On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$

In this paper we use the topological vertex formalism to derive a structural formula for the leading terms of the generating series of Gromov--Witten invariants of local $\mathbb P^2$. Then we give two applications of this formula. First we use this result to derive an explicit formula for the leadi