Dynamic Closure Theory: A minimal framework for critical transitions across scales

Critical transitions — abrupt reorganizations triggered by smooth parameter changes — appear in systems ranging from protein folding to anesthesia-induced loss of consciousness. We introduce Dynamic Closure Theory (DCT), a minimal framework for single-order-parameter, near-equilibrium transitions that makes an energy-dependent closure functional CE(E), linking internal energy to the fraction of modes capable of sustaining coherent dynamics, explicit and mathematically tractable. From a maximum-entropy derivation of CE(E) we obtain a logistic (Fermi–Dirac-type) closure function with a half-closure threshold and a scaling law relating the critical coherence level Φc to an effective resource density ρeff.