  We prove that on any log Fano pair of dimension $n$ whose stability threshold
is less than $\frac{n+1}{n}$, any valuation computing the stability threshold
has a finitely generated associated graded ring. Together with earlier works,
this implies: (a) a log Fano pair is uniformly K-stable (resp. reduced
uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the
K-moduli spaces are proper and projective; and combining with the previously
known equivalence between the existence of K\"ahler-Einstein metric and reduced
uniform K-stability proved by the variational approach, (c) the
Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano
pairs.
