The Born Rule as Thermodynamic Equilibrium: Observer Convergence under Informational Work Constraints

The Born rule, which prescribes the probability of measurement outcomes as ||², has long been regarded as a foundational postulate of quantum mechanics. Rather than attempting to derive why quantum amplitudes are squared—a question we treat as analogous to asking why entropy increases—this study asks a different question: under what dynamical conditions is any physically constrained observer guaranteed to converge to the Born distribution? We introduce Thermodynamic Information Dynamics (TID), a framework that models the observer as a resource-limited physical system embedded in a thermal environment. Taking the Born weights W = \| (xᵢ) |²\ as an empirical given, we hypothesize that maintaining an empirical distribution deviating from W requires dissipating a minimum additional work proportional to the Kullback–Leibler divergence: W₄ₗₓₑ₀ kB T D₊₋ (\| W). Under this constraint, the Born distribution emerges as the unique stable fixed point of the observer's dynamical evolution. Using high-precision stochastic simulations (N = 2, 000, 000) across Metropolis and Glauber transition engines, we numerically confirm this convergence and demonstrate a robust scaling law T^-1/2 for steady-state fluctuations—an inevitable consequence of the competition between thermal noise and informational stiffness. Most significantly, at ultra-low temperatures (e. g. , 2. 7 K), we identify an Informational Glass Phase characterized by severe ergodicity breaking, where insufficient thermal energy prevents the observer from correcting deviations from the Born distribution, leading to convergence failure and dynamical freezing. These results provide a quantitative, falsifiable account of why observers robustly see the Born rule in warm environments, and predict measurable breakdowns in extreme thermal conditions—without claiming to explain the deeper origin of the ||² form itself. This work suggests that quantum measurement may be understood not as instantaneous collapse, but as a thermodynamic relaxation process toward a state of minimum informational expenditure.