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.