Non-idle question

Suppose for a moment that the government of the US were under the command of a madman. Someone wethepeople had reason to suspect was batshit crazy - you know, like Adolf or Charles.

Do wethepeople have the right to have the ruling headpiece checked? Or is the President entitled to some sort of immunity from psychiatric testing, the way foreign diplomats are immune from parking tickets?

Are we required to continue to obey the chief executive as though he were sane, with no recourse to verifying the hypothetical, because the Constitutors didn't happen to provide for any alternative contingency?

What is the law regarding our rights vis a vis the presidential mental?