Highest Weight Classification Theorem for Compact Connected Lie Groups identifies how weights organize finite-dimensional representations of compact Lie groups. In concrete terms, it formalizes the statement that let be a compact connected Lie group with gT G tX^(T):=(T,U(1))T g C:= g R C t C:= t R CR X^(T) g C t C t CR^+ R R^+ R^^+(G,T,R^+) X^(T) ,^ Z 0:G GL(V)|TTX^(T) - Z 0R^+^+(G,T,R^+)[.