The author previously defined the spectral invariants, denoted by $ ho(H;;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $ ho(H;;a)$ states that the mini-max value is a critical value of the action functional ${cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $omega$). We also prove that the spectral invariant function ${ ho}_a$ : $H;{mapsto}; ho(H;;a)$ can be pushed down to a continuous function defined on the universal (${acute{e}}tale$) covering space $widetilde{HAM}$(M, $omega$) of the group Ham((M, $omega$) of Hamiltonian diffeomorphisms on general (M, $omega$). For a certain generic homotopy, which we call a Cerf homotopy ${cal{H}};=;{H^s}_{0{leq}s{leq}1}$ of Hamiltonians, the function ${ ho}_a;{circ};{cal{H}}$ : $s;{mapsto};{ ho}(H^s;;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.