  For classical discrete systems under constant composition, statistical
mechanics tells us that a set of microscopic state dominantly contributing to
thermodynamically equilibrium state should depend on temperature as well as on
many-body interaction (i.e. thermodynamic information), through Boltzamann
factor of exp(-bE). Despite this fact, our recent study reveals that a single
(and a few additional) microscopic state (called projection state: PS), whose
structure can be known a priori without r equiring thermodynamic information,
can universally characterize equiibrium properties for disordered states, where
their sturctures depends on configurational geometry before applying many-body
interaction to the system. Although mathematical condition for the structures
of PS have been rigorously established, practically effective condition for
constructing the stuructures, especially for which set of a finite structural
degree of freedoms (SDF) should be selected for considered coordination has not
been clarified so far. We here tuckle this problem, proposing a quantitative
and systematic criteria for an optimal set of SDFs. The present proposal
enables to effectively constructing PSs for a limited system size, and also
providing new insight into which set of SDFs should generally affects
equilibrium properties along a chosen coordination, without using any
thermodynamic information.
