Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral and Araki Entropies with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms

Recent developments in holography and information geometry suggest that entropy plays a foundational role in gravitational dynamics. The pseudo‑entropy framework of Takayanagi, Kusuki, and Tamaoka shows that a non‑Hermitian extension of entanglement entropy in CFT 2 satisfies a first law whose holographic dual reproduces the linearized Einstein equation in dS 3 . Variations of pseudo‑entropy also obey a Klein–Gordon equation on kinematic dS 2 , revealing an emergent temporal structure from boundary data. Independently, Bianconi’s formulation of gravity from metric relative entropy demonstrates that curvature and gravitational response can arise from the informational divergence between spacetime metrics. This paper presents a unified perspective in which both frameworks appear as limiting or boundary‑restricted manifestations of the Theory of Entropicity (ToE). ToE posits that entropy 𝑆 ( 𝑥 ) is a fundamental dynamical field governed by two complementary variational principles: the Local Obidi Action (LOA), which yields nonlinear entropic field equations, and the Spectral Obidi Action (SOA), which encodes global geometric and informational constraints through the spectrum of an entropy operator. Together, these actions generate the Master Entropic Equation, entropic geodesics, irreversible temporal flow, and a unified entropic geometry encompassing Tsallis and Rényi entropies, the Fisher–Rao and Fubini–Study metrics, and the Amari–Čencov 𝛼 -connections. We show that pseudo‑entropy corresponds to the boundary‑projected, linearized limit of the Master Entropic Equation, while Bianconi’s relative‑entropy action emerges from the spectral sector of ToE. This synthesis positions ToE as a universal entropic field theory from which holographic pseudo‑entropy, metric relative entropy, classical information geometry arise naturally.