Abstract: In this talk, I’ll survey recently-developed geometric methods for EFTs relevant to the Higgs sector, including a new understanding of the connection between EFT geometry and observables; a new notion of ​‘functional geometry’ invariant under derivative field redefinitions; and a new off-shell recursion relation for EFT amplitudes that follows from this functional geometry. These developments sharpen an open question to address at the LHC and future colliders: is electroweak symmetry linearly realized by the known particles of the Standard Model?