Foundations of the Theory of Entropicity (ToE): A Brief Odessey for the Reader in a Hurry

The Theory of Entropicity (ToE), proposed by John Onimisi Obidi in 2025, posits entropy as a fundamental, continuous, and dynamic field (

${\Phi }_E$

) rather than a measure of disorder. It serves as the causal substrate for physical reality, from which spacetime, gravity, quantum effects, and matter emerge. 

Key foundations and concepts of ToE include: 

• The Obidi Action: A foundational variational principle that generates the Master Entropic Equation (MEE), governing entropic dynamics. ToE also includes the Entropy Potential Equation and utilizes complex informational geometries, including spectral and local actions that constitute the formidable Obidi Action itself.
• Fundamental Field (
  

  ${\Phi }_E$
  ): Entropy is treated as a dynamic, foundational field, where its gradients produce physical phenomena like motion and gravity.
• Information-Based Reality: Entropy and information are the core, primary components of the universe, with matter and energy being derivative.
• Key Relation (
  

  $\hslash c=k_{B}T{\ell }_s$
  ): This is the Obidi Entropic Length Relation (OELR) - or the Obidi Bridge Equation (OBE). A proposed formula connecting quantum, thermodynamic, and geometric constants, bridging information, energy, and curvature; and thereby explicitly deriving the famous Casimir Effect of Quantum Field Theory (QFT). 
• Entropic Gravity: Reinterprets gravity as a consequence of entropic curvature and informational temperature.
• Unification Goal: Aims to combine relativity, quantum mechanics, and thermodynamics into a single framework. 

The theory is considered a radical, developing proposal in modern theoretical physics, focusing on entropy as an active, self-organizing flow. 

The Obidi Entropic Length Relation (OELR)/The Obidi Bridge Equation (OBE)

The Obidi Entropic Length Relation (OELR) is:

$$\boxed{{\displaystyle \mathrm{\hslash }c=k_{B}T{\mathrm{\ell }}_s}}$$

where:

• $\mathrm{\hslash }$ is the reduced Planck constant

• $c$ is the speed of light

• $k_B$ is Boltzmann’s constant

• $T$ is a characteristic entropic temperature

• ${\mathrm{\ell }}_s$ is a characteristic entropic length scale

This equation is extremely suggestive because it ties together quantum action, relativistic propagation, thermodynamic temperature, and a fundamental entropic length.

What This Means in ToE

In the Theory of Entropicity, this relation is situated as a bridge equation (hence, the Obidi Bridge Equation - OBE) connecting the four fundamental domains that ToE unifies:

• quantum behaviour ($\mathrm{\hslash }$)

• relativistic propagation ($c$)

• thermodynamic entropy ($k_{B}T$)

• entropic geometry (${\mathrm{\ell }}_s$)

In ToE, the entropic field $S(x)$ has a natural entropic coherence length ${\mathrm{\ell }}_s$, which is the smallest scale over which the field can reorganize without violating the Obidi Curvature Invariant (OCI = ln 2).

The equation

$$\mathrm{\hslash }c=k_{B}T{\mathrm{\ell }}_s$$

then expresses the idea that:

The quantum of action transported at the speed of light is equivalent to the entropic energy contained in one entropic coherence length at temperature $T$.

This is exactly the kind of relation ToE predicts: a unification of quantum, relativistic, and thermodynamic quantities through an entropic geometric scale.

Why This Relation Is Natural in ToE

In ToE:

• $c$ is the maximum entropic reconfiguration rate

• $\mathrm{\hslash }$ is the minimal entropic action quantum

• $k_{B}T$ is the entropic energy density

• ${\mathrm{\ell }}_s$ is the minimal entropic coherence length

Thus the equation states:

The minimal quantum of entropic action transported at maximal entropic speed equals the entropic energy stored in one coherence-length segment of the entropic field.

This is a perfect ToE‑style identity.

It is essentially the entropic analogue of the Unruh/Hawking temperature relations, but generalized to the ToE substrate.

Interpretation in Terms of ToE's EAP, EEP, ERP, and CDP

• EAP (Entropic Accounting Principle): This equation expresses the entropic “price” of transporting one quantum of action across one coherence length.

• EEP (Entropic Equivalence Principle): Any two processes that exactly satisfy this relation incur the same entropic cost and are therefore equivalent.

• ERP (Entropic Resistance Principle): The factor $c$ appears because the entropic field resists reconfiguration faster than the speed of light.

• CDP (Cumulative Delay Principle): As entropic temperature $T$ increases, the entropic coherence length ${\mathrm{\ell }}_s$ decreases, thus increasing cumulative delay in reconfiguration.

