An unknown point $x^\star\in\mathbb{R}^d$ is queried through unit linear functionals, each answered with adversarial additive error at most $\delta$. Recent work determined the adaptive minimax error. Its excess above the infinite-query Jung limit $J_d\delta=\sqrt{2d/(d+1)} \delta$ vanishes doubly exponentially in the number of queries, and it left the nonadaptive game open. We characterize that game. For every fixed $d\geq2$, its minimax excess is $\Theta_d(\delta T^{-2/(d-1)})$. Thus adaptivity changes the convergence law from polynomial to doubly exponential. We also resolve the previously open exponential-budget regime in high dimension. If $\log T/d\to\alpha\in(0,\infty)$, then the nonadaptive minimax error satisfies $\frac{\operatorname{OPT}^{na}_d(T,\delta)}{\delta} \longrightarrow \sqrt{\frac{2}{1-e^{-2\alpha}}}.$ The upper bounds combine spherical coverings with Jung's theorem. The lower bounds use a common transcript that hides every vertex of a suitably rotated and expanded regular simplex. This construction is what makes classical spherical-cap geometry sharp for point reconstruction. Finally, a batched strategy shows that each additional adaptive batch can double the polynomial exponent, interpolating between the two convergence laws.
Keywords: nonadaptive queries, adversarial noise, linear queries, adaptivity gap, Jung's theorem, query complexity