The dynamical influence of binary companions is expected to profoundly influence planetary systems. However, the difficulty of identifying planets in binary systems has left the magnitude of this effect uncertain; despite numerous theoretical hurdles to their formation and survival, at least some binary systems clearly host planets. We present high-resolution imaging of 382 Kepler Objects of Interest (KOIs) obtained using adaptive-optics imaging and nonredundant aperture-mask interferometry (NRM) on the Keck-II telescope. Among the full sample of 506 candidate binary companions to KOIs, we super-resolve some binary systems to projected separations of <5 AU, showing that planets might form in these dynamically active environments. However, the full distribution of projected separations for our planet-host sample more broadly reveals a deep paucity of binary companions at solar-system scales. For a field binary population, we should have found 58 binary companions with projected separation rho < 50 AU and mass ratio q > 0.4; we instead only found 23 companions (a 4.6 sigma deficit), many of which must be wider pairs that are only close in projection. When the binary population is parametrized with a semimajor axis cutoff a_cut and a suppression factor inside that cutoff S_bin, we find with correlated uncertainties that inside a_cut = 47 +59/-23 AU, the planet occurrence rate in binary systems is only S_bin = 0.34 +0.14/-0.15 times that of wider binaries or single stars. Our results demonstrate that a fifth of all solar-type stars in the Milky Way are disallowed from hosting planetary systems due to the influence of a binary companion.
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