the space of real $n \times
n$-matrices. In [@Drton07] this model is studied from an algebraic point of view. In particular, the ideal of polynomials vanishing on $F_{k,n}$ is determined for $k=2,3$ and $n \leq 9$. The case where $k=1$ had already been done in [@deLoera95]. The authors of [@Drton07] pose some very intriguing finiteness questions. In particular, one might hope that for fixed $k$ the ideal of $F_{k,n}$ stabilises, as $n$ grows, modulo its natural symmetries coming from simultaneously permuting rows and columns. For $k=1$ this is indeed the case, and for arbitrary $k$ it is true in a weaker, set-theoretic sense [@Draisma08b]. In this paper we prove that the ideals of $F_{2,n}$ stabilise at $n=6$. To state our theorem we denote by $y_{ij}$ the coordinates on the space of symmetric $n \times n$-matrices; we will identify $y_{ji}$ with $y_{ij}$. Recall from [@Drton07] that the ideal of $F_{2,5}$ is generated by a single polynomial $$P:=\frac{1}{10} \sum_{\pi \in {\operatorname{Sym}}(5)} \sgn(\pi)
y_{\pi(1),\pi(2)}y_{\pi(2),\pi(3)}y_{\pi(3),\pi(4)}y_{\pi(4),\pi(5)}y_{\pi(5),\pi(1)},$$ called the [*pentad*]{}. The normalisation factor is important only because it ensures that all coefficients are $\pm 1$—indeed, the stabiliser in ${\operatorname{Sym}}(5)$ of each monomial in the pentad is the dihedral group of order $10$. We consider $P$ an 