A Sharp Dynamic Haussler Lemma

Haussler's covering lemma (JCTA 1995) gives the sharp (C/δ)^d metric-entropy bound for binary VC classes under normalized \ell_1 distance. We prove a sharp global dynamic analogue for bounded real-valued classes in normalized \ell_p, for every 1\le p<\infty. Writing d=\operatorname{sfat}_{\varepsilon/32}(\mathcal F), we obtain a global dynamic cover of size at most (C/\varepsilon^p)^d for every horizon and 0<\varepsilon<1/2. Here a single family of deterministic causal experts, fixed independently of the instance sequence, covers every target on every sequence. For 0<\varepsilon<1/4, a matching worst-case lower bound (c/\varepsilon)^{pd} holds already for binary classes on one fixed sequence, even with real-valued experts. The constants are universal, independent of p. For p=2, this resolves the horizon-independent sequential \ell_2 covering problem of Rakhlin, Sridharan, and Tewari (PTRF 2015) in a stronger global form. For p=1, it gives the sharp (C/δ)^{\operatorname{Ldim}(\mathcal H)} global dynamic covering bound for binary classes, resolving questions of Alon, Ben-Eliezer, Dagan, Moran, Naor, and Yogev (STOC 2021) and Malliaris and Moran (Trans. AMS 2024). Our proof combines rank-decreasing correction schedules and accurate sampled prefixes with exact continuations after target identification. Shared-parent routing distributes these continuations among eligible prefixes, eliminating the logarithmic overhead.