Scale-Invariant Resonance: A Unified Criterion for Pattern Emergence in Physical and Cognitive Systems

The distinction between genuine emergent structures and stochastic fluctuations remains a fundamental challenge in the study of complex systems. While traditional theories often define order through local thermodynamic stability or free-energy minimization, they rarely account for the persistence of structure across varying resolutions. In this work, we propose Scale-Invariant Resonance as a unified criterion for pattern emergence. We posit that true self-organizing forms are characterized by their structural robustness under coarse-graining, whereas noise is intrinsically "scale-fragile." To operationalize this framework, we apply a multi-scale renormalization operator to two distinct dynamical systems: the Kuramoto model (physical layer) and the Hopfield network (cognitive layer). Our simulations demonstrate that resonant states maintain thermodynamic stability (order parameter R→1) and low algorithmic complexity (approximated via Lempel-Ziv compression) across multiple scales of observation. In contrast, stochastic states, even when dynamically stable, rapidly degenerate into disorder at intermediate scales. These findings establish scale-invariance as a quantifiable signature of meaningful information, bridging energetic cost and algorithmic compressibility in self-organizing systems.

Version 1.0 – November 2025. Uploaded as a conceptual preprint on Zenodo to establish authorship and public timestamp.