The convex geometric framework of positivity bounds allows us to explore the ultraviolet (UV) states in new physics models from the bottom up. The UV states in three types of seesaw models for tiny Majorana neutrino masses, as irreducible representations of the $SU(2)_L$ gauge group, naturally fit into this framework. Since neutrino masses arising from the dimension-five Weinberg operator imply that the UV states may couple to the left-handed lepton doublet $L$ and the Higgs doublet $H$, we construct a convex cone of positivity bounds to constrain the dimension-eight operators consisting of 𝐿 or 𝐻. Such a construction offers a novel way to distinguish between different seesaw models and derive lower bounds on the seesaw scale.