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)

This paper reformulates Ginestra Bianconi’s Gravity from Entropy within the universal framework of the Theory of Entropicity (ToE). Whereas Bianconi interprets gravity as the quantum relative entropy between two metrics, ToE reveals that her formulation is a limiting case of the broader Obidi Action, the foundational variational principle of the entropic field. By expanding the Obidi Action around equilibrium, Bianconi’s relative-entropy functional emerges naturally as the quadratic approximation of ToE’s entropic potential, showing that her informational dual-metric model corresponds to the weak-gradient regime of the universal entropy field. In ToE, entropy is not statistical but ontological—a fundamental field whose gradients generate curvature, motion, and the flow of time. The theory unifies local and global formulations through its Local and Spectral Obidi Actions, bridging differential dynamics and operator geometry. This duality demonstrates that entropy is the common source of matter and geometry. ToE extends beyond Bianconi’s model by establishing spectral operator actions as the universal foundation of physics, integrating bosonic and fermionic dynamics within one entropic–spectral framework. Furthermore, ToE clarifies Bianconi’s auxiliary G-field as the modular operator whose spectral excitations account for dark matter and yield a small positive cosmological constant through entropy-flux constraints. By resolving Bianconi’s open challenges—canonical quantization and dark-energy interpretation—ToE advances a unified, self-consistent entropic field theory where gravity, quantum mechanics, and thermodynamics emerge from a single spectral principle that defines the structure and evolution of the universe.