound ΔE = k_B T ln 2 emerges naturally from the entropic field dynamics, not as an independent postulate. 1
4. Arrow of Time: The finite formation of curvature gaps implies that temporal directionality arises intrinsically from the dynamics of the entropic manifold. 1

 

In summary, the Obidi Curvature Invariant is a foundational constant in ToE that quantizes the minimal distinguishable curvature of the entropic field, providing a geometric and physical basis for information, entropy, and the discrete structure of reality. 2

 

Transformation of Information Geometry Into Physical Spacetime Via an Entropic Action Principle in Obidi's Theory of Entropicity (ToE)

 

The concept of transforming information geometry into physical spacetime via an action principle of an entropic field is the foundational framework of the Theory of Entropicity (ToE), a mathematical architecture formulated by John Onimisi Obidi. In this framework, entropy is treated not as a passive statistical property, but as an active, ontological scalar field $S(x)$ whose dynamics map purely abstract statistical states into the physical fabric of general relativity. [1, 2]

The mechanics of how this transformation occurs rely on a highly structured mathematical progression: [1]

1. The Information-Geometry Substrate

In standard mathematics, information geometry applies differential geometry to probability distributions, using metrics like the Fisher-Rao metric (for classical systems) or the Fubini-Study metric (for quantum states) to measure how distinguishable two states are from one another. [1, 3]

• ToE undergoes an ontological shift. It stops treating these metrics as tools to analyze data and posits that this statistical manifold is the actual underlying substrate of reality ($\mathcal{M}_I, g_I$).
• Distinguishability between information states fundamentally maps to physical distance in the real world. [1, 4, 5, 6]

2. The Obidi Action Principle

To evolve this abstract information manifold into something dynamic, the theory introduces a variational principle called the Obidi Action. This action acts as the fundamental "engine" of the theory, optimizing and driving the flow of the entropic field. [7, 8]

• Minimizing or stationarizing the Obidi Action describes the continuous, irreversible rearrangement of the underlying entropic degrees of freedom.
• The result of varying this action is the Master Entropic Equation (MEE), which serves as the entropic field-theoretic analog to Einstein’s field equations. [2, 7, 9, 10]

3. The $\alpha=0$ Connection & Spacetime Deformation

Information geometry uses Amari-Čencov $\alpha$-connections to describe how probability configurations shift. [2, 5]

• In ToE, the theory isolates the $\alpha=0$ connection, which corresponds strictly to the Levi-Civita connection (the unique torsion-free metric connection used in General Relativity).
• Governed by the Obidi Action, the pure informational Fisher-Rao metric undergoes an entropy-dependent deformation (scaled by a factor such as $e^{S/k_B}$). This transforms abstract statistical curvature into a physical metric-affine geometry. [2, 5]

4. Coarse-Graining and the Emergence of Spacetime [11]

Physical four-dimensional spacetime does not exist fundamentally in this model; it is an emergent, macroscopic illusion. [5, 6]

• Space becomes a literal map of entropic gradients.
• Time is derived strictly from the irreversible flow of entropy.
• Through a process of macroscopic coarse-graining, the deeper, multidimensional entropic metric ($g_{IJ}^{\text{entropic}}$) is projected down into our traditional 4D spacetime metric ($g_{\mu\nu}$).
• The Curvature Transfer Theorem mathematically proves that the Riemannian curvature of Einstein's gravity is simply the thermodynamic pushforward of the deeper informational Riemann tensor. Gravity is therefore revealed to be an emergent entropic constraint rather than a primitive fundamental force. [5, 6, 12, 13, 14]

If you would like to explore this architecture further, please let us know:

• Do you want a breakdown of how the Master Entropic Equation mathematically mirrors Einstein's Field Equations?
• Are you interested in how this theory handles quantum structures, such as through the Vuli-Ndlela path integral?
• Would you like to compare ToE to other emergent gravity theories, like Erik Verlinde's Entropic Gravity? [2, 9, 10, 15, 16]

