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Extensions of the photon and graviton soft theorems are derived in 4D local effective field theories with massless particles of arbitrary spin. We prove that effective operators can result in new terms in the soft theorems at subleading order for photons and subsubleading order for gravitons. The new soft terms are unique, and we provide a complete classification of all local operators responsible for such modifications. We show that no local operators can modify the subleading soft graviton theorem. The soft limits are taken in a manifestly on-locus manner using a complex double deformation of the external momenta. In addition to the new soft theorems, the resulting master formula yields consistency conditions, such as the conservation of electric charge, the Einstein equivalence principle, supergravity Ward identities, and that particles with spin greater than two cannot couple to those with spin less than or equal to two.