The Wu-Wei Inference Theorem: Effortless Cognition as a Thermodynamic Optimum Between Forcing and Drift

Paper supported by machine-verified Lean 4. An inferring agent must decide how hard to update its beliefs on incoming evidence—how much gain, or precision, to place on prediction error. Intuition and much of machine practice treat more precision as strictly better: trust the data, minimize error, force the model onto the world. We show this intuition is thermodynamically and epistemically wrong. Modeling an agent that tracks a fluctuating environment with a constant-gain (precision g) filter, and reading its dissipation through the Still–Sivak–Bell–Crooks thermodynamics of prediction—dissipated work equals the non-predictive information the agent retains—we prove that optimal cognition sits at an interior point of restraint. First, the Bayes/Kalman-optimal gain is strictly less than one: fully trusting noisy evidence degrades tracking, so even error-minimizing inference already "lets go" (g_K<1). Second, and paradoxically, driving the gain past g_K toward maximal precision lowers the predictive information the agent holds about the future—a measurable forcing tax, dI_rm pred/dg<0 for g>g_K. Third, once the metabolic cost of maintaining precision is priced, the net-predictive optimum—the Wu-Wei gain g^ast—falls strictly below the Bayes gain, g^ast, and decreases monotonically as effort becomes dearer. The effortless first-order condition equates the marginal predictive gain to the marginal cost of forcing, dI_rm pred/dg=2kappa g^ast. We name the Intelligence-Bound self-application the Harmonizer: the useful predictive-intelligence rate is the raw bound throttled by an effortlessness eta=I_rm pred/I_rm memin(0,1), so the bound is spent on prediction only at the harmonized gain. All results are numerically verified (21/21) against a Monte-Carlo of the exact filter. We give three falsifiable predictions spanning attention neuroscience, machine regularization, and ecological monitoring.