Planets larger than Neptune have elevated eccentricities

NASA's Kepler mission identified over 4000 extrasolar planets that transit (cross in front of) their host stars. This sample has revealed detailed features in the demographics of planet sizes and orbital spacings. However, knowledge of their orbital shapes - a key tracer of planetary formation and evolution - remains far more limited. We present measurements of eccentricities for 1646 Kepler planets, 92% of which are smaller than Neptune. For all planet sizes, the eccentricity distribution peaks at e=0 and falls monotonically toward zero at e=1. As planet size increases, mean population eccentricity rises from $\langle e \rangle = 0.05 \pm 0.01$ for small planets to $\langle e \rangle = 0.20 \pm 0.03$ for planets larger than $\sim$ 3.5 Earth-radii. The overall planet occurrence rate and planet-metallicity correlation also change abruptly at this size. Taken together, these patterns indicate distinct formation channels for planets above and below $\sim$ 3.5 Earth-radii. We also find size dependent associations between eccentricity, host star metallicity, and orbital period. While smaller planets generally have low eccentricities, there are hints of a noteworthy exception: eccentricities are slightly elevated in the ``radius valley,'' a narrow band of low occurrence rate density which separates rocky ``super-Earths'' (1.0-1.5 Earth-radii) from gas-rich ``sub-Neptunes'' (2.0-3.0 Earth-radii. We detect this feature at $2.1\sigma$ significance. Planets in single- and multi-transiting systems exhibit the same size-eccentricity relationship, suggesting they are drawn from the same parent population.