= Hilbert-Mumford criterion for the affine null cone
{c}
For complex $SL_n$, 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.
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