Shokurov's bounded klt complements conjecture for Fano type contractions with finite rational coefficients

Theorem 1.1: for every $d$ and rational $\epsilon>0$ there is $N(d,\epsilon)$ such that every projective contraction $f:X\to Z$ over an algebraically closed field of characteristic zero with $X$ $\epsilon$-lc of dimension $d$, $K_X$ $\mathbb Q$-Cartier and $-K_X$ ample over $Z$ has, near each closed point of $Z$, a boundary $B$ with $(X,B)$ klt and $N(K_X+B)\sim0$. Corollary 1.2 (over $\mathbb C$): the finite-rational-coefficient form for Fano type pairs with nef anti-log-canonical divisor, with monotone complements. The companion proves that some Cartier divisor through any base point has pulled-back lc threshold at least $\tau(d,\epsilon)$, also for real boundaries over $\mathbb C$. Not shown: $\epsilon'$-lc complements, DCC coefficient sets, or effective values of $N$.

Theorem 1.1: for every $d$ and rational $\epsilon>0$ there is $N(d,\epsilon)$ such that every projective contraction $f:X\to Z$ over an algebraically closed field of characteristic zero with $X$ $\epsilon$-lc of dimension $d$, $K_X$ $\mathbb Q$-Cartier and $-K_X$ ample over $Z$ has, near each closed point of $Z$, a boundary $B$ with $(X,B)$ klt and $N(K_X+B)\sim0$. Corollary 1.2 (over $\mathbb C$): the finite-rational-coefficient form for Fano type pairs with nef anti-log-canonical divisor, with monotone complements. The companion proves that some Cartier divisor through any base point has pulled-back lc threshold at least $\tau(d,\epsilon)$, also for real boundaries over $\mathbb C$. Not shown: $\epsilon'$-lc complements, DCC coefficient sets, or effective values of $N$.