Half-wormholes in nearly AdS$_2$ holography

What I wanted to say is that $z$ is an overall shift in the SYK definition of energy: for a generic tensor $J_{ijkl}$, the sum $\sum_{ijkl} J_{ijkl} \psi_i \psi_k \psi_k \psi_l$ can be separated into two parts. The antisymmetric part of $J_{ijkl}$ gives the standard SYK hamiltonian.
The symmetric part of $J_{ijkl}$ yields a constant, since fermions anticommute: $\{\psi_i, \psi_j\} = \delta_{ij}$. Since all fermion operators disappear, the residual sum over $ijkl$ yields $z$. This is a purely analytic argument. Interestingly, eq. (4.8) is indeed the shift in the ground state energy, after cancelling $\arctan$ and $\tan$ and factors of $T$. So one does not need to perform a numerical fit to obtain a precise relation between $j_0$ and $z$(modulo subtleties of taking a $\log$ of a complex $Z$, presumably this is why you have $\arctan(\ tan)$ in eq. (4.8) ). I strongly believe that Section 6 should emphasize this point.