The inverse local-relative correspondence and the Nekrasov--Shatashvili vanishing blowup equation

We prove the Nekrasov--Shatashvili vanishing blowup equation for the local Gromov--Witten theory of smooth weak del Pezzo surfaces, by identifying its unique formal solution with the inverse of the local-relative correspondence. We construct this inverse through an associative set-partition convolution and establish the identification using Zagier's recursion for the Bloch--Okounkov $n$-point function. The proof applies in all genera and includes non-toric surfaces.