The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS$_3$

In spread complexity, the average position of an operator along its Krylov chain, recovers the right radial momentum of an infalling particle in AdS, yet it is a measure of the first moment, irrespective of the spread of the wavepacket away from its classical trajectory. The rate of a normalized Krylov-Wigner negativity can be proposed as a diagnostic of the second moment of the boundary state that captures this spreading. Starting with the \emph{seed-normalized} Krylov-Wigner distribution -- that is, the Wigner transform of the descendant cloud, with the decaying return amplitude divided out -- we obtain an analytic Bessel form in the macroscopic limit and compute its total negativity explicitly. Retaining the Bessel variable all the way through, we find that the negativity goes as $\sinh^{4\Delta}(\pi t/\beta)$, while the raw, seed-normalized state negativity saturates, as dictated by the $O(\sqrt{D})$ bound. Using the exact negative binomial statistics of the Krylov chain and the momentum dictionary of Caputa et al.~\cite{Caputa:2024}, the rate of the negativity scales as the growth rate of the Krylov variance at late time asymptotics \emph{if and only if} $\Delta=1$, the Breitenlohner-Freedman saturating dimension in $AdS_3$. This dimensionality is special in that the negativity rate is the product of the proper radial position and momentum, $\mathcal{R} \propto \mathcal{C} P_\rho$, i.e., the rate of the tidal stretch of nearby geodesic falling into the horizon. We comment on the direction for future research, in particular the interpretation of the transverse string size operator in terms of the Krylov number operator through the common $SU(1,1)$ discrete series.