  The Large Hadron Collider provides us new opportunities to search for the
origin of neutrino mass. Beyond the minimal see-saw models a plethora of models
exist which realise neutrino mass at tree- or loop-level, and it is important
to be sure that these possibilities are satisfactorily covered by searches. The
purpose of this paper is to advance a systematic approach to this problem.
Majorana neutrino mass models can be organised by SM-gauge-invariant operators
which violate lepton number by two units. In this paper we write down the
minimal ultraviolet completions for all of the mass-dimension 7 operators. We
predict vector-like quarks, vector-like leptons, scalar leptoquarks, a charged
scalar, and a scalar doublet, whose properties are constrained by neutrino
oscillation data. A detailed collider study is presented for $O_3=LLQ\bar dH$
and $O_8 = L\bar d\bar e^\dagger \bar u^\dagger H$ completions with a
vector-like quark $\chi\sim(3, 2, -\frac{5}{6})$ and a leptoquark
$\phi\sim(\bar 3,1,\frac{1}{3})$. The existing LHC limits extracted from
searches for vector-like fermions and sbottoms/stops are $m_\chi \gtrsim 620$
GeV and $m_\phi\gtrsim 600$ GeV.
