How does the Theory of Entropicity (ToE) use the Amari-Čencov α-connections?

In ToE, the **Amari–Čencov $$\alpha$$**-connections are used to give the entropic manifold a **geometric structure** that links probability, information, and ordinary spacetime physics. The idea is that entropy is the primary field, and the $$\alpha$$-connection encodes how probability and information “curve” in that space, thereby generating physical dynamics [1][5].

## What they are doing geometrically

ToE treats the space of probability densities (or information states) as an entropic manifold and then equips that manifold with an **information geometry** built from the **Fisher–Rao metric** plus the **Fubini–Study metric**, using the Amari–Čencov $$\alpha$$-connection as the affine connection [1][5]. In this setup, different values of $$\alpha$$ correspond to different ways of relating information–entropic structure to affine geometry, which in turn shapes how entropic gradients drive motion, gravity, and time [1][6].

## How $$\alpha$$ becomes physical

A key and audacious move in the Theory of Entropicity (ToE) is to **tie the $$\alpha$$** parameter to thermodynamic or information deformation, for example via a relation such as $$\alpha = 2(1 - q)$$, where $$q$$ is the Tsallis nonextensivity parameter [6]. This makes $$\alpha$$ not just a statistical label but a **physical deformation index** that governs how entropy, probability, and curvature are coupled in the entropic manifold [6][7]. In this language, the $$\alpha$$-connection effectively becomes the **physical connection coefficients** of the entropic spacetime, replacing (or underpinning) the Christoffel symbols one would otherwise use in general relativity [1][7].

## Role in the overall derivation

By integrating Fisher–Rao and Fubini–Study through the Amari–Čencov $$\alpha$$-connection, ToE claims to construct a **unified information‑geometric foundation** for entropy‑driven dynamics: entropic geodesics, mass increase, time dilation, and Lorentz symmetry all arise as consequences of this geometry, without positing spacetime geometry first [1][5]. In this sense, the $$\alpha$$-connections are the **bridge between statistical information and relativistic physics** within the entropic paradigm [5][7].