What is the Alpha Connection in the Theory of Entropicity (ToE)? How Obidi's Audacious and Revolutionary Insight Transformed the Alpha Parameter of Information Geometry into the Physical Principle of Spacetime 

In the Theory of Entropicity (ToE), the α‑connection is the affine connection that fuses classical and quantum information geometry into a single “entropic” geometric structure and encodes the directionality (arrow) of entropy flow and time.[1][5]

### Role of the α‑connection

- ToE works on an **entropic manifold** where states are points and entropy/information define the geometry.[1][5]

- On this manifold, the Fisher–Rao metric (classical probabilities) and the Fubini–Study metric (quantum states) are combined using the Amari–Čencov α‑connection formalism.[1][5]

- The α‑connection $$\nabla_{(\alpha)}$$ and its dual $$\nabla_{(-\alpha)}$$ define a dual affine geometry; their asymmetry for $$\alpha \neq 0$$ is taken to be the geometric origin of irreversibility and the entropic arrow of time.[2][5]

### Entropic meaning of α

- In extensions of ToE, α is promoted to an **entropic order parameter** linking generalized entropies (e.g., Tsallis/Rényi) to geometric deformation.[5][6]

- A constitutive relation such as $$\alpha = 2(1 - q)$$ connects the non‑extensivity index $$q$$ of Tsallis‑type entropy to the degree of affine asymmetry, so α measures how far the entropic dynamics depart from standard extensive, time‑reversible behavior.[6]

- Thus, changing α simultaneously deforms the entropy functional and the underlying information geometry, providing a single knob that tunes entropic curvature, non‑extensivity, and temporal asymmetry.[5][6]

### Why this matters in ToE

- With the Obidi Action defined on this α‑deformed information geometry, entropic geodesics and the Master Entropic Equation encode both classical and quantum evolution within one formalism.[1][5]

- The asymmetry between $$\nabla_{(\alpha)}$$ and $$\nabla_{(-\alpha)}$$ means forward and backward “entropic transport” are not equivalent, so microscopic dynamics already select a preferred temporal direction without adding an external arrow of time.[2][5]

As a compact summary: the α‑connection in ToE is the **geometric bridge** that (i) unifies Fisher–Rao and Fubini–Study metrics, (ii) ties generalized entropy (via $$q$$) to curvature, and (iii) makes the arrow of time a property of the entropic manifold itself.[1][5][6]

Citations:

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

[2] The Theory of Entropicity (ToE) Lays Down ... https://johnobidi.substack.com/p/the-theory-of-entropicity-toe-lays

[3] The entropic brain: a theory of conscious states informed by ... https://pmc.ncbi.nlm.nih.gov/articles/PMC3909994/

[4] A theory of entropic bonding - PMC https://pmc.ncbi.nlm.nih.gov/articles/PMC8795519/

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

[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] Psychedelics, entropic brain theory, and the taxonomy of conscious states https://pmc.ncbi.nlm.nih.gov/articles/PMC10072236/