What does this equation mean?

The dossier for this paper flags something the two equations above do not, by themselves, settle: the periodic-evolution bound above was derived for a Hamiltonian with a genuinely time-independent spectrum. A real computation drives H($\tau$) through a sequence of different instantaneous Hamiltonians, one per gate, and nothing guarantees in advance that a chain of instantaneously applied bounds sums to a valid bound on the whole chain. The condition that licenses treating $N_\perp$[$\Gamma$] as a legitimate partition into instantaneously static Levitin-Toffoli steps is a quasi-static one: over one local orthogonalization time $\tau_{\rm ML}(\tau)$ = $\pi$$\hbar$/[2($\langle$ H($\tau$)$\rangle$-$E_0(\tau)$)] , the energy gap itself must change by a small fraction,