  In a generic random system, the coexistence of extended and localized states
can be evidenced by the subextensive width of energy distribution of a physical
initial state in, for example, the quantum quenches which involving the local
Hamiltonian. The robust thermalization is also evidenced in terms of the
microscopic canonical ensemble average in the thermalization
limit\cite{Shiraishi}, which satisfies the weak eigenstate thermalization
hypothesis (ETH). In this article, we study the method of two-particle
self-consistency for a system without condensation, i.e., without the
inaccessible localizations that violating the ETH. We provide another
pespective that considering the local conservation of an nonintegrable system
the stubborn correlations between the three kinds of decompositions for the
four-point functions, which can be regarded as the elements in a product of the
self-energy and Green function matrices (i.e., the two-particle correlations in
Kadanoff and Baym notation\cite{Vilk Y M}).
