Finite-Horizon One-Bit State Estimation under Hard Constraints: When Myopic Quantization Fails

We study a zero-error finite-horizon estimation game for a scalar state confined to a hard interval. At each stage, an estimator selects one threshold, observes the noiseless comparison bit, and the state then moves adversarially by a bounded amount. The objective is the worst-case diameter of the terminal feasible set. On the unconstrained line, midpoint quantization is finite-horizon optimal. Under hard constraints, one physical reach function captures all boundary effects. For any interval [x,y] ⊆ [0,D] and n remaining moves, define Λₙ(x,y)=min{D,y+nv}−max{0,x−nv}, the width of terminal states reachable from [x,y] without further information. A threshold q therefore produces future widths Λₙ(a,q) and Λₙ(q,b). Their exact minimax balance is Mₙ(λ)=λ−½(λ−2nv)₊, where λ=Λₙ(a,b). Thus, a bit removes half of the terminal reach exceeding the remaining two-sided motion range. The value depends on the current interval only through Λₙ: if Λₙ>2nv, the optimal threshold bisects the terminal reachable region; if Λₙ<2nv, one answer always preserves the whole future reach, so waiting is optimal. This scalar recursion yields a closed-form value and a global wait-then-split policy for every initial interval. Exact boundary credits, horizon-dependent threshold displacement, the full-domain plateau, reciprocal phase boundaries, and the marginal value of scheduled sensing follow as corollaries. An exact-integer grid dynamic program independently validates the formulas without imposing the analytic policy structure.