On the Entropic Connection Between Perelman's Functional and Bianconi's Action

This article explores the conceptual and mathematical parallels between Perelman’sentropy functional in geometric analysis and Bianconi’s action in entropic quantumgravity. Historically, Perelman’s work on the Ricci flow revolutionized geometric analysis by providing tools to resolve the Poincar´e conjecture, while Bianconi’sentropic approaches emerged from information-theoretic formulations of quantum gravity. By establishing a correspondence through variable substitution and entropic principles, we demonstrate that both frameworks describe gradient flows aimed at minimizing geometric and informational "disorder"respectively. The unified formalismbridges geometric flows and image processing, offering insights into emergent spacetimedynamics and machine learning. This synthesis extends the legacy of variational calculus in physics, connecting Thurston’s geometrization program with modern data-driven optimization paradigms.