  We provide a proof that all polynomial higher-derivative effective field
theories of vacuum gravity admit a well-posed initial value formulation when
augmented by suitable regularising terms. These regularising terms can be
obtained by field redefinitions and allow to rewrite the resulting equations of
motion as a system of second-order nonlinear wave equations. For instance, our
result applies to the quadratic, cubic, and quartic truncations of the
effective field theory of gravity that have previously appeared in the
literature. The regularising terms correspond to fiducial massive modes,
however, their masses can be chosen to be non-tachyonic and heavier than the
cutoff scale and hence these modes should not affect the dynamics in the regime
of validity of effective field theory. Our well-posed formulation is not
limited to the weakly coupled regime of these theories, is manifestly covariant
and does neither require fine tuning of free parameters nor involves
prescribing arbitrary equations.
