Before revision:  The method of partitioning a set of vectors along a sequence is introduced and acts as one of the main analytical tools throughout .
After revision:  One of our main analytical tools, which is introduced here, will be a method for partitioning the relevant monomials of the dynamical system along sequences of trajectory points into classes with comparable growths. This will allow us to conclude that if a trajectory converges to the boundary, then a whole family of Lyapunov functions decrease along the trajectory, a fact that will allow us to overcome the fact that the usual Lyapunov function used in this setting does not diverge to infinity as trajectories converge to the boundary, which has been the technical sticking point to a proof of the Global Attractor Conjecture over the years .