Constructive Lyapunov Functions via Topology-Preserving Neural Networks

We prove that ONN achieves order-optimal performance on convergence rate (μ propto λ2), edge efficiency (E = N for minimal connectivity k = 2), and computational complexity (O(N d2)). Empirical validation on 3M-node semantic networks demonstrates 99.75% improvement over baseline methods, confirming exponential convergence (μ = 3.2 × 10-4) and topology preservation. ORTSF integration into transformers achieves 14.7% perplexity reduction and 2.3 faster convergence on WikiText-103. We establish deep connections to optimal control (Hamilton-Jacobi-Bellman), information geometry (Fisher-efficient natural gradient), topological data analysis (persistent homology computation in O(KN)), discrete geometry (Ricci flow), and category theory (adjoint functors). This work transforms Massera's abstract existence theorem into a concrete, scalable algorithm with provable guarantees, opening pathways for constructive stability analysis in neural networks, robotics, and distributed systems.