Hyperbolic Neural Cellular Automata: A Geometric Framework for Emergent Complexity

This repository contains the source code (LaTeX) and figures for the paper "Hyperbolic Neural Cellular Automata: A Geometric Framework for Emergent Complexity". Abstract We introduce Hyperbolic Neural Cellular Automata (H-NCA), addressing two fundamental limitations in classical cellular automata: the Geometric Capacity Bottleneck and the Temporal Binding Problem. By situating cellular automata on hyperbolic tessellations (pentagrid {5,4}) and integrating phase-oscillator dynamics via Hyperbolic Artificial Kuramoto Oscillatory Neurons (H-AKORN), we achieve exponential capacity growth and intrinsic temporal coherence. We formalize H-NCA as a Riemannian dynamical system where updates correspond to Euler discretization of natural gradient flows on Lorentz manifolds. Toy-model simulations demonstrate emergent phase synchronization and geometric structure preservation.