We report Spitzer Space Telescope observations during predicted transits of the exoplanet Proxima Centauri b. As the nearest terrestrial habitable-zone planet we will ever discover, any potential transit of Proxima b would place strong constraints on its radius, bulk density, and atmosphere. Subsequent transmission spectroscopy and secondary-eclipse measurements could then probe the atmospheric chemistry, physical processes, and orbit, including a search for biosignatures. However, our photometric results rule out planetary transits at the 200~ppm level at 4.5$~{mu}m$, yielding a 3$sigma$ upper radius limit of 0.4~$R_rm{oplus}$ (Earth radii). Previous claims of possible transits from optical ground- and space-based photometry were likely correlated noise in the data from Proxima Centauris frequent flaring. Follow-up observations should focus on planetary radio emission, phase curves, and direct imaging. Our study indicates dramatically reduced stellar activity at near-to-mid infrared wavelengths, compared to the optical. Proxima b is an ideal target for space-based infrared telescopes, if their instruments can be configured to handle Proximas brightness.
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