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The passage around this formula

​ : $N_\perp = \frac{2}{\pi\hbar}E_\infty\,\Delta t = \frac{2}{\pi\hbar}\left(\sqrt f\,E_0'\right)\frac{\Delta\tau}{\sqrt f} = \frac{2}{\pi\hbar}E_0'\,\Delta\tau$. 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…

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