Thermodynamic-Complexity Duality in Constrained Equilibrium Ensembles

Many complex systems, particularly glasses and disordered materials, exhibit energy landscapes with exponentially many metastable states. Such landscape structure strongly influences equilibrium behavior but is not explicitly represented in standard thermodynamic state spaces. We develop a constrained equilibrium framework in which configurational complexity, defined as the logarithmic density of metastable basins, is treated as an additional macroscopic coordinate. Starting from maximum entropy with simultaneous constraints on energy and complexity, we obtain a generalized Gibbs ensemble characterized by a conjugate bias parameter. Standard thermodynamic structure remains intact, with extended relations arising as constrained equilibrium identities. A mean-field glassy example with explicit complexity function demonstrates how complexity bias shifts the saddle-point structure of the partition function and modifies equilibrium response functions. The geometric formulation further provides a diagnostic of landscape reorganization within an enlarged state space. This framework offers a systematic equilibrium description of how energy-landscape structure influences thermodynamic behavior in systems with rugged configuration spaces.