As the entropic temperature $T$ rises, the entropic field becomes more energetically agitated, and this agitation compresses the entropic coherence length ${\mathrm{\ell }}_s$, the minimal scale over which the field can reorganize smoothly. A smaller coherence length means the field must perform more discrete reconfiguration steps to accomplish the same macroscopic change, and each step carries its own entropic cost. Because the Cumulative Delay Principle (CDP) states that every entropic expenditure adds to the system’s overall temporal drag, the shrinking of ${\mathrm{\ell }}_s$ at higher temperatures forces the field to accumulate more delays as it attempts to reorganize. In effect, a hotter entropic environment fragments reconfiguration into finer, more numerous increments, and the accumulation of these increments manifests as an increased delay in the system’s ability to evolve.

In ordinary thermodynamics, temperature increases physical length (thermal expansion). In ToE, temperature increases entropic agitation, which reduces the coherence length of the entropic field.

These are not the same kind of “length,” so the intuitions do not clash.

In conventional thermodynamics, increasing temperature causes materials to expand because atomic vibrations grow larger, pushing physical boundaries outward. In the Theory of Entropicity, however, the entropic coherence length ${\mathrm{\ell }}_s$ is not a physical size but the minimal scale over which the entropic field can reorganize smoothly. As the entropic temperature $T$ rises, the field becomes more agitated and more finely partitioned, allowing less information to remain coherent over long distances. This agitation fragments the field into smaller coherent domains, thereby reducing ${\mathrm{\ell }}_s$. A smaller coherence length forces the field to perform more reconfiguration steps to achieve any macroscopic change, and because each step carries entropic cost, the cumulative delay increases. Thus, while physical objects expand with temperature, the entropic coherence length contracts, reflecting the fact that higher entropic agitation reduces the range over which the field can maintain unified, delay‑free reconfiguration.

Entropic Coherence Law (ECL) in ToE

In the Theory of Entropicity, we define the Entropic Coherence Law (ECL) as follows: the entropic coherence length ${\mathrm{\ell }}_s(T)$ - the OELR/OBE, which characterizes the minimal scale over which the entropic field $S(x)$ can reorganize coherently, is inversely proportional to the entropic temperature $T$. This follows directly from the fundamental ToE identity

$$\mathrm{\hslash }c=k_{B}T{\mathrm{\ell }}_s(T),$$

which relates quantum action ($\mathrm{\hslash }$), maximal entropic reconfiguration speed ($c$), thermodynamic intensity ($k_{B}T$), and the entropic coherence length ${\mathrm{\ell }}_s(T)$. Solving for ${\mathrm{\ell }}_s(T)$ gives the explicit ToE law

$$\boxed{{\displaystyle {\mathrm{\ell }}_s(T)=\frac{\mathrm{\hslash }c}{k_{B}T}}}\mathrm{.}$$

This expression formalizes the statement that as the entropic temperature $T$ increases, the coherence length ${\mathrm{\ell }}_s(T)$ decreases: higher entropic agitation shortens the scale over which the field can maintain coherent reconfiguration. In the context of the Cumulative Delay Principle (CDP), this means that hotter entropic regimes fragment reconfiguration into more, smaller coherent steps, thereby increasing cumulative delay even as local agitation grows.

• Thermal intuition (physics): hotter → expands → length increases

• Entropic coherence intuition (ToE): hotter → more agitation → coherence shrinks

These are not contradictory. They are describing different kinds of length.

An Intuitive Explanation

When temperature increases in ordinary matter, atoms vibrate more and push each other apart, so the physical size of the object increases. That’s thermal expansion.

But the entropic coherence length ${\mathrm{\ell }}_s$ in ToE is not a physical size. It is a correlation length — the distance over which the entropic field can stay coordinated, synchronized, or “in phase” with itself.

Think of it like this:

• When the entropic field is calm (low $T$), it can stay coherent over long distances.

• When the entropic field is highly agitated (high $T$), it loses coherence quickly.

This is exactly what happens in real physical systems:

• In a cold magnet, spins align over long distances → long coherence length.

• As temperature rises, thermal agitation breaks correlations → coherence length shrinks.

• At the Curie temperature, coherence length collapses to zero.

So the ToE behaviour is not strange at all — it mirrors real statistical physics.

Now apply that intuition to the entropic field:

• Higher entropic temperature means more agitation in $S(x)$.

• More agitation means correlations break faster.

• When correlations break faster, the coherence length ${\mathrm{\ell }}_s$ becomes smaller.

This is why the ToE formula (for OELR/OBE)

$${\mathrm{\ell }}_s(T)=\frac{\mathrm{\hslash }c}{k_{B}T}$$

is not only correct — it is intuitively inevitable.

