  We exhibit the geometric structure of the convex cone in the linear space of
the Wilson coefficients for the dimension-8 operators involving the left-handed
lepton doublet $L$ and the Higgs doublet $H$ in the Standard Model effective
field theory (SMEFT). The boundary of the convex cone gives rise to the
positivity bounds on the Wilson coefficients, while the extremal ray
corresponds to the unique particle state in the theory of ultra-violet
completion. Among three types of canonical seesaw models for neutrino masses,
we discover that only right-handed neutrinos in the type-I seesaw model show up
as one of extremal rays, whereas the heavy particles in the type-II and
type-III seesaw models live inside the cone. The experimental determination of
the relevant Wilson coefficients close to the extremal ray of type-I seesaw
model will unambiguously pin down or rule out the latter as the origin of
neutrino masses. This discovery offers a novel way to distinguish the most
popular seesaw model from others, and also strengthens the SMEFT as an
especially powerful tool to probe new physics beyond the Standard Model.