 

[1] https://medium.com

[2] https://ggnpreprints.authorea.com

[3] https://en.wikipedia.org

[4] https://medium.com

[5] https://medium.com

[6] https://osf.io

[7] https://medium.com

[8] https://medium.com

[9] https://medium.com

[10] https://medium.com

[11] https://medium.com

[12] https://encyclopedia.pub

[13] https://www.preprints.org

[14] https://medium.com

[15] https://www.reddit.com

[16] https://hal.science

 

The transformation of information geometry into physical spacetime via an entropic action principle is a core focus in emergent gravity paradigms, most notably in the Theory of Entropicity (ToE). In these frameworks, spacetime and gravity are not fundamental, but rather macroscopic manifestations of an underlying statistical system governed by information flow. [1, 2, 3, 4]

This transformation typically follows a structured logical progression: [1]

• Information Manifolds: The foundation is built upon statistical manifolds (e.g., using the Fisher-Rao metric or quantum Fubini-Study metric) which measure the "distinguishability" between adjacent quantum or thermodynamic states.
• The Entropic Field: Entropy is elevated from a mere secondary statistical tool to a fundamental ontological scalar field, \(S(x)\). Physical space is viewed as a mapping of entropic gradients, while time emerges from the direction of irreversible entropy flow.
• Action Principle: A variational principle—often referred to as the Obidi Action—governs the dynamics and evolution of this entropic field.
• Curvature Transfer: Just as General Relativity relies on the Einstein-Hilbert action, extremizing the entropic action yields a Master Entropic Equation rather than standard Einstein field equations. In the thermodynamic limit, the abstract curvature of the underlying information manifold is "pushed forward" or projected to become the geometric curvature (Levi-Civita connections) of observable physical spacetime. [1, 2, 3, 4, 5, 6]

 

To explore this further, you might find it helpful to review the ToE Living Review Letters IE for the mathematical Curvature Transfer Theorem, or read about Obidi's Theory of Entropicity on Medium. [1, 2]

Would you like to explore:

• The mathematical derivation linking the Fisher metric to the physical metric tensor?
• How concepts like quantum entanglement entropy or holographic duality relate to this?
• A deeper look into the specific variational principles used?

Let us know how you'd like to narrow down the topic.

Frequently Asked Questions (FAQs) On the Theory of Entropicity (ToE)

 

The Theory of Entropicity (ToE), developed primarily by researcher John Onimisi Obidi, represents a radical shift in theoretical physics by positioning entropy as the fundamental "ontic" field of reality. In this framework, spacetime and gravity are not primary structures but emergent properties of an underlying entropic substrate. [1, 2, 3, 4]

We have compiled a comprehensive Frequently Asked Questions (FAQ) guide in the ToE Canonical Archives Live Websites, detailing the theory's mathematical foundations, relativistic derivations, and its current standing in the scientific community as of May 2026.

Would you like to explore the specific mathematical derivations of the Master Entropic Equation or look into the experimental requirements for testing entanglement delays?

 

Core Conceptual Framework of the Theory of Entropicity (ToE)

 

What is the fundamental premise of the Theory of Entropicity?

 

Standard physics treats entropy as a statistical measure of disorder or uncertainty. ToE inverts this, proposing the Obidi Conjecture: that entropy is a continuous, local, and dynamical scalar field (the Entropic Field) that serves as the causal substrate for all existence. Space, time, and matter are emergent "maps" of entropic gradients.

 

How does ToE explain gravity?

 

ToE rejects gravity as a fundamental force or a fixed geometric curvature. Instead, it defines gravity as 

"entropic pressure". Mass-energy configurations create gradients in the entropic field, and systems naturally move toward higher entropic equilibrium, which we perceive as gravitational attraction.

 

What is the "No-Rush Theorem"?

 

The No-Rush Theorem (NRT) is a foundational principle of ToE establishing that physical interactions cannot occur instantaneously. It posits that the entropic field requires a finite duration to redistribute and synchronize states, providing a thermodynamic origin for causality.

