  With an expected rate of about one event per 100,000 top-quark pairs, four
top-quark final states very rarely arise at the LHC. Though scarce, they offer
a unique window onto top-quark compositeness, self-interactions and more
generically, onto any top-philic new physics. By employing simplified models
featuring heavy resonances, we study the range of validity of effective theory
interpretations of current four top-quark analyses at the LHC and establish
their future reach at the HL-LHC. We find that for the class of models under
consideration, the effective field theory interpretations are not applicable.
We therefore present the most up-to-date limits obtained from public CMS
analyses using simplified models. Finally, we put forward a novel recasting
strategy for the experimental results based on the production of top quarks
with large transverse momentum.
