  We derive the scale of unitarity violation from the geometry of Effective
Field Theory (EFT) extensions of the Standard Model Higgs sector. The
high-energy behavior of amplitudes with more than four scalar legs depends on
derivatives of geometric invariants with respect to the physical Higgs field
$h$, such that higher-point amplitudes begin to reconstruct the scalar manifold
away from our vacuum. In theories whose low-energy limit can be described by
the Higgs EFT (HEFT) but not the Standard Model EFT (SMEFT), non-analyticities
in the vicinity of our vacuum limit the radius of convergence of geometric
invariants, leading to unitarity violation at energies below $4\pi v$. Our
results unify approaches to the HEFT/SMEFT dichotomy based on unitarity,
analyticity, and geometry, and more broadly illustrate the sense in which
observables probe the geometry of an EFT. Along the way, we provide novel
basis-independent results for Goldstone/Higgs boson scattering amplitudes
expressed in terms of geometric covariant quantities.