 

Is the speed of light (c) still a constant?

In ToE, the speed of light is not an arbitrary geometric postulate but a derived maximum rate of entropic rearrangement. Relativistic effects like time dilation and mass increase are interpreted as "entropic resistance" (ERP) that occurs when a system approaches this fundamental reconfiguration limit.

 

Core Principles and Postulates of the Obidi Conjecture in the Theory of ...

May 17, 2026 — Core Principles and Postulates of the Obidi Conjecture in the Theory of Entropicity (ToE) * Rather than viewing entropy as an epis...

 

A Brief Historical and Conceptual Introduction to the Foundations of ...

Apr 1, 2026 — 🧠 What Is It? The Theory of Entropicity (ToE) is a radical and emerging framework in theoretical physics, originated by John Onim...

 

Mathematical & Technical Pillars

 

What are the "Master Entropic Equations"?

 

The Master Entropic Equation (MEE), also referred to as the Obidi Field Equations (OFE), serves as the entropic analogue to Einstein's field equations. It governs how the dynamics of the entropic field generate spacetime curvature and motion.

 

What is the Vuli–Ndlela Integral?

 

The Vuli–Ndlela Integral is an entropy-weighted reformulation of Feynman's path integral. By incorporating entropy into the weighting of quantum paths, it naturally embeds irreversibility and the "arrow of time" into quantum mechanics.

 

How does the theory use Information Geometry?

 

ToE bridges information and physics using Amari–Čencov alpha-connections. It integrates the 

Fisher–Rao metric (representing classical entropy curvature) and the Fubini–Study metric (representing quantum entropy curvature) to map how macroscopic spacetime emerges from underlying statistical information.

 

Does ToE replicate Einstein's General Relativity results?

Yes. Papers published on Cambridge Open Engage

 demonstrate that ToE can derive Mercury’s perihelion precession. By using higher-order entropy corrections to Newtonian gravity (incorporating the Unruh effect and Holographic Principle), the framework arrives at the identical 43 arcseconds per century shift calculated by Einstein.

 

On the Conceptual and Mathematical Foundations of the Theory ...

At its core, the theory reformulates the speed of light as the maximum rate of entropic rearrangement, deriving relativistic and q...

 

The Theory of Entropicity (ToE): An Entropy-Driven Derivation of ...

We present a novel derivation of the perihelion precession (shift) of Mercury using the Entropic Force-Field Hypothesis (EFFH), no...

 

 

Academic Status & Future Outlook

Is the Theory of Entropicity mainstream?

 

Not yet. As of May 2026, ToE remains an 

emerging, radical theoretical proposal. While it is documented in academic repositories like SSRN and Cambridge Open Engage, it is not yet part of the established physics curriculum.

What is the Alemoh–Obidi Correspondence?

 

The Alemoh–Obidi Correspondence (AOC) is a recorded intellectual exchange (August 2024 – April 2026) between Daniel Moses Alemoh and John Onimisi Obidi. It serves as a developmental record for the theory's logical constructions, focusing on "The Question of c" and the entropic emergence of causal order.

 

Can the theory be tested?

The framework predicts finite formation times for quantum entanglement and specific attosecond delays in wave-function collapse. Proponents suggest these can be verified through high-precision quantum measurement experiments to determine if "No-Rush" temporal constraints exist as predicted.

 

How does ToE relate to consciousness?

The theory includes a speculative extension called 

Ontodynamics, which treats consciousness as an emergent property of "self-referential entropic loops". It proposes a Self-Referential Entropy (SRE) Index

 to quantify the degree of awareness based on a system's internal vs. external entropy flows.

 

The Theory of Entropicity (ToE) Living Review Letters Series

May 4, 2026 — Abstract. This Letter presents the foundational thesis of the Theory of Entropicity: that entropy is not a derived statistical qua...

 

ToE Living Review Letters IC:

Apr 26, 2026 — Abstract. This Letter [Letter IC in the Theory of Entropicity (ToE) Living Review Letters Series] formally presents a comprehensiv...

