  Heavy neutrinos $(N)$ remain one of most promising explanations for the
origin of neutrinos' tiny masses and large mixing angles. In light of broad
advances in understanding and modeling of hadron collisions at large momentum
transfer, we revisit the long-standard search strategy for heavy $N$ decaying
to multiple charged leptons $(\ell)$, $pp \to N\ell X \to 3\ell \nu X$. For
electroweak and TeV-scale $N$, we propose a qualitatively new collider analysis
premised on a dynamic jet veto and discriminating, on an event-by-event basis,
according to the relative amount of hadronic and leptonic activity. We report
that the sensitivity to $N$ at the Large Hadron Collider (LHC) can be improved
by roughly an order of magnitude over the collider's lifetime. At $\sqrt{s}=14$
TeV with $\mathcal{L}=3~{\rm ab}^{-1}$, active-sterile mixing as small as
$\vert V_{\ell N}\vert^2 = 10^{-2} ~(10^{-3})~[5\times10^{-4}]$ can be probed
at $95\%$ CL for Dirac neutrinos masses $m_N \lesssim 1200~(300)~[200]$ GeV,
well beyond present constraints for such heavy states. The improvement holds
also for Majorana $N$, and is largely independent of whether charged lepton
flavor is conserved or violated. The analysis, built almost entirely from
inclusive, transverse observables, is designed to be robust across increasing
collider energies, and hence serves as a basis for searches at future
colliders: With $\mathcal{L}=15~{\rm ab}^{-1}$ at $\sqrt{s}=27$ TeV, one can
probe mixing below $\vert V_{\ell N}\vert^2 = 10^{-2}
~(10^{-3})~[2\times10^{-4}]$ for $m_N \lesssim 3500~(700)~[200]$ GeV. At a
hypothetical 100 TeV $pp$ collider with $\mathcal{L}=30~{\rm ab}^{-1}$, one can
probe below $\vert V_{\ell N}\vert^2 = 9\times10^{-5}~(10^{-3})~[10^{-2}]$ for
$m_N \lesssim 200$ GeV ($4$ TeV) [$15$ TeV]. We anticipate these results can be
further improved with detector-specific tuning and application of machines
learning techniques.
