 Hilbert-Mumford criterion for the affine null cone

For complex $S{L}_n$, choose a sequence taking an unstable vector to zero and apply singular value decomposition to each group element. Compactness gives a limiting right-unitary factor. Every nonzero torus-weight component of the transformed vector forces a strict linear inequality on the diagonal exponents. A rational solution of these finitely many strict inequalities scales to integral exponents with sum zero, producing an algebraic one-parameter subgroup that takes the vector to zero. The converse follows immediately from the subgroup limit.