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

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 𝑆(𝑥) 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. 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. 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.