A proof of the Brian\c{c}on-Iarrobino Conjecture in three dimensions

We resolve the 1978 Brian\c{c}on-Iarrobino Conjecture regarding the maximum singularity of $\mathcal{H}=\mathrm{Hilb}^{l}(\mathbb{A}^3)$, where $l$ is a tetrahedral number, by refining the work of Ramkumar-Sammartano in \cite{Ramkumar-Sammartano}. This also immediately implies the conjectural necessary condition for a point of $\mathcal{H}$ to have the maximal singularity, suggested by the second-named author in \cite{Rezaee-23-Conjectures}. In a sequel to this article, \cite{Mackenzie-Rezaee2}, we prove a generalized version of this conjecture for certain non-tetrahedral $l$, via proving the conjectural necessary condition.