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, etc.) arise from **three core entropic principles**, independent of $$\alpha$$-connection details:

1. **Entropic Resistance Principle (ERP)**: Moving systems face "resistance" to spatial entropy redistribution, forcing compensatory temporal entropy buildup.

2. **Entropic Accounting Principle (EAP)**: Total entropy budget $$S_0 = S_t(v) + S_x(v)$$ is conserved, yielding $$\gamma(v) = 1/\sqrt{1-v^2/c^2}$$.

3. **No-Rush Theorem**: $$c$$ is maximum rate of entropic rearrangement, imposing causal bounds.

These produce Einstein's transformations **without geometric postulates**.[2][1]

## $$\alpha$$-Connection's supporting role

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

$$

\frac{d^2\theta^k}{d\lambda^2} + \Gamma^{(\alpha)}_{ij}{}^k \frac{d\theta^i}{d\lambda}\frac{d\theta^j}{d\lambda} = 0.

$$

**Key contributions**:

- **$$\alpha = 0$$ limit**: Levi-Civita (Fisher-Rao) geodesics recover classical entropic paths where resistance effects → 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 from the Theory of Entropicity (ToE)

```

Entropic field S(x) → Information manifold (θ^i) 

             ↓ [α-connection governs paths]

Entropic geodesics θ^i(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-quantum regimes, while the **entropic budget conservation** (independent of connection) enforces the relativistic kinematics along those paths.[2][3][1]

**Bottom line**: $$\alpha$$-connection provides the rails; entropic resistance provides the relativistic physics.

Citations:

[1] (PDF) The Theory of Entropicity (ToE) Derives Einstein's ... https://www.academia.edu/144796856/The_Theory_of_Entropicity_ToE_Derives_Einsteins_Relativistic_Speed_of_Light_c_as_a_Function_of_the_Entropic_Field_ToE_Applies_Logical_Entropic_Concepts_and_Principles_to_Derive_Einsteins_Second_Postulate_Version_2_0

[2] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[3] An Alternative Path toward Quantum Gravity and the Unification of ... http://www.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c

[4] Relativistic Dynamics https://www2.oberlin.edu/physics/dstyer/Modern/RelativisticDynamics.pdf

[5] The Feynman Lectures on Physics Vol. I Ch. 17: Space-Time https://www.feynmanlectures.caltech.edu/I_17.html

[6] Quantum physics: Theory of everything | Geometric algebra https://toe-physics.org/

[7] The Einstein Connection of the Unified Theory of Relativity https://pmc.ncbi.nlm.nih.gov/articles/PMC1063575/

[8] α-connections in generalized geometry https://www.sciencedirect.com/science/article/pii/S0393044021000747