What does this equation mean?

$N_\perp$[$\Gamma$] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and d$\tau$ is a time, the integral has action units and $N_\perp$ is dimensionless. It is invariant under reparametrizing $\Gamma$ ’s own curve parameter, since only proper time enters, and invariant under any passive coordinate change of the ambient spacetime, since both d$\tau$ and a comoving-frame energy are scalars. It is not invariant under boosting the local tetrad used to evaluate $\langle$ H($\tau$)$\rangle$ — a different local observer at the same event assigns a different local energy — which is why the convention must be pinned to the processor’s own rest frame and stated explicitly, not left implicit the way it usually is in flat-space quantum speed limit work.