Non-proportional wall crossing for K-stability

In this paper, we present a general wall crossing theory for K-stability and K-moduli of log Fano pairs whose boundary divisors can be non-proportional to the anti-canonical divisor. Along the way, we prove that there are only finitely many K-semistable domains associated to the fibers of a log bounded family of couples. Under the additional assumption of volume bounded from below, we show that K-semistable domains are semi-algebraic sets (although not necessarily polytopes). As a consequence, we obtain a finite semi-algebraic chamber decomposition for wall crossing of K-moduli spaces. In the case of one boundary divisor, this decomposition is an expected finite interval chamber decomposition. As an application of the theory, we prove a comparison theorem between GIT-stability and K-stability in non-proportional setting when the coefficient of the boundary is sufficiently small.