What does this equation mean?

Define the object this paper needs: for a worldline $\Gamma$ traced by a processor, with instantaneous mean energy above its own ground state $\langle$ H($\tau$)$\rangle$ - $E_0(\tau)$ measured in the processor’s own comoving rest frame at proper time $\tau$ , the cumulative orthogonalization count is $N_\perp[\Gamma] = \frac{2}{\pi\hbar}\int_\Gamma\left[\langle H(\tau)\rangle - E_0(\tau)\right]d\tau$. $N_\perp$[$\Gamma$] 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.