Diagonal trace identities for bosonic and fermionic matrices

We determine the second stabilised diagonal of finite-rank fermionic trace relations in the $\partial\psi$ model. In this paper, the formal letters of the model are treated as independent odd matrix letters, denoted $\psi_0,\psi_1,\psi_2,\ldots$. For every $N\geqslant 2$, the charge-$2N+2$ relation space at rank $N$ has odd dimension $4$ and even dimension $4$. More precisely, it has a basis consisting of the eight signed polarised Cayley--Hamilton trace identities obtained by inserting copies of the even word $\psi_0^2$. This proves a conjecture of Eleftheriou, Ji, and Murthy.