What does this equation mean?

a standard identity in quantum metrology [ 12 ] . Before scrambling, this is fixed by how the compass itself was built — a large, well-prepared gyroscope has a large charge variance and a small intrinsic uncertainty on its own heading. After scrambling, that fixed total has to be reconstructed piecemeal from radiation, and each individually emitted quantum, to the extent it is only weakly and independently correlated with g once the hole’s own state is traced over, contributes an addition to the total that adds like an independent sample rather than like a decoded codeword. This is the standard-quantum-limit regime of parameter estimation: for N independent, identically prepared probes each with per-probe Fisher information $f_1$ , the achievable variance scales as 1/(N $f_1$) , in contrast to the quadratically better Heisenberg scaling available only when probes are used coherently together [ 13 ] . Modeling the collected radiation up to time t as contributing $N_\gamma(t)$ $\sim$ t/$\beta$ roughly independent quanta gives, as a stated phenomenological ansatz rather than a first-principles evaporation calculation, $F_Q(t) \;\approx\; f_1\,\frac{t}{\beta}, \qquad t_G(\delta) \;\approx\; \frac{\beta}{f_1\,\delta^2}$. 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 directly in their symmetry-constrained analysis of the Hayden-Preskill protocol: a residue that a purely logical decoder never has to wait for, because it was never entitled to it in the first place [ 8 ] .