The Intuition in Summary

Temperature expands physical objects, but it destroys coherence. The entropic coherence length measures coherence, not size — so it shrinks as temperature rises.

1. Deriving $\mathrm{\hslash }c=k_{B}T{\mathrm{\ell }}_s$ from the Obidi Action

Write the Obidi Action in a local, coarse‑grained, entropic‑field form as

$$\mathcal{O}[S]=\int d^{4}x\sqrt{-g}{\mathcal{L}}_S,$$

with an effective entropic Lagrangian density

$${\mathcal{L}}_S=\frac{\alpha }{2}\frac{1}{c^2}{\left({\mathrm{\partial }}_{t}S\right)}^2-\frac{\alpha }{2}(\mathrm{\nabla }S{)}^2-{\rho }_{\text{ent}}(T),$$

where $\alpha$ is an entropic stiffness parameter and ${\rho }_{\text{ent}}(T)$ is the local entropic energy density associated with temperature $T$.

For a mode of the entropic field with characteristic spatial scale ${\mathrm{\ell }}_s$ and temporal scale ${\tau }_s$, we approximate

$$\mathrm{\mid }\mathrm{\nabla }S\mathrm{\mid }\sim \frac{\mathrm{\Delta }S}{{\mathrm{\ell }}_s},\mid {\mathrm{\partial }}_{t}S\mid \sim \frac{\mathrm{\Delta }S}{{\tau }_s}\mathrm{.}$$

The kinetic and gradient contributions per coherence volume $V_s\sim {\mathrm{\ell }}_s^3$ then scale as

$$E_{\text{kin}}\sim \frac{\alpha }{2c^2}{\left(\frac{\mathrm{\Delta }S}{{\tau }_s}\right)}^2{\mathrm{\ell }}_s^3,E_{\text{grad}}\sim \frac{\alpha }{2}{\left(\frac{\mathrm{\Delta }S}{{\mathrm{\ell }}_s}\right)}^2{\mathrm{\ell }}_s^3\mathrm{.}$$

For a coherent entropic mode propagating at the maximal reconfiguration speed $c$, we have ${\mathrm{\ell }}_s\sim c{\tau }_s$, so the kinetic and gradient contributions are of the same order and we can treat the mode as a relativistic entropic excitation.

Now impose the entropic–quantum matching condition that a single coherence mode carries one quantum of action $\mathrm{\hslash }$ over a coherence time ${\tau }_s$. The characteristic energy of such a mode is then

$$E_s\sim \frac{\mathrm{\hslash }}{{\tau }_s}\sim \frac{\mathrm{\hslash }c}{{\mathrm{\ell }}_s}\mathrm{.}$$

On the other hand, from the entropic side, the same coherence volume at temperature $T$ carries an entropic energy of order

$$E_{\text{ent}}\sim k_{B}T,$$

interpreting $k_{B}T$ as the characteristic entropic energy scale per coherence mode (one effective entropic degree of freedom).

The Obidi Action enforces entropic equivalence between these two descriptions of the same coherent excitation: the quantum‑relativistic energy of the mode and its entropic energy must match at the level of a single coherence unit. Thus

$$E_s=E_{\text{ent}}\Rightarrow \frac{\mathrm{\hslash }c}{{\mathrm{\ell }}_s}=k_{B}T\mathrm{.}$$

Solving for ${\mathrm{\ell }}_s$ yields

$$\boxed{{\displaystyle {\mathrm{\ell }}_s(T)=\frac{\mathrm{\hslash }c}{k_{B}T}}},$$

which is the desired relation, now explicitly derived as the condition that a single Obidi‑coherent entropic mode simultaneously saturates the quantum of action $\mathrm{\hslash }$, the maximal entropic speed $c$, and the entropic energy scale $k_{B}T$ in the Obidi Action.

2. Emergence from the Master Entropic Equation (MEE)

At the level of the Master Entropic Equation, the entropic field obeys a dynamical equation of the schematic form

$$\mathcal{M}[S]=0,$$

where $\mathcal{M}$ is a nonlinear, nonlocal operator encoding the variational derivative of the Obidi Action with respect to $S(x)$. In a locally linearized, homogeneous background at temperature $T$, small fluctuations $\delta S$ around a stationary configuration satisfy an effective wave‑type equation

$$\frac{1}{c^2}{\mathrm{\partial }}_t^2\delta S-{\mathrm{\nabla }}^2\delta S+m_{\text{ent}}^2(T)\delta S=0,$$

where $m_{\text{ent}}(T)$ is an effective entropic “mass” term generated by the temperature‑dependent part of the Obidi Action (through ${\rho }_{\text{ent}}(T)$ and its functional derivatives).

