Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement

This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations. First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, formally proved to preserve isometric knowledge embedding (Theorem~refthm:isometric). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem~refthm:critical) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma~reflem:surgery). Third, we establish an AdS/CFT-type holographic duality (Theorem~refthm:ads) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1× convergence acceleration and 63% topological simplification while maintaining O(Nlog N) complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem~refthm:converge) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.