Finite generation for valuations computing stability thresholds and applications to K-stability

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