$\Theta$-reductivity and $S$-completeness for adjoint Fano foliated structures

We prove the valuative criteria of $\Theta$-reductivity and $S$-completeness for the moduli problem of $t$-K-semistable adjoint Fano foliated structures.

We develop a mixed Ding theory for arbitrary linearly bounded multiplicative filtrations, prove inversion of adjunction with arbitrary ideals for adjoint foliated structures, and establish a relative extraction and finite generation theorem.

Together, these results yield the required relative extension theorems for families.

As applications, we prove uniqueness of $t$-K-polystable degenerations, reductivity of the automorphism group of $t$-K-polystable adjoint Fano foliated structures, and finiteness of the automorphism group in the $t$-K-stable case.