PANMENZ Framework: Hierarchical Coupling, Hybrid Structural Order, and the Energy–Coherence Trade-off

PANMENZ² formulates a universal law describing the fundamental trade‑off between energetic expenditure and structural coherence in adaptive systems. The framework demonstrates that systems—independent of substrate, architecture, or complexity—are constrained by a shared energy‑coherence boundary characterized by the constant λ₀. This constant defines the structural limit within which stable, efficient, and learning‑capable dynamics can emerge.

The present version focuses on the theoretical formulation of the law, the multiscale organization of adaptive processes, and the derivation of core stability criteria. The mathematical operator algebra underlying PANMENZ², including the definitions of operators and meta‑operators (Z₄), is being developed in parallel and will be detailed in a dedicated follow‑up paper. The current preprint therefore concentrates on the unified formulation of hybrid structural order and hierarchical coupling.

This work further situates PANMENZ² in relation to established theoretical paradigms such as the Free Energy Principle (FEP), Integrated Information Theory (IIT), and Global Workspace Theory (GWT). Exemplary system analyses illustrate the applicability of the law across cognition, consciousness studies, artificial intelligence, biological regulation, and social or organizational dynamics. A set of empirical validation protocols supports reproducible testing of the proposed universality of λ₀.

PANMENZ² is intended for researchers in cognitive science, artificial intelligence, neuroscience, systems theory, complexity science, and organizational studies, and is presented as an open, extensible contribution to the foundational study of adaptive systems.