The key is to produce these particles and watch them decay. The steps taken are to first accelerate particles up to a desired energy and have them
collide. We produce a whole bunch of stuff and we observe tracks and decays.
The proposed 4th generation of lepton I think is ruled out based on a wealth of experimentation.
to produce it you need a centre of mass energy of about ~21 GeV
Leptons are easy to search for because their production tends to be very clean, put in double the kinetic energy and get out a particle -
anti-particle pair. This is how the tau was discovered, and given we have electron - positron beams that were performing spectrum sweeps through the
20 GeV energy region, it is something that would have perhaps been observed.
Accelerators that were capable of observing this are
Technically any accelerator using an electron-positron beam can do it. Though higher design energy beams will typically not search low energy regions,
these regions will be then searched by other, smaller, fixed target experiments.
So I think a 4th generation lepton is ruled out, and was ruled out as far back as the 80s or 90s that is not to say people are not looking for it
This arxiv paper is one example i found just by dropping the phrase into google.
The search for 4th generation of quarks is also on going, but if the Standard Model is to be believed, the discovery of the higgs where it is,
actually limits the existence of a forth generation of quarks. Production of higher mass fermions that would contain a 4th generation have not been
observed as yet, and the existence would affect the mass of the higgs since that 4th generation would strongly couple to it.
It would be a similar affect as the charm coupling suppressing particle production at lower energy. The same would happen, existence of a forth
generation at higher energy would be hinted in the data we have already, even at 16TeV.
Regardless, it is something that I don't think can be ruled out, and I don't believe we need mathematical beauty in all this. We could have a weird
system where we have 3 generations of leptons but 4 of quarks.
The LHC will turn on again this year and ramp up to ~14 TeV, it would not be enough to produce those particles you 'predict' (i put it in inverted
commas because if it is based on the model you posted previously then it is not clear that the model should have any predictive power at all) but it
would get close and so probably produce a spectrum with hints of the coupling (if it exists)