  We study, from theoretical and phenomenological angles, the Higgs boson
oblique parameter $\hat{H}$, as the hallmark of off-shell Higgs physics.
$\hat{H}$ is defined as the Wilson coefficient of the sole dimension-6 operator
that modifies the Higgs boson propagator, within a Universal EFT.
Theoretically, we describe self-consistency conditions on Wilson coefficients,
derived from the K\"{a}ll\'{e}n-Lehmann representation. Phenomenologically, we
demonstrate that the process $gg\to h^\ast \to VV$ is insensitive to propagator
corrections from $\hat{H}$, and instead advertise four-top production as an
effective high-energy probe of off-shell Higgs behaviour, crucial to break flat
directions in the EFT.
