Thin monodromy in $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$

We explore the thinness of hypergeometric groups of type $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$ by applying a new approach of computer-assisted ping pong. We prove the thinness of $17$ hypergeometric groups with maximally unipotent monodromy in $\mathrm{Sp}(6)$, completing the classification of all $40$ such groups into arithmetic and thin cases. In addition, we establish the thinness of further $46$ hypergeometric groups in $\mathrm{Sp}(6)$, and of $3$ hypergeometric groups in $\mathrm{Sp}(4)$, completing the classification of all $\mathrm{Sp}(4)$ hypergeometric groups. To the best of our knowledge, this article produces the first $63$ examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.