Classification in Active Dimension 2 for Weighted Residual Dynamics

We study weighted residual dynamics associated with a rank-one projection in finite dimension. The iteration reduces, after finitely many steps, to a nonlinear recursion on a stabilized active subspace. We prove that this recursion can be classified when the active dimension is two: either a transverse reducing direction persists unchanged, or the coupled part collapses completely. As a consequence, we obtain a description of the limit in the active two-dimensional case and identify the threshold beyond which higher-dimensional behavior becomes more flexible.