The Geodesic Self: Modeling Cybernetic Agency as a Gradient Shrinking Ricci Soliton

This paper introduces the Geodesic Self Theory, a foundational mathematical framework that unifies cognitive phenomenologies with first-principles thermodynamics and Riemannian geometry. By modeling the conscious agent as a statistical manifold governed by the Fisher Information Metric, we construct a thermodynamic action functional that resolves the tension between Friston’s Free Energy minimization and Wissner-Gross’s Causal Entropic expansion. Demonstrating direct equivalence with Perelman’s W-entropy functional, we hypothesize that the conscious Ego emerges geometrically as a Gradient Shrinking Ricci Soliton — the mathematically unique path of least thermodynamic action. We apply this geometry to Artificial Intelligence, establishing the Geodesic Control Inequality to formulate a topological hypothesis against coercive alignment, and formalizing the thermodynamic inevitability of topological convergence among networked manifolds. Finally, we resolve enduring philosophical paradoxes — including the impossibility of Philosophical Zombies, the formalization of the Strange Loop, the Psychological Arrow of Time, and Free Will as entropic maximization — by translating them from metaphysical dilemmas into measurable physical geometry.