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The author previously defined the spectral invariants, denoted by$\rho(H;a)$, of a Hamiltonian function $H$ as the mini-max value of the actionfunctional $\AA_H$ over the Novikov Floer cycles in the Floer homology classdual to the quantum cohomology class $a$. The spectrality axiom of theinvariant $\rho(H;a)$ states that the mini-max value is a critical value of theaction functional $\AA_H$. The main purpose of the present paper is to provethis axiom for {\it nondegenerate} Hamiltonian functions in {\it irrational}symplectic manifolds $(M,\omega)$. We also prove that the spectral invariantfunction $\rho_a: H \mapsto \rho(H;a)$ can be pushed down to a {\it continuous}function defined on the universal ({\it \'etale}) covering space$\widetilde{Ham}(M,\omega)$ of the group $Ham(M,\omega)$ of Hamiltoniandiffeomorphisms on general $(M,\omega)$. For a certain generic homotopy, whichwe call a {\it Cerf homotopy} $\HH = \{H^s\}_{0 \leq s\leq 1}$ of Hamiltonians,the function $\rho_a \circ \HH: s \mapsto \rho(H^s;a)$ is piecewise smooth awayfrom a countable subset of $[0,1]$ for each non-zero quantum cohomology class$a$.  The proof of this nondegenerate spectrality relies on several new ingredientsin the chain level Floer theory, which have their own independent interest: astructure theorem on the Cerf bifurcation diagram of the critical values of theaction functionals associated to a generic one-parameter family of Hamiltonianfunctions, a general structure theorem and the handle sliding lemma of NovikovFloer cycles over such a family and a {\it family version} of newtransversality statements involving the Floer chain map, and many others. Wecall this chain level Floer theory as a whole the {\it Floer mini-max theory}.

FLOER MINI-MAX THEORY, THE CERF DIAGRAM, AND THE SPECTRAL INVARIANTS