Abstract: Higher-order time derivatives naturally occur in effective field theory (EFT) but are usually avoided due to an expectation of catastrophic instability. 
In the first part of this talk, I highlight an unscrutinized inference step in folk no-go theorems: While it is proven that generic higher-derivative systems exhibit unbounded energy, their dynamics need not be fatally unstable. Specifically, I will present proven stability statements for exemplary models in classical mechanics, quantum mechanics, and classical field theory. Throughout, I will emphasize more general physical insights that these counterexamples teach us.

In the second part, I connect these insights to recent progress in numerical relativity, enabling well-posed time evolution in EFTs of gravity. In particular, this is crucial for systematic constraints from gravitational-wave observations of black-hole binaries.