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The Palomar High-precision Astrometric Search for Exoplanet Systems (PHASES) monitored 51 subarcsecond binary systems to evaluate whether tertiary companions as small as Jovian planets orbited either the primary or secondary stars, perturbing their otherwise smooth Keplerian motions. Twenty-one of those systems were observed 10 or more times and show no evidence of additional companions. A new algorithm is presented for identifying astrometric companions and establishing the (companion mass)–(orbital period) combinations that can be excluded from existence with high confidence based on the PHASES observations, and the regions of mass–period phase space being excluded are presented for 21 PHASES binaries.