Permutation-equivariant quantum K-theory VII. General theory

We introduce K-theoretic GW-invariants of mixed nature: permutation-equivariant in some of the inputs and ordinary in the others, and prove the ancestor-descendant correspondence formula. In genus 0, combining this with adelic characterization, we derive that the range ${\mathcal L}_X$ of the big J-function in permutation-equivariant theory of a target space $X$ is overruled.