<i>On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity</i>

<p dir="ltr">On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity</p><p dir="ltr">In developing the Theory of Entropicity (ToE), several foundational concepts come into play. Among them are the α-connections of information geometry and the Amari–ˇCencov framework, which formalize dual affine structures. The role of Christoffel symbols and Levi–Civita connections highlights the geometric underpinnings that parallel General Relativity (GR) and its Einstein field equations. From the quan tum side, both Quantum Mechanics (QM) and Quantum Field Theory (QFT) are invoked, with the Schr¨odinger equations providing the canonical limit. The Entropic Field itself, together with the Obidi Action, defines the dynamical heart of the theory, while Entropic Geodesics and the Vuli–Ndlela Integral extend the notion of trajectories and integrals into the entropic domain. Supporting structures include the Fisher–Rao and Fubini–Study metrics, which capture classical and quantum distinguishability, re spectively, and the broader framework of Information Geometry. Finally, the theory situates itself within Pre-geometric Physics and Thermodynamic Field Theory, where entropy is not merely a measure but a generative principle shaping the very metrics of classical and quantum distinguishability.</p><p><br></p><p dir="ltr">The Theory of Entropicity (ToE) establishes entropy not as a statistical consequence of disorder, but as the fundamental field and causal substrate of physical reality. In this formulation, entropy S(x) is elevated to a dynamic, local, and continuous field that generates the phenomena of motion, gravitation, time, and information flow through its spatial and temporal gradients. The framework introduces the Obidi Action, a variational principle encoding the dynamics of the entropy f ield, from which the Master Entropic Equation (MEE), Entropic Geodesics, and Entropy Potential Equation emerge. By integrating the information geometry of Fisher-Rao’s classical distinguishability and Fubini-Study’s quantum distinguishability through the Amari–ˇ Cencov α connection formalism, ToE provides a rigorous geometric and probabilistic foundation for the evolution of physical systems within an entropic manifold. At its core, the theory reformulates the speed of light (c) as the maximum rate of entropic re arrangement, deriving relativistic and quantum phenomena as constraints imposed by finite entropy propagation. The No-Rush Theorem establishes a universal time-limit to all interactions, while the Vuli Ndlela Integral, an entropy-weighted reformulation of Feynman’s path integral, introduces irreversibility and temporal asymmetry into quantum mechanics. Together, these constructs unify thermodynamics, relativity, and quantum theory within a single entropy-driven continuum, resolving long-standing paradoxes of simultaneity, causality, and measurement. What we have here achieved with the Theory of Entropicity (ToE) is conceptually equivalent to a unification of General Relativity (GR) and Quantum Mechanics (QM) — but through entropy geometry rather than through quantization or string/brane compactification. The implications of ToE extend beyond physics into cosmology, computation, and cognition, proposing that mass, energy, spacetime, and consciousness arise as emergent constraints of the en tropic field. By redefining entropy as the universal field of interaction, ToE offers a new mathematical ontology—one where geometry, force, and information are not separate entities but projections of a single entropic reality.</p>