Hyper-K\"ahler manifolds from Riemann-Hilbert problems I: Ooguri-Vafa-like model geometries

We construct model hyper-K\"ahler geometries that include and generalize the multi-Ooguri-Vafa model using the formalism of Gaitto, Moore, and Neitzke. This is the first paper in a series of papers making rigorous Gaiotto--Moore--Neitzke's formalism for constructing hyper-K\"ahler metrics near semi-flat limits. In that context, this paper describes the assumptions we will make on a sequence of lattices $0 \to \Gamma_{f} \to \widehat{\Gamma} \to \Gamma \to 0$ over a complex manifold $\mathcal{B}'=\mathcal{B} - \mathcal{B}''$ near the singular locus, $\mathcal{B}''$, in order to define a smooth manifold $\mathcal{M} \to \mathcal{B}$ and hyper-K\"ahler model geometries on neighborhoods of points of the singular locus. In follow-up papers, we will use a modified version of Gaiotto-Moore-Neitzke's iteration scheme starting at these model geometries to produce true global hyper-K\"ahler metrics on $\mathcal{M}$.