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We establish an algebraic approach to prove the properness of moduli spacesof K-polystable Fano varieties and reduce the problem to a conjecture ondestabilizations of K-unstable Fano varieties. Specifically, we prove that ifthe stability threshold of every K-unstable Fano variety is computed by adivisorial valuation, then such K-moduli spaces are proper. The argument relieson studying certain optimal destabilizing test configurations and constructinga Theta-stratification on the moduli stack of Fano varieties.

On properness of K-moduli spaces and optimal degenerations of Fano varieties