Newton-Okounkov Bodies and Jet Separation: Canonical-Free and Multipoint Generalizations

We establish three generalizations of the K\"uronya-Lozovanu jet separation criterion via Newton-Okounkov bodies: if an inverted standard simplex of size $n+k+\varepsilon$ is contained in all infinitesimal Newton-Okounkov bodies at $x$, then $K_X+D$ separates $k$-jets at $x$. We prove (1) a canonical-free version with a computable multiple $m(D)$; (2) a multipoint extension for simultaneous jet separation; and (3) a combination of both. Proofs use Trusiani's framework and Nadel vanishing. We conclude with explicit computations for a double cover of a product of elliptic curves.