One Functional, Three Principles

Abstract

Three sciences carry a variational principle at their foundations, and the three do not appear to be the same principle. Nonequilibrium thermodynamics near equilibrium has Prigogine's theorem that stationary states minimize entropy production. Climate science, plasticity theory and parts of biology use the opposite rule — that driven systems maximize it. Theoretical neuroscience holds that anything which persists minimizes a variational free energy. Each principle works somewhere; none has an accepted derivation from more fundamental physics; and the literatures that use them rarely explain why a living cell, which falls under all three at once, is not pulled in three directions.

This supplement does three things. First, it reviews each principle on its own terms — the history, the exact mathematical statement, the complete proof where one exists, and the precise point at which every attempted general derivation has failed. Second, it collects what the critical literature actually established: each principle is exact only in a restricted regime, the restrictions are of the same three kinds, and no entropy-production extremum can be exact beyond linear order.

Third, it derives all three from one object. The program's open-system boundary law, projected from the six-dimensional bulk, carries a single constrained functional; read with the path, the currents, or the beliefs held primary, it returns the least-time law, the flux–force law with the minimum and maximum principles as dual conventions, and the natural-gradient inference rule. The domains of validity are derived rather than assumed, each classical counterexample lands where the derivation says it must, and every step is graded — derived, conditional, or open.