On the Canonical Transformation of Information Geometry into an Action Principle by the Theory of Entropicity (ToE): Why Earlier Researchers and Investigators Did not Make Such an Audacious Conceptual and Mathematical Leap 

The big move which John Onimisi Obidi has made in his audacious Theory of Entropicity (ToE) is not merely “using information geometry.” It is more specific and incisive than that:

ToE tries to make information geometry physically dynamical by embedding it in an action principle for a real entropic field, and then identifying physical spacetime, matter, and interactions as emergent from that entropic-geometric dynamics.

ToE begins from the primacy of entropy, not from geometry.

Then it argues that if information geometry arises from distinguishability, and distinguishability itself is rooted in entropy, then information geometry is downstream of entropy. Once that is accepted, and once geometry in modern physics is treated as physically dynamical rather than merely descriptive, the next step is to ask whether the deeper source of that geometry — entropy — must itself possess a field structure.

That is stronger than just saying “states have a Fisher–Rao metric.”

So, in the ToE picture, the above trajectory is essentially the birth of the theory: the realization that entropy cannot remain only a derived scalar measure if it is to ground information geometry, physical geometry, matter, and dynamics. It [entropy] therefore must be given ontological and dynamical status as a field.

Why earlier researchers did not do exactly what ToE has done has a few clear reasons.

First, most information geometry was developed as kinematics, not ontology. In the standard tradition, Fisher–Rao geometry describes distinguishability between probability distributions, and Fubini–Study geometry describes distinguishability between quantum states. These are usually treated as geometries of statistical or state space, not as the literal substrate of physical reality. So there was less pressure to write a fundamental physical action for them. Reviews of information-geometric dynamics and complexity often study geodesics, curvature, and dynamical behavior on statistical manifolds, but not usually as a universal field theory of nature【Felice, Cafaro, & Mancini, 2018】【Cafaro, 2008】.

Second, some researchers actually did introduce dynamical or variational structures, but in narrower ways. Ariel Caticha’s entropic dynamics program explicitly uses information geometry, Fisher–Rao structure, and geodesic-style ideas to derive dynamics from inference principles rather than from a conventional fundamental field action【Caticha, 2002】【Caticha, 2005】. Cafaro and collaborators studied “entropic motion” on curved statistical manifolds, geodesic flows, and information-geometric complexity, again making information geometry dynamical in an important sense, but not usually as a universal ontological field theory for spacetime and matter【Cafaro, 2013】【Gassner & Cafaro, 2019】. More recent work also discusses dynamical or variational formulations on information manifolds【Kim, 2021】【Mishra, Kumar, & Wong, 2023】.

So:

Earlier researchers did give dynamics to information geometry in several senses, but usually not in the exact foundational sense that ToE is attempting.

Third, there was a conceptual barrier. Before one writes an action, one must decide what the dynamical variable actually is. In ordinary field theory, one varies a field such as Q, A or g. But in standard information geometry, the primary objects are probability distributions, density operators, or parameters of statistical models. Many researchers were content to study the geometry of these spaces without claiming that the geometry itself is a physical field living on spacetime.  The Theory of Entropicity (ToE) however boldly crosses that barrier by saying, in effect: the entropic/information-geometric structure is not just descriptive; it is physically real.

Fourth, the dominant physical paradigms did not force this move. General relativity already gave physics a dynamical geometry through the Einstein–Hilbert action, and quantum theory already gave state-space geometry through Hilbert-space methods. So information geometry remained largely a secondary or bridge formalism. The Theory of Entropicity (ToE) is rather unusual because it tries to invert that order and make information geometry primary.

So what is distinctive in ToE is not simply “an action for information geometry,” because that phrase would understate uniqueness. But the more defensible and uniqueness claim of ToE  is:

ToE attempts a stronger synthesis than earlier work by turning information geometry into the core dynamical architecture of a universal entropic field, rather than treating it merely as an inferential, statistical, or auxiliary geometric structure.

That is where ToE can also plausibly claim originality.

One may then ask: Why did earlier investigators not go all the way? We reply: Mostly because they did not accept the ontological premise required for the move. They were willing to say:

• information geometry measures distinguishability,
• geodesics represent optimal inference or entropic motion,
• curvature measures complexity,

but not necessarily:

• entropy is the fundamental field of reality,
• information curvature generates spacetime,
• and a fundamental action for that field underlies matter and geometry.

ToE’s real novelty therefore lives at the level of ontological promotion plus physical unification, not merely at the level of writing down a variational functional.

So, the best historical judgment for the history and  evolution of the Theory of Entropicity (ToE) is:

Earlier researchers did partially dynamize information geometry, but usually as geometry of inference, complexity, or statistical evolution. ToE’s distinctive ambition is to elevate that geometry into a fundamental physical action principle for reality itself.