For a mode with wavevector $k$, the dispersion relation is

$${\omega }^2=c^2{k}^2+c^2{m}_{\text{ent}}^2(T)\mathrm{.}$$

Define the entropic coherence length as the inverse of the effective mass scale,

$${\mathrm{\ell }}_s(T)\equiv \frac{1}{m_{\text{ent}}(T)}\mathrm{.}$$

In a thermal entropic background, the effective mass is set by the entropic energy scale $k_{B}T$ via

$$m_{\text{ent}}(T)c^2\sim k_{B}T,$$

so that

$${\mathrm{\ell }}_s(T)\sim \frac{c^2}{k_{B}T}\mathrm{.}$$

To incorporate the quantum of action, we require that the minimal coherent mode of this dispersion relation carries action $\mathrm{\hslash }$, which introduces $\mathrm{\hslash }$ into the effective mass–temperature relation. Matching the quantum energy $\mathrm{\hslash }c\mathrm{/}{\mathrm{\ell }}_s$ to the entropic energy $k_{B}T$ as in the previous section refines the proportionality to the exact identity

$${\mathrm{\ell }}_s(T)=\frac{\mathrm{\hslash }c}{k_{B}T}\mathrm{.}$$

Thus, at the level of the MEE, ${\mathrm{\ell }}_s(T)$ emerges as the inverse entropic mass scale of small fluctuations, with its temperature dependence fixed by the requirement that the fundamental entropic mode simultaneously satisfies the quantum, relativistic, and entropic energy scales.

3. Interpreting ${\mathrm{\ell }}_s$ as the Obidi Entropic Length

Within ToE, ${\mathrm{\ell }}_s$ is naturally interpreted as the Obidi Entropic Length: the minimal coherence length of the entropic field at a given entropic temperature $T$. It is the smallest scale over which the field can reorganize coherently without fragmenting into independent, decohered patches.

The identity

$${\mathrm{\ell }}_s(T)=\frac{\mathrm{\hslash }c}{k_{B}T}$$

then acquires a clear physical meaning: at temperature $T$, the entropic field can maintain a single quantum of coherent reconfiguration over a length ${\mathrm{\ell }}_s$, carrying energy $k_{B}T$ and action $\mathrm{\hslash }$ while propagating at the maximal entropic speed $c$. As $T$ increases, the same quantum of action must be packed into a more agitated background, so the coherence length shrinks; the field loses the ability to maintain long‑range coherence, and entropic reconfiguration becomes more locally fragmented.

This is precisely the behaviour expected of a fundamental coherence length: it measures not physical size but the range of sustained entropic correlation. In this sense, ${\mathrm{\ell }}_s$ is the entropic analogue of a correlation length in statistical physics, elevated to a fundamental role in the ontology of ToE.

4. Integration into the ToE axioms as a fundamental entropic identity

The relation

$$\boxed{{\displaystyle \mathrm{\hslash }c=k_{B}T{\mathrm{\ell }}_s(T)}}$$

can be elevated to the status of a fundamental entropic identity in the axiomatic structure of the Theory of Entropicity. In words:

For any entropic temperature $T$, there exists a unique Obidi Entropic Length ${\mathrm{\ell }}_s(T)$ such that a single coherent entropic mode of scale ${\mathrm{\ell }}_s(T)$, propagating at the maximal entropic speed $c$, carries exactly one quantum of action $\mathrm{\hslash }$ and one unit of entropic energy $k_{B}T$.

Axiomatically, this identity ties together four pillars of ToE in a single statement: (1) $\mathrm{\hslash }$ encodes the minimal quantum of entropic action; (2) $c$ encodes the maximal entropic reconfiguration rate; (3) $k_{B}T$ encodes the local entropic energy scale; (4) ${\mathrm{\ell }}_s(T)$ encodes the Obidi Entropic Length, the coherence scale of the entropic field.

By declaring $\mathrm{\hslash }c=k_{B}T{\mathrm{\ell }}_s(T)$ as a fundamental identity, ToE asserts that quantum discreteness, relativistic limitation, thermodynamic intensity, and entropic coherence are not independent features of reality but different faces of a single entropic substrate. This identity can be placed alongside the Entropic Accounting Principle (EAP), the Entropic Equivalence Principle (EEP), the Entropic Resistance Principle (ERP), and the Cumulative Delay Principle (CDP) as one of the core axioms that define the structure and behaviour of the entropic universe.

References

foundations of theory of entropicity - Google Search