On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE)

On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy

A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE)

(Version 1.0)

Abstract

This work presents a rigorous reformulation of Ginestra Bianconi’s Gravity from Entropy within the universal framework of the Theory of Entropicity (ToE). Whereas Bianconi interprets gravity as emerging from the quantum relative entropy between two metrics — a spacetime metric and a matter-induced metric — the present study demonstrates that her construction, essentially geometric in an operator-theoretic sense rather than information-geometric, can be fully recovered from the ToE under near-equilibrium expansion and appropriate correspondence between the ToE’s entropy field and Bianconi’s informational metrics. In this view, Bianconi’s model appears as a specific limiting case within the broader entropy–geometric structure of ToE, rather than a separate theory.

In ToE, entropy is not a statistical descriptor but a fundamental physical field whose gradients generate curvature, motion, and temporal flow. By expanding the ToE’s variational principle, known as the Obidi Action, around an equilibrium configuration, Bianconi’s relative-entropy functional naturally emerges as its quadratic approximation. This reveals that her formulation corresponds to the weak-gradient or quasi-equilibrium regime of the universal entropic field, in which informational and operator geometries become equivalent.

ToE introduces two complementary formulations of physical law: the Local Obidi Action, which describes the differential dynamics of the entropy field, and the Spectral Obidi Action, which expresses the same physics globally through operator traces. The spectral formulation connects equilibrium geometry and its matter-deformed counterpart through the modular operator, establishing a bridge between local field equations and global spectral consistency. This duality between local and spectral dynamics distinguishes ToE from prior entropy-based theories.

Through this unification, ToE shows that entropy is not merely a comparative measure, as in Bianconi’s dual-metric approach, but the ontological substrate of reality itself — the single field from which both matter and geometry arise recursively. Moreover, ToE extends the framework to a universal principle: spectral operator actions are not optional reformulations but fundamental components of physical law. By integrating bosonic and fermionic dynamics into a single entropic–spectral variational structure, ToE provides a common origin for the Einstein–Hilbert, Yang–Mills, Klein–Gordon, and Dirac actions, demonstrating that they are not disparate constructs but natural projections of one universal field theory.

Additionally, ToE clarifies the physical significance of Bianconi’s auxiliary G-field and her emergent cosmological constant. Both are shown to arise from the global conservation of entropy flux — a principle that naturally produces a small positive cosmological constant and explains the dark-matter energy density as a spectral property of the entropic field.

Taken together, these results confirm that Bianconi’s framework is entirely contained within ToE, just as classical mechanics is contained within quantum theory. ToE also encompasses earlier thermodynamic and information-theoretic gravitation models proposed by Jacobson, Padmanabhan, and Verlinde, demonstrating that all such approaches are boundary cases of a single entropic field theory. By offering a consistent canonical quantization of the Obidi Actions and resolving the physical meaning of the G-field, ToE not only fulfills Bianconi’s open challenges but also establishes itself as a breakthrough framework — a unifying principle where entropy, information, and geometry converge to describe the fundamental structure of reality.

Keywords

Amari — Čencov $\alpha$ — connections; Araki Relative Entropy; Atiyah — Singer Index Theorem; Bekenstein — Hawking Entropy; Bosons; Canonical Quantization; Dark Matter; Dirac — Kähler Fermions; Dirac Spinors; Einstein — Hilbert Action; Entropic Field; Entropy Geometry; Fermions; Fisher — Rao Metric; Fubini — Study Metric; G — Field; Ginestra Bianconi; Information Geometry; Jacobson Thermodynamics; Local Obidi Action (LOA); Obidi Actions; Padmanabhan Entropic Gravity; Quantum Gravity; Relative Entropy; Rényi Entropy; Shannon Information; Small Positive Cosmological Constant; Spectral Action; Spectral Geometry; Spectral Obidi Action (SOA); Spectral Theories; Theory of Entropicity (ToE); Tsallis Entropy; Verlinde Emergent Gravity; Vuli — Ndlela Integral; Yang — Mills Theory.

References

Obidi, John Onimisi (12 Nov. 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Figshare. https://doi.org/10.6084/m9.figshare.30596129.v1

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