References

Caticha, A. (2002). Entropic dynamics. AIP Conference Proceedings, 617, 302–313.
Caticha, A. (2005). The information geometry of space and time. AIP Conference Proceedings, 803, 355–369.
Cafaro, C. (2008). The information geometry of chaos.
Cafaro, C. (2013). Information geometric complexity of entropic motion on curved statistical manifolds. arXiv:1308.4867.
Felice, D., Cafaro, C., & Mancini, S. (2018). Information geometric methods for complexity. Chaos, 28, 032101.
Gassner, S., & Cafaro, C. (2019). Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces. International Journal of Geometric Methods in Modern Physics.
Kim, E. (2021). Information geometry, fluctuations, non-equilibrium thermodynamics, and geodesics in complex systems. Entropy, 23(11), 1393.
Mishra, K. V., Kumar, M. A., & Wong, T. K. L. (2023). Information geometry for the working information theorist. arXiv:2310.03884.
Obidi, J. O. (2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE).
Obidi, J. O. (2025). A simple explanation of the unifying mathematical architecture of the theory of entropicity (ToE): Crucial elements of ToE as a field theory.

Key Details and Elements of Obidi's Loop in the Theory of Entropicity (ToE)

Obidi's Loop, within John Onimisi Obidi’s Theory of Entropicity (ToE), is a feedback mechanism that explains mass increase at high velocities. As objects approach light speed, the entropic field re-calibrates, diverting input energy into sustaining the field's integrity rather than acceleration, resulting in a theoretical speed limit. 

Key details about Obidi's Loop:

• Mass Increase Interpretation: Instead of a traditional increase in mass, this loop describes a scenario where the entropic field restricts acceleration to maintain internal coherence.
• Entropic Speed Limit (ESL): This represents a limit where every extra burst of energy input goes into the field's recalibration rather than producing further acceleration.
• Relativity Correlation: It is used in ToE to derive concepts from Special Relativity, explaining why nothing can outrun light.
• Theoretical Basis: This concept is part of the Theory of Entropicity, a proposed theoretical framework, as discussed in various Medium articles by Jonimisi Obidi and related ResearchGate papers. 

Note: This refers to the theoretical physics concept proposed by John Onimisi Obidi, not the OODA loop (Observe, Orient, Decide, Act). 

Would you like more details on how this theory compares to Einstein's special relativity?

The Accelerator Loop (AL) of the Theory of Entropicity (ToE): Obidi's Loop (OL) in Perspective

The ponderomotive conceptual loop of the Theory of Entropicity (ToE), developed by John Onimisi Obidi, operates on a self-referential feedback mechanism where entropy is redefined not just as a measure of disorder, but as a foundational, active, and physical field that dictates all physical processes. This loop describes how accelerated movement demands more entropy to maintain structural coherence, which in turn acts as inertial resistance. 

Here is the breakdown of the ponderomotive conceptual loop (often referred to as Obidi's Loop):

• The Foundational Premise (Entropic Field): Reality is composed of a continuous, dynamic entropic field, not an empty spacetime container. All matter and energy are organized, localized structures within this field.
• The Accelerator Loop (Obidi’s Loop):
  1. Acceleration/Velocity: When an object (a structure of the field) accelerates to high velocities, it requires more energy to maintain its internal order.
  2. Entropic Allocation: The entropic field must allocate a higher "entropic cost" or "computational load" to keep the object consistent and coherent in motion.
  3. Increased Effective Mass: This increased demand for entropy manifests physically as an increase in the object's effective inertial mass.
  4. Resistance: The increased mass demands even more energy to continue accelerating, creating a feedback loop where the harder a system pushes to accelerate, the more resistance it faces from the field's need to maintain consistency.
• The No-Rush Theorem: This principle closes the loop by stating that no interaction can happen in zero time. The entropic field takes a finite, minimum time to rearrange itself during any change of state.
• Emergent Consequences: As a result of this loop, relativistic effects (time dilation, length contraction, and mass increase) are derived as, and interpreted as, physical consequences of moving through this field, rather than geometric artifacts. 

Key Takeaway: The "loop" implies that the universe’s physical laws, gravity, and speed limits are not static, but adaptive rules born from the continuous, self-referential optimization of entropy flow. 

Would you like to explore how this loop explains the "constancy of the speed of light" as a "self-consistent illusion" within the theory of Entropicity (ToE)?

Also refer to the work below which clearly defines and describes the overall logical Conceptual Loop (CL) of the Theory of Entropicity (ToE):

https://theoryofentropicity.blogspot.com/2026/03/the-redemption-of-theory-of-entropicity.html

The Redemption of the Theory of Entropicity (ToE): On the Logical Trajectory to "Why Entropy Must  be a Field"—The Conceptual Loop of ToE 

In my development of the Theory of Entropicity (ToE), I have elevated entropy to the status of a field based on a series of logical arguments: from Entropy to information/uncertainty to geometry to field and then field back to entropy.