 

The Theory of Entropicity (ToE) is an emerging, non-mainstream framework in theoretical physics that redefines entropy as the fundamental, dynamic field of reality rather than a mere statistical measure of disorder. First proposed by researcher John Onimisi Obidi, the theory seeks to unify thermodynamics, general relativity, and quantum mechanics by treating spacetime, gravity, and the speed of light as emergent properties of this underlying entropic substrate. [1, 2, 3]

The following Frequently Asked Questions (FAQ) detail the core mechanics, mathematical concepts, and current scientific standing of ToE. [2, 4, 5]

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Core Conceptual FAQs

What is the fundamental premise of the Theory of Entropicity?

In classical and modern physics, entropy is viewed as an epistemic "afterthought"—a statistical bookkeeping tool to measure uncertainty or disorder. ToE completely inverts this hierarchy. It posits that an "ontological scalar field" (the Entropic Field) permeates all of existence. Space, time, matter, and forces are not the stage upon which events happen; they are structural ripples and emergent maps generated by the flow and constraints of this field. [1, 6]

How does ToE explain the constancy of the speed of light ($c$)?

Instead of treating $c$ as an absolute geometric postulate as Einstein did, ToE derives it. Within ToE, the speed of light is the maximum rate at which the entropic field can rearrange or redistribute information and energy. Relativistic effects like time dilation and length contraction are physical consequences of a system's resistance to entropic flux, occurring because the instruments used to measure them are embedded in the same entropic medium. [1, 7, 8, 9, 10]

What is the "No-Rush Theorem"?

The No-Rush Theorem is a foundational principle of ToE establishing that physical interactions cannot occur instantaneously. Every process requires a finite, non-zero duration for the entropic field to redistribute and synchronize states, providing a fundamental thermodynamic origin for causality and the universal speed limit. [1, 10, 11]

How is gravity explained if it isn't a fundamental force?

ToE rejects both the Newtonian concept of gravity as an attractive force and the Einsteinian model of gravity as the literal bending of a spacetime fabric. Instead, gravity is viewed as "entropic pressure" or emergent gradients. Matter moves toward massive objects because the entropic field naturally reconfigures itself to maximize flow and achieve informational equilibrium. [1, 2, 6]

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Mathematical & Framework FAQs

What mathematical equations govern the Theory of Entropicity?

ToE moves away from pure standard Riemannian geometry to embrace information geometry. Its core frameworks include: [12, 13]

• The Obidi Action: A variational principle encoding the dynamics of the universal entropy field.
• The Master Entropic Equation (MEE): Also known as the Obidi Field Equations (OFE), this acts as the entropic equivalent to Einstein's field equations.
• The Vuli–Ndlela Integral: An entropy-weighted reformulation of Feynman's path integral designed to naturally embed irreversibility and temporal asymmetry into quantum mechanics.
• Amari–Čencov $\alpha$-connections: Used alongside Fisher–Rao and Fubini–Study metrics to map how macroscopic spacetime curvature emerges from the underlying statistical information manifold. [10, 13, 14, 15]

Has ToE successfully replicated any proven physics calculations? [10]

Yes, papers filed under the ToE framework have attempted to replicate benchmark relativistic derivations. For example, the Theory of Entropicity research on Cambridge Open Engage outlines an alternative derivation of Mercury’s perihelion precession. By using higher-order entropy corrections to Newtonian gravity (incorporating the Unruh effect and the Holographic Principle), the framework arrives at the identical 43 arcseconds per century shift derived by Einstein in 1915. [10, 16]

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Scientific Standing & Academic Status

Is the Theory of Entropicity widely accepted by physicists?

