Mathematical Operations and Maneuvers of the Amari-Čencov Alpha Connections in the Theory of Entropicity (ToE)

What is the Amari-Čencov alpha-connection and its role in the Theory of Entropicity (ToE)?

The Amari-Čencov $$\alpha$$-connection in the Theory of Entropicity (ToE) is the unifying affine connection that interpolates between Fisher-Rao (classical statistical) and Fubini-Study (quantum projective) geometries on the entropic manifold, enabling seamless classical-quantum transitions via an entropic deformation parameter $$\alpha$$.

## Definition and formula

On the statistical manifold $$\mathcal{P}$$ of entropic densities parameterized by $$\theta^i$$, the $$\alpha$$-connection coefficients are

$$

\Gamma^{(\alpha)}_{ij}{}^k = \Gamma^{(0)}_{ij}{}^k + \frac{1-\alpha}{2} T_{ij}{}^k + \frac{1+\alpha}{2} T_{ji}{}^k,

$$

where $$\Gamma^{(0)}$$ are the Levi-Civita symbols of the Fisher-Rao metric $$g_{ij}$$, and $$T_{ijk}$$ is the Amari-Čencov (skewness) tensor:

$$

T_{ijk} = g_{kl} \frac{\partial^2_\theta \ln p}{\partial\theta^i \partial\theta^j} \frac{\partial \ln p}{\partial\theta^k}.

$$

## Key Amari-Čencov alpha-connection limits in the Theory of Entropicity (ToE)

- $$\alpha = 0$$: Levi-Civita connection of Fisher-Rao metric (classical entropic geodesics).

- $$\alpha = +1$$: Exponential connection (∇^{(1)}), governs Fubini-Study quantum geodesics on pure states.

- $$\alpha = -1$$: Mixture connection (∇^{(-1)}), dual to exponential connection.

- $$\alpha \in (-1,1)$$: Interpolates between classical/quantum regimes via entropic order parameter.

The duality relation is $$\nabla^{(\alpha)} + \nabla^{(-α)} = 2\nabla^{(0)}$$ w.r.t. Fisher-Rao metric.[2][1]

## Role of the Amari-Čencov alpha-connections in ToE dynamics

In the Obidi Action, entropic geodesics follow

$$

\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,

$$

where $$\alpha$$ is coupled to generalized entropies (Rényi/Tsallis): $$\alpha = 2(1-q)$$.

- **Classical limit** ($$\alpha \to 0$$): Recovers entropy-maximizing paths in configuration space.

- **Quantum limit** ($$\alpha \to 1$$): Generates projective geodesics on entropic Hilbert space, recovering Schrödinger evolution.

- **OCI interaction**: The $$\ln 2$$ invariant sets discrete curvature thresholds along these $$\alpha$$-geodesics, quantizing stable configurations regardless of $$\alpha$$-deformation.[2]

## Unification mechanism of the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE) uses $$\alpha$$-connections to make **gravity emergent (appear or disappear) from entropic information geometry**: Einstein's Levi-Civita connection appears as the $$\alpha \to 0$$, weak-field limit of entropic $$\nabla^{(\alpha)}$$ geodesics, where spacetime curvature $$R[g]$$ encodes Fisher-Rao curvature in the entropic metric $$g_{ij}[S]$$.[2]

This whole non-trivial mechanism therefore provides ToE's core mathematical bridge: one deformable connection family governs all dynamics—from thermodynamic flows ($$\alpha \approx 0$$) to coherent quantum evolution ($$\alpha \approx 1$$) to classical (and relativistic) gravitational geodesics.

Citations:

[1] A Riemannian viewpoint on the Amari–Čencov α https://arxiv.org/html/2508.00371v1

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

[3] The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections https://par.nsf.gov/biblio/10490304-fisherrao-metric-amaricencov-alpha-connections

[4] A characterization of the alpha-connections on ... https://www.emergentmind.com/papers/2302.07471

[5] Information Geometry Part 2: Duality, Divergences, and Natural Gradient https://jiha-kim.github.io/crash-courses/information-geometry/2-duality-divergences/

[6] A Simple Explanation of the Unifying Mathematical ... https://www.authorea.com/users/896400/articles/1348176-a-simple-explanation-of-the-unifying-mathematical-architecture-of-the-theory-of-entropicity-toe-crucial-elements-of-toe-as-a-field-theory

[7] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5673430

[8] A Riemannian viewpoint on the Amari-Cencov $\alpha$-connections and Proudman-Johnson equations https://www.arxiv.org/pdf/2508.00371.pdf

[9] A Riemannian viewpoint on the Amari-Cencov $α$-connections and Proudman-Johnson equations http://arxiv.org/abs/2508.00371