  K2-19b and c were among the first planets discovered by NASA's K2 mission and
together stand in stark contrast with the physical and orbital properties of
the solar system planets. The planets are between the size of Uranus and Saturn
at 7.0$\pm$0.2 R_E and 4.1$\pm$0.2 R_E, respectively, and reside a mere 0.1%
outside the nominal 3:2 mean-motion resonance. They represent a different
outcome of the planet formation process than the solar system, as well as the
vast majority of known exoplanets. We measured the physical and orbital
properties of these planets using photometry from K2, Spitzer, and ground-based
telescopes, along with radial velocities from Keck/HIRES. Through a joint
photodynamical model, we found that the planets have moderate eccentricities of
$e \approx0.20$ and well-aligned apsides $\Delta \varpi \approx 0$ deg. The
planets occupy a strictly non-resonant configuration: the resonant angles
circulate rather than librate. This defies the predictions of standard
formation pathways that invoke convergent or divergent migration, both of which
predict $\Delta \varpi \approx 180$ deg and eccentricities of a few percent or
less. We measured masses of $M_{p,b}$ = 32.4$\pm$1.7 M_E and $M_{p,c}$ =
10.8$\pm$0.6 M_E. Our measurements, with 5% fractional uncertainties, are among
the most precise of any sub-Jovian exoplanet. Mass and size reflect a planet's
core/envelope structure. Despite having a relatively massive core of $M_{core}
\approx15$ $M_E$, K2-19b is envelope-rich, with an envelope mass fraction of
roughly 50%. This planet poses a challenge to standard models core-nucleated
accretion, which predict that cores $\gtrsim 10$ $M_E$ will quickly accrete gas
and trigger runaway accretion when the envelope mass exceeds that of the core.
