Application of complements

In the last decade unexpectedly, some expected properties of thresholds related to singularities and intersection properties were established. In its turn, the acc for lc thresholds implies the uniform stability of lc singularities and of the $\mathbb{R}$-complementary property. We present a different approach to these and other results related to thresholds based on the theory of complements. We start from an inversion of both uniform stabilities. After that we explained how the inversion can be applied to obtained some of already known and new results, including but not only, lc and Fano index thresholds, lc indices, $a$, $\alpha$- invariants.