$$F_Q^{(0)} = 4\operatorname{Var}\hat J/\hbar^2$$

Equation 104 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

This model has an explicit conservation ceiling built in: it cannot be extended past the compass’s own total intrinsic Fisher information $F_Q^{(0)}$ = 4$\operatorname{Var}$$\hat J$/$\hbar^2$ , since no amount of radiation can reveal more about g than the original state ever carried. A tolerance $\delta$ tighter than 1/$\sqrt{F_Q^{(0)}}$ is not a slower recovery — it is a request for better precision than the compass itself possessed, which no observer, inside or outside any horizon, could ever meet. Within that ceiling, $t_G(\delta)$ grows toward the hole’s full evaporation lifetime as $\delta$ is tightened, which is a specific, checkable form of the “information remnant” Nakata, Wakakuwa, and Koashi found…

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