Title: Stability of Valuations: Higher Rational Rank,
Abstract: Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial
valuation $v$ with a finitely generated associated graded ring is the minimizer
of the normalized volume function $\widehat{\rm vol}_{(X,D),x}$, if and only if
$v$ induces a degeneration to a K-semistable log Fano cone singularity.
Moreover, such a minimizer is unique among all quasi-monomial valuations up to
rescaling. As a consequence, we prove that for a klt singularity $x\in X$ on
the Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds, the
intermediate K-semistable cone associated to its metric tangent cone is
uniquely determined by the algebraic structure of $x\in X$, hence confirming a
conjecture by Donaldson-Sun.