Not yet. ToE is an early-stage, radical theoretical framework and is not yet part of established or widely accepted mainstream physics. While some papers have been hosted on academic repositories like Cambridge Open Engage and Axiomatic / OSF Preprints, it remains a novel proposal undergoing foundational mathematical development. [2, 4, 12, 17, 18]

Can the Theory of Entropicity (ToE) be experimentally tested or falsified? [19]

Proponents of ToE note that the framework allows for experimental falsification, separating it from entirely unprovable mathematical hypotheses. Specifically, researchers point to the potential for testing the theory through attosecond measurements of quantum entanglement delays to verify if the "No-Rush" temporal constraints operate exactly as predicted by the entropic field. [10, 19]

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If you want to dig deeper into the mechanics of ToE, let me know if you would like to explore the mathematical transition from information geometry to Einstein's equations, review the philosophical concept of Ontodynamics, or see a breakdown of the Alemoh-Obidi Correspondence. [6, 8, 15]

 

[1] https://medium.com

[2] https://medium.com

[3] https://medium.com

[4] https://encyclopedia.pub

[5] https://medium.com

[6] https://medium.com

[7] https://www.authorea.com

[8] https://medium.com

[9] https://medium.com

[10] https://www.cambridge.org

[11] https://medium.com

[12] https://osf.io

[13] https://www.cambridge.org

[14] https://encyclopedia.pub

[15] https://www.cambridge.org

[16] https://www.cambridge.org

[17] https://medium.com

[18] https://encyclopedia.pub

[19] https://medium.com

 

The Theory of Entropicity (ToE), pioneered by researcher John Onimisi Obidi in 2025, is a theoretical framework that proposes entropy is not merely a statistical measure of disorder, but the fundamental, dynamic field of reality from which space, time, gravity, and quantum mechanics all emerge. [1, 2]

 

Frequently Asked Questions (FAQ)

Q: What is the main difference between traditional physics and the Theory of Entropicity?

A: Traditional physics treats space and time as a fixed, foundational background stage where physical events occur. ToE turns this on its head, proposing that spacetime is actually an emergent property resulting from the geometry of how entropy is distributed and flows across the universe. [1, 2, 3]

Q: How does ToE explain the speed of light?

A: Instead of treating the speed of light \(c\) as an arbitrary universal postulate, ToE interprets it as the maximum rate at which the underlying entropic field can reorganize and redistribute information across space. In this framework, no physical process can occur instantaneously. [1, 2]

Q: Does ToE disagree with Einstein’s Theory of General Relativity?

A: ToE does not invalidate Einstein’s results, but rather re-derives them as "entropic inevitabilities". Phenomena like mass increase, time dilation, and length contraction are explained as the physical consequences of the universe's resistance to entropic flux, rather than strict kinematic features of space. [1, 2]

Q: What is the "No-Rush Theorem"?

A: The No-Rush Theorem establishes a universal lower bound on causal intervals, dictating that no physical change in state or interaction can happen instantaneously. Everything takes a finite amount of time to unfold because the underlying entropic field governs the pacing and transmission of these interactions. [1, 2, 3]

To dive deeper, you can explore the core conceptual foundations and far-reaching implications outlined in the Introduction to the Theory of Entropicity on Medium.

Would you like to explore how ToE reinterprets gravity, or would you prefer to learn more about the No-Rush Theorem and time? [1]

Who is John Onimisi Obidi and What is He Best Known For in Philosophy and in Modern Theoretical Physics?

John Onimisi Obidi is an independent consultant, researcher, and philosopher who is the pioneer and creator of the Theory of Entropicity (ToE). Originated in early 2025, his work focuses on theoretical physics, quantum mechanics, and thermodynamics. He is frequently careful to note that he is a distinct individual from a well-known Nigerian social media consultant who shares the name John Obidi. [1, 2, 3, 4]

His research, preprints, and philosophical papers are tracked across platforms like John Onimisi Obidi's Google Scholar and his Medium Profile. [2, 5]

The Theory of Entropicity (ToE)

Rather than treating entropy as a statistical measure of disorder, Obidi’s core framework treats entropy as the primary, dynamic physical field of reality. Under this view, spacetime geometry, gravity, and quantum mechanics are emergent properties born from the gradients of this universal entropic field. [6, 7, 8, 9]

