  The concept of information scrambling elucidates the dispersion of local
information in quantum many-body systems, offering insights into various
physical phenomena such as wormhole teleportation. This phenomenon has spurred
extensive theoretical and experimental investigations. Among these, the
size-winding mechanism emerges as a valuable diagnostic tool for optimizing
signal detection. In this work, we establish a computational framework for
determining the winding size distribution in large-$N$ quantum systems with
all-to-all interactions, utilizing the scramblon effective theory. We obtain
the winding size distribution for the large-$q$ SYK model across the entire
time domain. Notably, we unveil that the manifestation of size winding results
from a universal phase factor in the scramblon propagator, highlighting the
significance of the Lyapunov exponent. These findings contribute to a sharp and
precise connection between operator dynamics and the phenomenon of wormhole
teleportation.
