You shall not pass! -- explicit diffeomorphism violation "no-go" constraints and discontinuities

This paper collects several results in the study of the explicit symmetry-breaking limit of the effective-field theory (EFT) description of diffeomorphism and local Lorentz-symmetry breaking, where we generalize a subset of the EFT framework (the ``minimal" sector). It is well known that no-go constraints may arise in cases of explicit symmetry-breaking in curved spacetime as a consequence of the Bianchi identities; we show in this work that certain terms in the action can be countenanced and used to cancel would-be no-go constraints, at least in the linearized gravity limit. Nonetheless, we go on to find more potential issues, and we show that one particular explicit breaking subset of the EFT, while evading direct no-go constraints, results in a discontinuity - unsuppressed additional polarizations for gravitational waves. In a general treatment of the explicit breaking EFT, but confined to linearized gravity, we explicitly show the existence of an extra degree of freedom, independent of coordinates. We find extra polarizations of gravitational waves in the solutions, with a scalar mode unsuppressed by any coefficient, which could render these cases ruled out by observations.