Supporting Role of the Amari–Čencov Alpha‑Connections in the Derivation of Einstein’s Relativistic Kinematics from Obidi’s Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE), the Amari–Čencov \( \alpha \)-connection does not directly derive relativistic effects. Instead, it provides the geometric framework within which entropic geodesics manifest those effects as emergent consequences of finite entropy propagation along \( \alpha \)-deformed paths.^[2][3]

Primary Mechanism: Entropic Resistance + No‑Rush Theorem

Relativistic effects (Lorentz factor, time dilation, length contraction) arise from three core entropic principles, independent of the \( \alpha \)-connection:

1. Entropic Resistance Principle (ERP): Moving systems face resistance to spatial entropy redistribution, forcing compensatory temporal entropy buildup.
2. Entropic Accounting Principle (EAP): The total entropy budget \( S_0 = S_t(v) + S_x(v) \) is conserved, yielding the Lorentz factor \( \gamma(v) = 1/\sqrt{1 - v^2/c^2} \).
3. No‑Rush Theorem: \( c \) is the maximum rate of entropic rearrangement, imposing causal bounds.

These principles reproduce Einstein’s transformations without geometric postulates. ^[2][1]

The α‑Connection’s Supporting Role

The \( \alpha \)-connection governs the trajectories of entropic configurations \( \theta^i(\lambda) \) through the information manifold:

Key Contributions

• \( \alpha = 0 \) limit: Fisher–Rao (Levi–Civita) geodesics recover classical entropic paths where resistance effects produce the Lorentz factor.
• \( \alpha \neq 0 \) deformations: Rényi/Tsallis entropies \( \alpha = 2(1 - q) \) modify path curvature, but the underlying \( S_0 \)-conservation and \( c \)-bound remain universal.
• Quantum regime (\( \alpha \to 1 \), Fubini–Study): Entropic wavefunctions follow projective geodesics, with relativistic effects encoded in phase‑evolution constraints.

How Geometry and Relativistic Behavior Emerge in ToE

Entropic field S(x) → Information manifold (θᶦ)
           ↓  [α‑connection governs paths]
Entropic geodesics θᶦ(λ) → Emergent spacetime metric g_μν[S]
           ↓  [No‑Rush + Resistance along paths]
Lorentz invariance + γ(v) as entropic inevitabilities

The \( \alpha \)-connection ensures path consistency across classical and quantum regimes, while entropic budget conservation (independent of the connection) enforces relativistic kinematics along those paths.^[2][3][1]

Bottom line: The \( \alpha \)-connection provides the rails; entropic resistance provides the relativistic physics.

Citations

• [1] ToE Derives Einstein’s Relativistic Speed of Light
• [2] ToE Derives and Explains Mass Increase
• [3] Alternative Path Toward Quantum Gravity
• [4] Relativistic Dynamics
• [5] Feynman Lectures: Space‑Time
• [6] Quantum Physics: Theory of Everything
• [7] Einstein Connection of the Unified Theory of Relativity
• [8] α‑Connections in Generalized Geometry

Derivation of the ToE Curvature Invariant ln 2

1. Entropy as a Physical Field

In the Theory of Entropicity (ToE), entropy \( S(x) \) is not a statistical quantity but a continuous physical field permeating spacetime. Information corresponds to a localized curvature or deformation of this field. Each configuration of information is described by an entropic density \( \rho(x) \) defined over a region \( \Omega \) of the entropic manifold, satisfying:

Two informational configurations are distinguishable only if their entropic curvature profiles differ by a finite geometric gap.

2. Distinguishability as Relative Entropic Curvature

ToE defines the measure of distinguishability between two entropic configurations \( \rho_A(x) \) and \( \rho_B(x) \) through the relative entropic curvature functional:

This functional, formally similar to the Kullback–Leibler divergence, is interpreted geometrically as the integrated curvature deformation required to transform one entropic configuration into another. It is non‑negative and invariant under smooth coordinate transformations of the informational manifold.

3. The Binary Curvature Symmetry of the Entropic Field

The simplest stable entropic distinction is binary: a localized region of the field can exist in two minimally distinct configurations, A and B, related by a curvature ratio of 2{:}1. Within their overlap:

This represents the smallest nontrivial deformation of the entropic field capable of supporting two distinct information‑bearing states.

4. Computing the Minimum Entropic Curvature Gap

Substituting the binary relation into the distinguishability functional gives:

Since \( \int_{\Omega} \rho_A(x)\, dV = 1 \), we obtain:

Thus, the smallest nonzero curvature separation between two distinguishable entropic configurations has magnitude:

5. Conversion from Curvature to Physical Entropy

In ToE, Boltzmann’s constant \( k_B \) acts as the universal conversion factor between the dimensionless curvature measure \( D \) and physical entropy \( S \). Hence, the minimal entropy change associated with the smallest distinguishable entropic deformation is:

6. Geometric and Physical Interpretation

Equation \( \Delta S_{\min} = k_B \ln 2 \) implies:

• The smallest distinguishable entropic curvature difference corresponds to a binary curvature gap of \( \ln 2 \).
• The quantity \( k_B \ln 2 \) is therefore the fundamental unit of entropic curvature in nature.
• Information is geometric: each bit corresponds to a curvature transition \( \rho_A \leftrightarrow \rho_B \) with ratio \( 2{:}1 \).

7. Operator‑Valued Generalization

At the spectral level, ToE reformulates this using the Araki relative entropy:

For the binary deformation \( \hat{\rho}_B = 2 \hat{\rho}_A \):

8. The ToE Curvature Invariant as Fundamental

In summary, the Theory of Entropicity (ToE) identifies \( \ln 2 \) as the minimal curvature invariant of the entropic manifold:

The factor \( \ln 2 \) quantifies the smallest possible geometric deformation between two distinguishable entropic field configurations. This derivation is independent of microstate counting, thermodynamic equilibrium, or probabilistic assumptions. It arises purely from the geometric structure of the entropic field and its binary curvature symmetry, marking a fundamental departure from classical and quantum statistical mechanics.