Abstract We present a comprehensive theoretical framework unifying recent observations of lowdimensional structure in overparameterized neural networks—including the Lottery Ticket Hypothesis [2], Neural Collapse [3], and Grokking [4]—through the lens of algebraic topology and dynamical systems theory. Building on our previous work on accelerated grokking via triadic phase-locking [1], we formalize the training process as a Morse flow on an augmented loss landscape, where a differentiable Triadic Phase-Locking (TPL) operator enforces synchronization on weight triplets. We prove that TPL acts as a topological catalyst, eliminating high-index critical points via saddle-node bifurcations and inducing rapid convergence to low-dimensional invariant manifolds with persistent topological cycles. Through persistent homology analysis, we characterize this process as a second-order phase transition and introduce the Leo Kim H1-metric as a computable early-warning signal for generalization. Quantitative predictions include: (1) critical coupling scaling λc ∼ N-1/2; (2) 100–500 epoch lead time for topological indicators; (3) 2–5× convergence acceleration on structured tasks; (4) post-training dimensionality reduction to deff < 0.1N. Beyond theoretical advances, this framework has practical implications for Green AI, potentially reducing training energy by orders of magnitude through rapid discovery of efficient subnetworks. This work is purely theoretical; we invite empirical validation from the research community under open collaboration terms.