On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics

<p dir="ltr">The Theory of Entropicity (ToE) reconceives entropy not as a statistical by‑product of disorder but as the fundamental field and causal substrate of physical reality. In this framework, entropy ��(��) is elevated to a continuous, dynamical field whose gradients generate motion, gravitation, time, and information flow. The Obidi Action, a variational principle for the entropy field, yields the Master Entropic Equation, Entropic Geodesics, and the Entropy Potential Equation, establishing a unified dynamical foundation. By integrating Fisher–Rao’s classical information metric and the Fubini–Study quantum metric through the Amari–Čencov α‑connection formalism, ToE provides a rigorous geometric and probabilistic structure for physical evolution within an entropic manifold.</p><p><br></p><p dir="ltr">At its core, the theory reformulates the speed of light as the maximum rate of entropic rearrangement, deriving relativistic and quantum constraints from finite entropy propagation. The No‑Rush Theorem imposes a universal temporal bound on 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.</p><p><br></p><p dir="ltr">Extending its foundations, ToE incorporates Rényi and Tsallis entropies, establishing a correspondence between generalized entropy measures and geometric structures. The entropic order parameter α emerges as a universal deformation index linking geometry, information, and entropy flow. In this synthesis, ToE reproduces Einstein’s field equations as a limiting case and subsumes Bianconi’s “Gravity from Entropy” as a special instance. Beyond physics, ToE suggests that mass, energy, spacetime, and consciousness arise as emergent constraints of a single entropic reality.</p><p><br></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. The present work extends the mathematical foundations of the Theory of Entropicity (ToE) by incorporating the R´enyi and Tsallis entropic frameworks into the unified field formalism. This synthesis establishes a profound correspondence between generalized entropy measures and geometric structures through the Amari–ˇ Cencov α–connections, the Fisher–Rao informa tion metric, and the Fubini–Study quantum metric. The entropic order parameter α now serves as a universal deformation index linking the informational and geometric domains, such that the geometry of space, the curvature of information, and the flow of entropy become different manifestations of one underlying entropic field. The α and q parameters of R´enyi and Tsallis generalized entropies are not incidental but constitute an integral embedding within the Theory of Entropicity (ToE). With these incorporations, the ToE not only reproduces the Einstein field equations as a limiting case but also subsumes Ginestra Bianconi’s recent “Gravity from Entropy” formulation as a special instance of its more general entropic–geometric dynamics. Bianconi’s G–field and her prediction of a small positive cosmological constant are not independent assumptions but derived features within the Theory of Entropicity (ToE), emerging directly from the entropic field equations of ToE. We also show that ToE is a superset of Ariel Caticha’s Statistical Geometrodynamics (SGD) and Entropic Dymanics (ED). 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, biology, neuro science, Artificial Intelligence (AI), cognition, and philosophy, proposing that mass, energy, space time, and consciousness arise as emergent constraints of the entropic field. By redefining entropy as the universal field of all interactions, ToE offers a new mathematical ontology—one where geometry, force, and information are not separate entities but projections of a single entropic reality.</p>