What is the luminosity needed for discovering new physics if the electroweak scale is to remain stable? In this work we study this question, with the pertinent example of a real singlet scalar which couples to the Higgs field at the renormalizable level. Observing that the electroweak scale remains stable if the two scalars couple in a see-sawic fashion through a mass-degeneracy-driven unification linkup among quartic couplings at a given scale, we show by detailed simulation studies of the pp -> (singlet scalar)-> ZZ -> 4l channel that the HL-LHC, which is expected to deliver an integrated luminosity of 3 ab-1, has no significant excess of signal over the background in the 800-2000 GeV mass range. The FCC-hh, on the other hand, can discover scalars up to a mass of 870 GeV with an integrated luminosity 20 ab(-1). Observation at 3 sigma (discovery at 5 sigma) of a new scalar with a minimum mass 800 GeV requires at least 2 ab(-1) (5.2 ab(-1)) integrated luminosity, showing that the new physics that does not destabilize the electroweak scale is accessible only at very high luminosities, and can be tested already in the early stages of the FCC-hh operation period.