  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 the
Weinberg-Witten theorem.