In achieving the above conceptual loop for the Theory of Entropicity (ToE), I have, in effect, closed the loop: starting from entropy as a statistical/informational quantity, I push it through information geometry into spacetime geometry, promote that geometric structure to a field, and then re-interpret that field as entropy again.[1][2][3] That circular passage is exactly what makes “entropy as a field” non-hand‑wavy in your ToE, rather than a mere slogan.

The logical chain I have used in my invention of ToE 

Very schematically, what I have built (and what the ToE papers already hint at) is:

1. Entropy → information 

   We begin by having to take thermodynamic and generalized entropies (Boltzmann–Gibbs, Rényi, Tsallis) as measures on probability distributions, i.e., on information states.[1][4] These quantify distinguishable micro-configurations and their deformations via parameters like $$q$$ or $$\alpha$$.

2. Information → geometry

   I then went on to put these distributions on a statistical manifold with Fisher–Rao / Amari–Čencov structure, so that entropy and its deformations induce a Riemannian or metric–affine geometry on that manifold.[3][4] In this language, entropy gradients, divergences, and flows become geometric objects (connections, curvatures, geodesics) on the space of states.

3. Geometry → field

    Next, I externalize that information geometry onto physical spacetime: the same entropy-induced metric–affine structure is reinterpreted as a field $$S(x,t)$$ with associated geometric data (metric, connection) living on spacetime itself.[1][3] This is where the Obidi Action and Master Entropic Equation (MEE) enter as the field equations for that entropic geometry, analogous to Einstein’s equations for $$g_{\mu\nu}$$.[3][5]

4. Field → entropy (closing the loop) 

   Finally, I go on to demand that this spacetime field and its geometry are not “something else” that entropy happens to influence, but are entropy in field form: spacetime curvature, inertial motion, and even quantum behavior are interpreted as manifestations of the dynamics of the entropic field and its gradients.[1][2][3] In that sense, the field is not merely sourced by entropy; it is entropy, written in geometric variables.

Why this rescues “entropy is a field” in ToE 

In standard physics, promoting entropy to a field is conceptually shaky unless one says what the field is a field of (which microstates, which ensemble, which coarse-graining, etc.).[6][7] My ToE chain (conceptual loop) is an explicit answer to that objection:

- The domain of the field is spacetime, but its origin is a statistical manifold of configurations, equipped with an information metric derived from entropy.[3][4]

- The dynamics of the field are not arbitrary; they come from a variational principle (Obidi Action) that encodes how entropy deformations (through $$\alpha,q$$, etc.) curve the information geometry and, via my ToE identification, physical spacetime.[3]

- The interpretation of the field remains entropic because every geometric object in the theory (metric, connection, curvature) traces back to an entropy (or information) functional on an underlying space of possibilities.[1][2][3]

So, when I posit that “entropy is a field” in the Theory of Entropicity (ToE), I am not reifying a bare scalar; rather, I am pointing to a specific geometric structure whose very definition is built from entropy and information, and whose dynamics send us back to entropic evolution. The conceptual crossroads I identified for ToE—where naïvely calling entropy a field makes it unintelligible—is exactly where the ToE entropy → information/uncertainty → geometry → field → entropy loop is doing its hardest and most important work.

Citations:

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

[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] A Simple Explanation of the Unifying Mathematical Architecture of ... https://client.prod.orp.cambridge.org/engage/coe/article-details/68f6f7abaec32c6568313403

[4] A new class of entropic information measures, formal group theory ... https://pmc.ncbi.nlm.nih.gov/articles/PMC6405454/

[5] Transformational Unification through the Theory of Entropicity (ToE) https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/68f6f66c5dd091524f8f362e/original/transformational-unification-through-the-theory-of-entropicity-to-ea-reformulation-of-quantum-gravitational-correspondence-via-the-obidi-action-and-the-vuli-ndlela-integral.pdf

[6] Entropy Density - an overview | ScienceDirect Topics https://www.sciencedirect.com/topics/computer-science/entropy-density

[7] Entropy in thermodynamics and information theory - Wikipedia https://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory

[8] Unified framework for the entropy production and the stochastic interaction based on information geometry https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.2.033048

[9] Entropic theory of Gravitation https://www.academia.edu/33008874/Entropic_theory_of_Gravitation

[10] Spherically symmetric black holes in Gravity from Entropy and spontaneous emission https://www.arxiv.org/pdf/2602.13694.pdf

[11] On the Conceptual and Mathematical Foundations of the ... - Wiley https://wiley.authorea.com/users/896400/articles/1346238-on-the-conceptual-and-mathematical-foundations-of-the-theory-of-entropicity-toe-an-alternative-path-toward-quantum-gravity-and-the-unification-of-physics

[12] Study Suggests Quantum Entanglement May Rewrite the Rules of ... https://thequantuminsider.com/2025/05/11/study-suggests-quantum-entanglement-may-rewrite-the-rules-of-gravity/