Thermodynamic-Complexity Duality in Constrained Equilibrium Ensembles

The thermodynamic behavior of many complex physical systems is strongly influenced by the structure of their underlying energy landscapes, particularly by the presence of exponentially many metastable states. While such landscape-level properties are central to the physics of glasses and disordered systems, they are not explicitly represented in standard thermodynamic state spaces. In this work, we develop a constrained equilibrium thermodynamic framework in which configurational complexity, defined as the logarithmic density of metastable states, is treated as an explicit macroscopic coordinate. Starting from the principle of maximum entropy, we construct equilibrium ensembles subject to simultaneous constraints on energy and configurational complexity, leading to a generalized Gibbs distribution characterized by a conjugate complexity control parameter. Within this framework, extended thermodynamic relations arise naturally as constrained equilibrium identities, without modification of the fundamental laws of thermodynamics. The formalism is illustrated through a worked example based on a mean-field glassy landscape, where configurational complexity can be computed explicitly. In this setting, complexity bias alters the saddle-point structure of the partition function, produces nontrivial response functions, and yields clear signatures of structural transitions. An extension to thermodynamic geometry further demonstrates how reorganizations of the energy landscape manifest as geometric features in an enlarged thermodynamic state space. The results presented here provide a systematic and physically grounded approach to incorporating landscape-level complexity into equilibrium thermodynamics. Rather than proposing universal energetic bounds, the framework offers a model-dependent tool for analyzing how configurational structure influences thermodynamic behavior in complex systems.