Equilibrium Condition chain:( Conditional Gibbs property: XR Given any R and Xg the Xalways follows XR Fix any o∈[g]R,x∈[g]R X京 y∈[g]'that yg'=t 4OF)x∑x·P(x,R),O,R) x∈lgV XR=GEquilibrium Condition chain: (x, R) ⟶ (y, R′) Given any R and X , the always follows . R XR μ XR R Conditional Gibbs property: P μτ R′ (yR′ ) ∝ x ∑ ∈ [q] V xR = σ μσ R(xR) ⋅ P((x, R), (y, R′)) Fix any σ ∈ [q] . R , τ ∈ [q] R′ ∀y ∈ [q] that : V yR′ = τ XR XR