Constraints, Retention, and the Phase Structure of Functional Complexity in Nonequilibrium Systems.

We propose a nonequilibrium framework in which persistent functional complexity depends not on energy flow alone, but on the joint action of accessible work, constraints, and retention. Accessible work sets the thermodynamic resource base; constraints determine which channels can convert that work into functionally useful organization; retention preserves successful configurations once they arise. The aim is not a universal theory of complexity, but a minimal criterion for when persistent functional organization becomes viable. To formalize this idea, we study two nested toy models. In a reduced analytically tractable limit, a sigmoidal benefit of complexity and a linear maintenance cost imply a discontinuous transition in the optimal coarse-grained complexity: once a critical control parameter is crossed, the optimum jumps from zero to a finite value. Below a lower saddle-node threshold, no nonzero complex state exists; between the two thresholds, the utility landscape contains a barrier-separated nonzero branch. Under local relaxational dynamics this yields bistability and hysteresis, so the conditions required to acquire complexity are stricter than those required to maintain it. We then introduce a minimal dynamical extension that keeps constraint quality Q and retention capacity R distinct. In that extension, Q lowers the acquisition barrier and improves directed work channeling, whereas R stabilizes already-achieved organization without creating it on its own. The resulting (Q, R) phase diagram preserves the simple/bistable/complex structure and shows that high-Q/low-R and low-Q/high-R systems are dynamically inequivalent. The framework yields falsifiable signatures for microbial evolution and information engines, including discontinuous onset, metastability, and hysteresis.