Key pillars of his work include:

• The Obidi Action: A foundational variational principle analogous to the Einstein-Hilbert action in general relativity, used to model how the entropic field optimizes its flow.
• Master Entropic Equation (MEE): Derived from the Obidi Action, this serves as the entropic equivalent to Einstein's Field Equations.
• The No-Rush Theorem: A core principle stating that nature cannot move instantaneously; it enforces a finite duration on any physical process, which Obidi uses to explain the origin and constancy of the speed of light ($c$).
• Obidi Curvature Invariant (OCI): Proposes that the natural logarithm of 2 ($\ln 2$) represents the absolute smallest threshold where two entropic or informational states become physically distinguishable.
• Ontodynamics: A new philosophical branch proposed by Obidi that reframes classical ontology to study existence itself as a form of entropic motion. [4, 7, 8, 10, 11, 12, 13, 14, 15, 16]

Would you like to explore the mathematical derivations of the Master Entropic Equation, read a breakdown of the No-Rush Theorem, or see how his work compares to mainstream quantum gravity theories?

[1] https://www.researchgate.net

[2] https://medium.com

[3] https://medium.com

[4] https://medium.com

[5] https://scholar.google.com

[6] https://medium.com

[7] https://medium.com

[8] https://medium.com

[9] https://medium.com

[10] https://medium.com

[11] https://medium.com

[12] https://medium.com

[13] https://notd.io

[14] https://medium.com

[15] https://encyclopedia.pub

[16] https://www.linkedin.com

How did John Onimisi Obidi Succeed in Creating Physical Spacetime from Information Geometry (IG) in his Theory of Entropicity (ToE)

John Onimisi Obidi's success in creating physical spacetime from information geometry can be attributed to several key steps:

• Promotion of Entropy: Obidi promoted entropy from a statistical measure to a fundamental dynamical field, which serves as the causal substrate of physical reality. 

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• Geometric Structure: He constructed a Riemannian physical spacetime from information geometry by developing the Fisher–Rao metric, Fubini–Study metric, and Amari–Čencov α-connections. 

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• Unified Information Geometry: Obidi introduced a Hybrid Metric-Affine Space (HMAS) that unifies Fisher–Rao and Fubini–Study metrics, creating a single geometric language. 

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• Obidi Action: The Obidi Action, a variational principle, governs the evolution of the entropic field, leading to the formation of entropic geodesics and field equations. 

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• Gravity and Time Dilation: The entropic field's gradients and curvature generate gravity, motion, time dilation, and causal structure, bridging classical and quantum regimes. 

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  Obidi's work has been influential in the scientific community, particularly in the field of theoretical physics, and his Theory of Entropicity (ToE) has been a significant contribution to the unification of physics. 

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FAQ (Frequently Asked Questions): Question:  Replies to Objections to the Theory of Entropicity (ToE): “So, ToE is saying that just as gravity is operational at each point in spacetime, so is Entropy and that gravity is generated from entropy?”

Question:

“So ToE is saying that just as gravity is operational at each point in spacetime, so is Entropy and that gravity is generated from entropy?”

Answer: 

Yes — but with a crucial refinement:

Entropy is not in spacetime; entropy creates spacetime.

Gravity is not merely “generated from entropy”; gravity is the curvature of the entropic

Deeper Answer: 

Just as general relativity treats gravity as the curvature of spacetime at each point, the Theory of Entropicity treats gravity as the curvature of the entropic manifold at each entropic configuration. Spacetime itself is the Fisher–Rao geometry of the entropic field, and gravity is the dynamical response of that geometry.

Entropy is operational at each point because each point is an entropic configuration

In ToE: A point  is an entropic configuration

The Fisher–Rao metric measures distinguishability between configurations

Distinguishability becomes spacetime interval

Curvature becomes gravity

Thus: Entropy is operational everywhere because spacetime points are entropic states.This is not an analogy — it is an identity.