Exact Entanglement-Depth Speed Frontier for Complete Quantum Charging

Complete quantum charging provides a sharp setting in which to ask how much multipartite entanglement is forced by speed itself. For a closed $N$-qubit battery evolving from $\ket{\downarrow}^{\otimes N}$ to $\ket{\uparrow}^{\otimes N}$ under a time-independent Hamiltonian, we exactly solve the pure-state depth-constrained speed problem. If the realized trajectory has entanglement depth at most $k$, then the largest possible QSL-normalized rate $\eta=\tau_{\rm QSL}/T$ is $\eta_{\max}(k)=\lceil N/k\rceil^{-1/2}$. Conversely, an observed rate $\eta$ certifies trajectory entanglement depth at least $\bigl\lceil N/\lfloor \eta^{-2}\rfloor\bigr\rceil$. The mechanism is block orthogonalization: under a fixed product partition, complete charging forces all blocks to orthogonalize simultaneously, and the quantum speed limit converts this counting constraint into the speed bound. Balanced cluster-flip evolutions saturate the bound, establishing an exact integer staircase frontier. Thus fast complete charging cannot be explained by many small independently charging blocks; in particular, crossing the threshold $\eta>1/\sqrt2$ certifies, for $N>1$, the generation of genuine $N$-partite entanglement.