The author previously defined the spectral invariants, denoted by

, of a Hamiltonian function H as the mini-max value of the action functional

over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant

states that the mini-max value is a critical value of the action functional

. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M,

). We also prove that the spectral invariant function

:

can be pushed down to a continuous function defined on the universal (

) covering space

(M,

) of the group Ham((M,

) of Hamiltonian diffeomorphisms on general (M,

). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;