What does this equation mean?

​ : $N_\perp = \frac{2}{\pi\hbar}E_\infty\,\Delta t = \frac{2}{\pi\hbar}\left(\sqrt f\,E_0'\right)\frac{\Delta\tau}{\sqrt f} = \frac{2}{\pi\hbar}E_0'\,\Delta\tau$. The two calculations agree exactly. This is the covariance check the construction has to pass: $N_\perp$[$\Gamma$] does not care which energy convention is used, provided the convention is applied consistently — local energy paired with proper time, or Killing energy paired with coordinate time, never a local energy stapled to a coordinate-time integral or the reverse. That last error is precisely the one the naive “free thoughts” argument makes. It takes the huge blueshift factor 1/$\sqrt f$ multiplying local energy and pairs it with the huge 1/$\sqrt f$ multiplying elapsed coordinate time, double-counting the same factor as though it were two independent windfalls rather than one factor and its own reciprocal. Priced honestly, in whichever single currency the experimenter actually holds, the redshift and the blueshift are not two effects. They are one identity, and an identity cannot fund a subsidy.