Thermodynamic-Complexity Duality: Embedding Computational Hardness as a Thermodynamic Coordinate

We propose a duality between thermodynamics and computational complexity, elevating the difficulty of a computational task to the status of a thermodynamic variable. By introducing a complexity measure C as a novel coordinate, we formulate an extended first law, dU = T dS - p dV + ... + lambda dC, capturing energy costs beyond classical bit erasures. This perspective unifies ideas from Landauer's principle with the combinatorial overhead of hard (e.g., NP-complete) problems, suggesting that algorithmic intractability can manifest as an additional contribution to thermodynamic potentials. We outline how this "complexity potential" might produce phase-transition-like signatures in spin glasses, random constraint satisfaction, or advanced computing hardware near minimal dissipation. We also discuss parallels with previous geometry-information dualities, emphasize the role of complexity in shaping energy landscapes, and propose experimental avenues (in reversible computing or spin-glass setups) to detect subtle thermodynamic signatures of computational hardness. This framework opens a route for systematically incorporating complexity constraints into physical modeling, offering a novel link between the fundamental cost of computation and thermodynamic laws.