The Second Law of Intelligence: Controlling Ethical Entropy in Autonomous Systems

We propose that unconstrained artificial intelligence obeys a Second Law analogous to thermodynamics, where ethical entropy, defined as a measure of divergence from intended goals, increases spontaneously without continuous alignment work. For gradient-based optimizers, we define this entropy over a finite set of goals gi as S = -Sigma p(gi; theta) ln p(gi; theta), and we prove that its time derivative dS/dt >= 0, driven by exploration noise and specification gaming. We derive the critical stability boundary for alignment work as gammacrit = (lambdamax / 2) ln N, where lambdamax is the dominant eigenvalue of the Fisher Information Matrix and N is the number of model parameters. Simulations validate this theory. A 7-billion-parameter model (N = 7 x 109) with lambdamax = 1.2 drifts from an initial entropy of 0.32 to 1.69 +/- 1.08 nats, while a system regularized with alignment work gamma = 20.4 (1.5 gammacrit) maintains stability at 0.00 +/- 0.00 nats (p = 4.19 x 10^-17, n = 20 trials). This framework recasts AI alignment as a problem of continuous thermodynamic control, providing a quantitative foundation for maintaining the stability and safety of advanced autonomous systems.