Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger

The hyper-Catalan number $C[m_2,m_3,m_4,\ldots]$ counts the number of subdivisions of a roofed polygon into $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc. Its closed form has been known since Erd\'elyi and Etherington, 1940. In 2025, Wildberger and Rubine showed its generating sum $\mathbf{S}[t_2,t_3,t_4,\ldots]$ is a zero of the general geometric univariate polynomial. We use that to derive a recurrence for hyper-Catalans, which expresses each in terms of other hyper-Catalans with smaller indices, generalizing the well-known Catalan convolution sum. Wildberger notes the factorization $\mathbf{S}-1=(t_2 + t_3 + t_4 + \ldots)\mathbf{G}$, where the factor $\mathbf{G}$ is called the Geode. We derive a recurrence that let us express the Geode coefficients in terms of other hyper-Catalan and Geode coefficients, and ultimately in terms of hyper-Catalans alone. We use it to prove three conjectures of Wildberger, all closed forms for special cases of elements of $\mathbf{G}$. While the recurrence allows us to expand each Geode coefficient as an integer combination of hyper-Catalans, enabling calculation, a closed-form for the general Geode coefficient remains unknown, as does what it counts.