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In a generic random system, the coexistence of extended and localized statescan be evidenced by the subextensive width of energy distribution of a physicalinitial state in, for example, the quantum quenches which involving the localHamiltonian. The robust thermalization is also evidenced in terms of themicroscopic canonical ensemble average in the thermalization limit[14], whichsatisfies the weak ETH. Here we provide another pespective that considering thelocal conservation of an nonintegrable system as the stubborn correlationsbetween the three kinds of decompositions for the four-point functions, whichcan be regarded as the elements in a product of the self-energy and Greenfunction matrices (i.e., the two-particle correlations in Kadanoff and Baymnotation[5]).

Two-particle self-consistency beyond the extensive and integrable  assumption: Thermalization in perspective of the system localization without  a lowest energy state