To see how much the minimal point contributes to the integral we can integrate it in its vicinity

I think you should be looking at the entire stable island, not just integrating from zero to one. I expect you could get a decent approximation with Lie transform perturbation theory, and this looks similar to the idea of macro-states in condensed matter physics, but I’m not knowledgeable in these areas.

Perhaps “fit”, from the Latin

fio(come about) + Englishfit(fit). An object must fit, survive, and spread.