Highest Weight Classification 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,S^1)T g C:= g R C t C:= t R CR X^(T) g C t CR^+ R R^(G,T,R^+)X^(T)^+:=\ X^(T): ,^ 0 R^+\:G GL(V)V X^(T)TVR^+ g C(G)G[V] V(.