The Temporal Integral of the Maximum Entropy Production Principle: Statefulness, Entropic Currents, and the Physics of Phase-Space Viability

Abstract
In theoretical physics, the Maximum Entropy Production Principle (MEPP) is typically formulated under the strict constraints of non-equilibrium steady states. However, when extrapolated into biophysical economics and systems ecology, MEPP is frequently misinterpreted as an unconstrained rate-maximising function, improperly conflating it with the Maximum Power Principle (MPP). This paper demonstrates that applying a rate-based maximisation to bounded, stateful Complex Adaptive Systems (CAS) guarantees premature structural collapse, paradoxically yielding a lower total quantity of entropy. We propose that a self-consistent formulation of MEPP must optimise the temporal integral of entropy production over the entirety of a system’s existence. Achieving this integral requires structural memory, defined here as Informational Density (ρI ), to navigate dynamic environmental limits. Applying optimal control theory to nested CAS recursion, we reframe ‘rogue agency’ not as a valid evolutionary strategy, but as an open-loop control failure. Anchoring this framework to Boltzmann’s equation (S = kB ln Ω), we demonstrate that entropic currents naturally bias bounded systems towards K-selected trajectories that preserve viable future phase-space (Ω_viable). We provide an explicit falsification condition, proving that homeostatic pacing is a rigorous thermodynamic requirement for systemic endurance.