  We conduct a rigorous examination of the nearby red supergiant Betelgeuse by
drawing on the synthesis of new observational data and three different modeling
techniques. Our observational results include the release of new, processed
photometric measurements collected with the space-based SMEI instrument prior
to Betelgeuse's recent, unprecedented dimming event. We detect the first radial
overtone in the photometric data and report a period of $185\pm13.5$ d. Our
theoretical predictions include self-consistent results from multi-timescale
evolutionary, oscillatory, and hydrodynamic simulations conducted with the
Modules for Experiments in Stellar Astrophysics (MESA) software suite.
Significant outcomes of our modeling efforts include a precise prediction for
the star's radius: $764^{+116}_{-62} R_{\odot}$. In concert with additional
constraints, this allows us to derive a new, independent distance estimate of
$168^ {+27}_{-15}$ pc and a parallax of $\pi=5.95^{+0.58}_{-0.85}$ mas, in good
agreement with Hipparcos but less so with recent radio measurements. Seismic
results from both perturbed hydrostatic and evolving hydrodynamic simulations
constrain the period and driving mechanisms of Betelgeuse's dominant
periodicities in new ways. Our analyses converge to the conclusion that
Betelgeuse's $\approx 400$ day period is the result of pulsation in the
fundamental mode, driven by the $\kappa$-mechanism. Grid-based hydrodynamic
modeling reveals that the behavior of the oscillating envelope is
mass-dependent, and likewise suggests that the non-linear pulsation excitation
time could serve as a mass constraint. Our results place $\alpha$ Ori
definitively in the core helium-burning phase near the base of the red
supergiant branch. We report a present-day mass of $16.5$--$19
~M_{\odot}$---slightly lower than typical literature values.
