Consistent Model Chasing Is Minimax Optimal: The Exact Value of Scalar Adversarial Adaptive Control under Large Parametric Uncertainty

We solve exactly a fundamental problem of adaptive control against adversarial disturbances: regulate the scalar system xt+1 = axt + ut + wt, x0=0, |w|_∞ ≤ 1, where the constant pole a ∈ [-Delta, Delta] is unknown in sign and magnitude and Delta is arbitrarily large. Elementary as the system looks, the least worst-case peak |x|_∞ that a causal controller can guarantee against an adversarial pair (a, w) (the value of this game) has, to our knowledge, never been determined for any adaptive control problem with parametric uncertainty of arbitrary size under this criterion; existing theory supplies stability certificates, gain bounds, and regret rates, not the value. That value is γ^star(Delta) = 1 + Delta for every Delta>0. The summand 1 is the irreducible price of the disturbance, and Delta the exact price of a single, unavoidable identification spike. The optimal policy is certainty-equivalent deadbeat control at the midpoint of the set-membership consistent interval, an instance of the robust oracle × consistent model chasing architecture. The architecture is forced, not merely sufficient: writing θt := -ut/xt exhibits every causal controller as an oracle-selector composition, and optimality pins the selector to the midpoint at the critical histories. The standard tools, classical and modern, each fail quantifiably: probing is punished before it pays, commitment is fatal at sub-disturbance excitation once adaptation is necessary, optimism degenerates to tie-breaking or pays asymptotically at least twice the optimum, and regret certificates are blind to the worst-case peak in both directions. The optimal law contains no exploration mechanism, its learning purely passive. These results give the first exact optimality certificate for consistent model chasing as a design principle for adversarial adaptive control.