  Contact binary star systems represent the long-lived penultimate phase of
binary evolution. Population statistics of their physical parameters inform
understanding of binary evolutionary pathways and end products. We use light
curves and new optical spectroscopy to conduct a pilot study of ten
(near-)contact systems in the long-period ($P$>0.5 d) tail of close binaries in
the Kepler field. We use PHOEBE light curve models to compute Bayesian
probabilities on five principal system parameters. Mass ratios and third-light
contributions measured from spectra agree well with those inferred from the
light curves. Pilot study systems have extreme mass ratios $q$<0.32. Most are
triples. Analysis of the unbiased sample of 783 0.15 d<$P$<2 d (near-)contact
binaries results in 178 probable contact systems, 114 probable detached
systems, and 491 ambiguous systems for which we report best-fitting and
16th/50th/84th percentile parameters. Contact systems are rare at periods
$P$>0.5 d, as are systems with $q$>0.8. There exists an empirical mass ratio
lower limit $q_{min}$($P$)$\approx$0.05--0.15 below which contact systems are
absent, supporting a new set of theoretical predictions obtained by modeling
the evolution of contact systems under the constraints of mass and angular
momentum conservation. Pre-merger systems should lie at long periods and near
this mass ratio lower limit, which rises from $q$=0.044 for $P$=0.74 d to
$q$=0.15 at $P$=2.0 d. These findings support a scenario whereby nuclear
evolution of the primary (more massive) star drives mass transfer to the
primary, thus moving systems toward extreme $q$ and larger $P$ until the onset
of the Darwin instability at $q_{min}$ precipitates a merger.
