  Neptune's dynamical history shaped the current orbits of Kuiper belt objects
(KBOs), leaving clues to the planet's orbital evolution. In the "classical"
region, a population of dynamically "hot" high-inclination KBOs overlies a flat
"cold" population with distinct physical properties. Simulations of
qualitatively different histories for Neptune -including smooth migration on a
circular orbit or scattering by other planets to a high eccentricity - have not
simultaneously produced both populations. We explore a general Kuiper belt
assembly model that forms hot classical KBOs interior to Neptune and delivers
them to the classical region, where the cold population forms in situ. First,
we present evidence that the cold population is confined to eccentricities well
below the limit dictated by long-term survival. Therefore Neptune must deliver
hot KBOs into the long-term survival region without excessively exciting the
eccentricities of the cold population. Imposing this constraint, we explore the
parameter space of Neptune's eccentricity and eccentricity damping, migration,
and apsidal precession. We rule out much of parameter space, except where
Neptune is scattered to a moderately eccentric orbit (e > 0.15) and
subsequently migrates a distance Delta aN=1-6 AU. Neptune's moderate
eccentricity must either damp quickly or be accompanied by fast apsidal
precession. We find that Neptune's high eccentricity alone does not generate a
chaotic sea in the classical region. Chaos can result from Neptune's
interactions with Uranus, exciting the cold KBOs and placing additional
constraints. Finally, we discuss how to interpret our constraints in the
context of the full, complex dynamical history of the solar system.
