What does this equation mean?

The honest reading of this is not that the inertial case is somehow paradoxical — physically, nothing is wrong: an inertial detector in the vacuum, left on forever, simply never clicks on the excitation branch, which is exactly the statement that the vacuum is empty from an inertial point of view. The failure belongs to the mathematical object $D_{\rm det}$ , not to the physics it is trying to summarize. A distance between two probability distributions cannot be computed when one of the two distributions does not exist. The fix available is the one the switching-function literature already worked out for a related reason: give the inertial detector a finite switch-on, so that transient, non-eternal excitation probability appears even where the eternal limit gives none. Louko and Satz showed that switching on and off necessarily produces exactly this kind of transient content, and that if the switching is not smooth — a sharp step rather than a gradual ramp — the instantaneous transition rate develops a logarithmic divergence right at the switching instants, an artifact of the switching function’s own discontinuous derivative rather than any statement about the trajectory [ 10 , 11 ] . Their fix — smoothing the switch, or regularizing with a spatial profile — restores a finite, comparable spectrum, but at the cost of making $D_{\rm det}$ depend explicitly on how the switching was smoothed, which is precisely the fourth treaty clause reasserting itself: match the switching profile, not merely its duration, or the comparison is measuring the shape of the shutter rather than the shape of spacetime.