  The sample of host stars with multiple transiting planets has illuminated the
orbital architectures of exoplanetary systems. These architectures may be
shaped mostly by formation conditions, be continually sculpted by ongoing
dynamical processes, or both. As more studies place planet occurrence within a
galactic context, evidence has emerged for variable planet multiplicity over
time. In this manuscript, we investigate the use of transit multiplicity as a
tool to constrain longer-timescale (>1 Gyr) dynamical sculpting. First, with a
suite of injection-and-recovery tests, we quantify sensitivity to sculpting
laws across different regimes. We employ a forward modeling framework in which
we generate synthetic planetary systems, according to a prescribed sculpting
speed and timescale, around the FGK dwarfs studied by the Kepler Mission. Some
sculpting scenarios are hypothetically detectable in the Kepler sample, while
others can be disfavored from Kepler transit statistics alone. Secondly, we
apply our analysis to reverse-engineer the sculpting laws consistent with the
true yield from Kepler. We confirm the present-day fraction of host stars
containing dynamically cool "systems with tightly-packed inner planets" (STIPs)
is 4-13%. A variety of Gyr-timescale sculpting laws successfully predict the
transit multiplicity of the Kepler sample, but none of these laws succeeds in
also producing a detectable trend with transit multiplicity and stellar age. An
improvement to measured stellar age precision may help uncover such a sculpting
law, but nevertheless reflects limitations in transit multiplicity as an
observable. Therefore other phenomena, apart from Gyr-timescale dynamical
sculpting, may be required to explain the Kepler yield.
