Bounded $\mathfrak{sp}_4$-laminations and their intersection coordinates

We introduce rational bounded $\mathfrak{sp}_4$-laminations on a marked surface $\boldsymbol{\Sigma}$ as a proposed topological model for the rational tropical points $\mathcal{A}_{Sp_4,\boldsymbol{\Sigma}}(\mathbb{Q}^{\mathsf{T}})$ of the Fock--Goncharov moduli space [FG06]. Our space consists of certain equivalence classes of $\mathfrak{sp}_4$-webs introduced by Kuperberg [Kup96], together with rational measures. We define tropical coordinate systems using the $\mathfrak{sp}_4$-case of the intersection number of Shen--Sun--Weng [SSW25], and establish a bijection using the framework of the graded $\mathfrak{sp}_4$-skein algebra. This provides a topological perspective for Fock--Goncharov duality for $\mathfrak{sp}_4$.