Complexity of the Obidi Action: a Clever Integration of Fisher-Rao and Fubini-Study Metrics with Amari-Čencov alpha-Connections and Generalized Tsallis and Renyi Entropies and Araki Relative Entropy and the Kullback-Leibler (Umegaki) Divergence

The Obidi Action is not merely a "clever integration" of those information-geometric and entropic tools, with the Theory of Entropicity (ToE) explicitly integrating the Fisher-Rao and Fubini-Study metrics via Amari-Čencov α-connections as its geometric foundation.[1][2]

## Core Structure of the Obidi Action

The Obidi Action is a variational principle for the entropy field $$S(x)$$, typically of the form

$$

A[S] = \int d^4x \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} \nabla_\mu S \nabla_\nu S - V(S) + \eta S T^\mu_\mu \right],

$$

where the kinetic term uses an information metric on the entropy manifold, the potential $$V(S)$$ encodes self-interactions (potentially generalized Tsallis/Rényi forms), and the coupling to matter stress-energy traces enables emergent gravity.[2][1] Varying yields the Master Entropic Equation (MEE), entropic geodesics, and an entropy potential equation, deriving GR limits and quantum effects.[1]

## Relation to Cited Concepts

ToE draws directly from information geometry:

- **Fisher-Rao metric**: Defines the Riemannian structure on probability/entropy manifolds, providing the natural kinetic term for $$S(x)$$.[1][2]

- **Fubini-Study metric**: Incorporated for quantum projective spaces, enabling quantum information geometry in the entropic framework.[1]

- **Amari-Čencov α-connections**: Unifies dual affine structures, allowing interpolation between divergences like KL (α=0, Umegaki relative entropy), reverse KL (α=1), and symmetric forms; central to ToE's "entropic α-connection".[1][2]

- **Generalized entropies/divergences**: Araki relative entropy (quantum extension of KL), Tsallis/Rényi (non-additive generalizations), and KL appear in distinguishability potentials or Vuli-Ndlela integrals, but as derived consequences rather than primitives.[1]

These are foundational, per ToE papers, but the Obidi Action innovates by promoting entropy to a dynamical spacetime field, yielding novel predictions like entropic Lorentz factors and No-Rush Theorem.[1]

## Beyond Concoction: Original Elements

| Aspect                  | Standard Info Geometry Role                  | ToE Innovation in Obidi Action [1] |

|-------------------------|----------------------------------------------|---------------------------------------|

| Metrics/Connections    | Statistical manifolds, divergences           | Entropy field dynamics, emergent metric $$g_{\mu\nu}$$ |

| Entropies/Divergences  | Measure info loss between distributions      | Causal/informational flow generating gravity/time |

| Scope                  | Probability spaces                           | Full unification (GR + QM + thermo) via MEE |

ToE positions these as recovered limits of a deeper entropic substrate, not ad hoc synthesis—e.g., speed of light as max entropic rearrangement rate.[1][2]

Citations:

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

[2] A Brief Note on Some of the Beautiful Implications ... https://johnobidi.substack.com/p/a-brief-note-on-some-of-the-beautiful

[3] Fisher-Rao Geometry in Statistical Models - Emergent Mind https://www.emergentmind.com/topics/fisher-rao-geometry

[4] Fubini-Study metric https://encyclopediaofmath.org/wiki/Fubini-Study_metric

[5] Further Properties of Tsallis Entropy and Its Application https://pmc.ncbi.nlm.nih.gov/articles/PMC9955289/

[6] [논문 리뷰] A numerical analysis of Araki-Uhlmann relative entropy in Quantum Field Theory https://www.themoonlight.io/ko/review/a-numerical-analysis-of-araki-uhlmann-relative-entropy-in-quantum-field-theory

[7] The Kullback–Leibler divergence between discrete ... https://blogs.sas.com/content/iml/2020/05/26/kullback-leibler-divergence-discrete.html

[8] A Simple Explanation of the Unifying Mathematical Architecture ... https://quarxiv.authorea.com/doi/full/10.22541/au.176099705.55607091/v1

[9] (27-4-25) Obi has introduced online portal and identity cards for Obidient Movement members https://www.youtube.com/watch?v=Qaz5pd7ch7E

[10] Entropy - Wikipedia https://en.wikipedia.org/wiki/Entropy