Log-local principle for BPS invariants

Assume that $S$ is a del Pezzo surface and $E$ is an anticanonical divisor. Let $P\in E(\beta)$ be $\beta$-primitive, where $m^P_\beta$ is the log BPS number at $P$ and $n_\beta$ is the genus-zero local BPS invariant of the class $\beta$. Log-local principle for BPS invariants.

$$n_\beta=(-1)^{w-1}w\,m^P_\beta.$$

This is the primitive-point version of the proposed relation between log and local BPS invariants. The paper identifies it as a BPS form of the log-local principle, known in some cases and connected with orbifold Gromov-Witten theory.