Law of Free Energy (LFE): An Effective Balance of Order, Dissipation, Entropy, and Predictive Memory in Driven Adaptive Systems

We formulate the Law of Free Energy (LFE) as an effective balance for driven adaptive systems with finite propagation and local feedback. The law is expressed through a functional F[Φ] = E[Φ] + λC[Φ] − θS[Φ] + κM[Φ], where E represents interaction or shape energy, C dissipative load, S an entropy or uncertainty proxy, and M predictive memory or temporal regularization. The purpose of this note is not to claim a universal final theory of adaptive dynamics, but to propose a compact operational law linking order, dissipation, entropy, and memory in a form precise enough to yield falsifiable predictions. Under a gradient-like stochastic dynamics, LFE predicts descent of F in expectation, reduced hysteresis under periodic forcing, OU-type relaxation signatures near stationary points, and robust operation near the stability boundary under bounded propagation and feedback. The present version focuses on the law itself and on its empirical implications, while treating multi-domain simulations as illustrative rather than as the primary basis of the claim.