Postulate 1 (Entropic Substrate)

There exists a real scalar field \( S(x^\mu) \), the entropic field, defined on a 4‑dimensional differentiable manifold \( \mathcal{M} \), such that all physical phenomena are determined by the configuration and evolution of \( S(x^\mu) \).

Postulate 2 (Emergence of Physical Observables)

Conventional physical observables—mass \( m \), energy \( E \), momentum \( p^\mu \), and spacetime metric \( g_{\mu\nu} \)—are emergent functionals of the entropic field and its derivatives:

\[ m = m[S],\quad E = E[S],\quad p^\mu = p^\mu[S],\quad g_{\mu\nu} = g_{\mu\nu}[S]. \]

Postulate 3 (Entropic Action Principle)

The dynamics of \( S(x^\mu) \) are obtained from a variational principle on an entropic action:

\[ \mathcal{A}_E[S] = \int_{\mathcal{M}} \mathcal{L}_E\big(S,\partial_\mu S,g_{\mu\nu}\big)\, \sqrt{-g}\,d^4x, \]

where physical evolution satisfies:

\[ \delta \mathcal{A}_E = 0, \]

yielding a Master Entropic Equation (MEE):

\[ \mathcal{E}[S,g_{\mu\nu}] = 0. \]

These constitute the Obidi Field Equations (OFE).

Postulate 4 (Entropic Geodesics)

The physical trajectory \( \gamma \) of any system corresponds to an extremal of an entropic length functional:

\[ \mathcal{L}[\gamma] = \int_{\gamma} \sqrt{G_{ij}(\theta)\,d\theta^i d\theta^j}, \]

where \( \theta^i \) are informational/entropic coordinates and \( G_{ij} \) is an information‑geometric metric.

\[ \frac{d^2\theta^k}{d\lambda^2} + \Gamma^{k}{}_{ij} \frac{d\theta^i}{d\lambda} \frac{d\theta^j}{d\lambda} = 0, \]

with \( \Gamma^{k}{}_{ij} \) an \( \alpha \)-connection.

Postulate 5 (Information Geometry & Generalized Entropy)

\[ g_{ij} = \frac{\partial^2}{\partial\theta^i\partial\theta^j} \mathcal{S}_\alpha(p), \]

where \( \mathcal{S}_\alpha \) is a generalized entropy (Rényi/Tsallis).

Postulate 6 (Maximum Entropic Rearrangement Rate)

\[ J_S^\mu J_{S\mu} \leq c^2 J_S^0 J_S^0. \]

Postulate 7 (Entropic Origin of Relativistic Effects)

\[ S_0 = S_t(v) + S_x(v), \]

\[ S_t(v) = \gamma(v)\,S_0,\quad S_x(v) = (\gamma(v)-1)S_0, \]

\[ \gamma(v) = \frac{1}{\sqrt{1 - v^2/c^2}}. \]

Postulate 8 (Causality & No‑Rush Bound)

\[ \Delta \tau(A,B) \geq \Delta \tau_{\min}. \]

Postulate 9 (Entropic Path Integral)

\[ \mathcal{Z}(S_f,S_i) = \int \mathcal{D}S\, \exp\!\left[ \frac{i}{\hbar}\mathcal{A}_E[S] - \Lambda\,\Delta \mathcal{S}[S] \right]. \]

Postulate 10 (Entropic Unification of Gravitation)

\[ \mathcal{G}[S,g_{\mu\nu}] = 8\pi G\,T_{\mu\nu}[S]. \]

\[ \mathcal{G}[S,g_{\mu\nu}] \to G_{\mu\nu}. \]

\[ G_{\mu\nu} = 8\pi G\,T_{\mu\nu}. \]

How Does \( \ln 2 \) Relate to Distinguishability Potential in ToE?

In ToE, \( \ln 2 \) is the first non‑zero minimum of the distinguishability potential.

Distinguishability Potential in ToE

\[ D(x) = S(x)\,\ln\!\frac{S(x)}{S_0(x)} - S(x) + S_0(x). \]

How \( \ln 2 \) Appears

\[ \rho_B(x) = 2\,\rho_A(x). \]

\[ \text{OCI} = \ln 2. \]

Physical Meaning

• The fundamental quantum of distinguishable curvature.
• The smallest curvature fold that counts as physically distinct.
• A curvature invariant linking information‑theoretic distinguishability to geometry.

