  It has long been understood that the light curve of a transiting planet
constrains the density of its host star. That fact is routinely used to improve
measurements of the stellar surface gravity and has been argued to be an
independent check on the stellar mass. Here we show how the stellar density can
also dramatically improve the precision of the radius and effective temperature
of the star. This additional constraint is especially significant when we
properly account for the 4.2% radius and 2.0% temperature systematic errors
inherited from photometric zero-points, model atmospheres, interferometric
calibration, and extinction. In the typical case, we can constrain stellar
radii to 3% and temperatures to 1.75% with our evolutionary-model-based
technique. In the best real-world cases, we can infer radii to 1.6% and
temperatures to 1.1% -- well below the systematic measurement floors -- which
can improve the precision in the planetary parameters by a factor of two. We
explain in detail the mechanism that makes it possible and show a demonstration
of the technique for a near-ideal system, WASP-4.
  We also show that both the statistical and systematic uncertainties in the
parallax from Gaia DR3 are often a significant component of the uncertainty in
$L_*$ and must be treated carefully. Taking advantage of our technique requires
simultaneous models of the stellar evolution, bolometric flux (e.g., a stellar
spectral energy distribution), and the planetary transit, while accounting for
the systematic errors in each, as is done in EXOFASTv2.
