The PANMENZ-Law of the Energy Coherence Trade-off

This preprint introduces the PANMENZ² Law of the Energy–Coherence Trade‑off, a minimal foundational formulation of a general dynamical constraint governing adaptive systems across neural, cognitive, agentive, social, and cultural scales. The law states that adaptive systems regulate their evolution such that an energy‑dimensioned functional, F(t) = E(t) − λ·C(t), does not increase over time. This inequality formalizes the principle that energy expenditure and structural complexity are permissible only when compensated by coherence gain. The PANMENZ² Law provides a unifying energetic perspective on stability, adaptation, and coherence across multiple domains and establishes the conceptual and mathematical core for future operator‑based and multi‑scale extensions. This work provides a theoretical and mathematical formulation rather than an empirical study.