What Are the Key Postulates of the Theory of Entropicity (ToE)?

A Beautiful and Concise Introduction to the Foundational Postulates of the Theory of Entropicity (ToE)

Here we provide a concise yet elegant list of the core postulates that define the Theory of Entropicity (ToE) as laid out in its foundational papers.^[1][2][4]

Ontological Postulate: Entropy as the Substrate

1. Entropy Is the Fundamental Physical Field

• Entropy \( S(\Lambda) \) is not a statistical by‑product but a continuous, dynamical field that forms the causal substrate of reality.^[1][4]
• Gradients and flows of this entropic field generate motion, gravitation, time, and information flow.^[1][4]

2. Physical Entities Are Emergent from Entropy

Mass, energy, spacetime geometry, and even consciousness arise as emergent constraints or patterns of a single entropic reality — not as independent primitives.^[4]

Dynamical Postulate: Obidi Action and Master Equations

3. The Obidi Action Governs Entropic Dynamics

• There exists an entropic action functional — the Obidi Action — for the entropy field. Extremizing this action yields the fundamental equations of motion.^[1][4]
• From this action follow:
  • The Master Entropic Equation (MEE),
  • Entropic Geodesics (entropy‑determined paths),
  • The Entropy Potential Equation.^[1][4]

4. Information Geometry Underlies Physical Law

• The dynamical arena is an informational–entropic manifold equipped with Fisher–Rao and Fubini–Study metrics unified via the Amari–Čencov α‑connection.^[1][4]
• Generalized entropies (Rényi, Tsallis) correspond to deformations of this geometry, with the entropic order parameter \( \alpha \) acting as a universal deformation index.^[4]

Relativistic Postulate: Speed of Light as Entropic Rate

5. \( c \) Is the Maximum Rate of Entropic Rearrangement

• The relativistic speed of light is not a primitive postulate; it is the characteristic propagation speed of disturbances in the universal entropic field.^[1][2]
• All causal processes are limited by this finite rate of entropy propagation, yielding Lorentz invariance as an entropic necessity.^[2]

6. Relativistic Effects Are Entropic Resistances

• Mass increase, time dilation, and length contraction arise from how the entropic field redistributes its “entropic budget” between motion and timekeeping.^[1][6]
• The Entropic Resistance Principle, Entropic Resistance Field, and Entropic Accounting Principle encode this redistribution and produce an entropic Lorentz factor that reproduces Einstein’s transformations without assuming geometric postulates.^[1][6]

Causality and Arrow‑of‑Time Postulate

7. No‑Rush Theorem and Causal Bounds

• There is a universal lower bound on causal intervals: the entropic field must first establish conditions before any interaction or information transfer can occur.^[1][2]
• This forbids superluminal processes and ties causality directly to the dynamics of the entropy field.^[2]

8. Fundamental Irreversibility via the Vuli–Ndlela Integral

• Quantum evolution is governed by an entropy‑weighted path integral — the Vuli–Ndlela Integral — a deformation of Feynman’s path integral.^[1][4]
• This introduces intrinsic irreversibility and a built‑in arrow of time at the fundamental level, rather than treating irreversibility as merely statistical.^[1][4]

Unification Postulate

9. Thermodynamics, Relativity, and Quantum Theory Form a Single Entropic Continuum

• When expressed in the entropic–informational language of ToE, Einstein’s field equations appear as a limiting geometric case of the more general entropic dynamics.^[2][4]
• Other “gravity from entropy” approaches (e.g., Bianconi‑style models) are recovered as special instances within this broader entropic framework.^[4]

In our next exposition, we shall wade through the sophisticated mathematical foundations of these postulates.

Postulate 7 — Entropic Origin of Relativistic Effects

For a system of rest entropic capacity \( S_0 \) moving with velocity \( v \), the effective temporal and spatial entropic budgets satisfy:

S_0 = S_t(v) + S_x(v)
  

with

S_t(v) = \gamma(v)\,S_0, 
S_x(v) = \big(\gamma(v) - 1\big) S_0
  

where the entropic Lorentz factor \( \gamma(v) \) is derived to be:

\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
  

This matches Einstein’s relativistic kinematics exactly, but is obtained here without invoking the geometric postulates of the Theory of Relativity (ToR). Time dilation, length contraction, and mass increase are manifestations of entropic redistribution encoded in \( \gamma(v) \).