  I explicitly construct a strong zero mode in the XYZ chain or, equivalently,
Majorana wires coupled via a four-fermion interaction. The strong zero mode is
an operator that pairs states in different symmetry sectors, resulting in
identical spectra up to exponentially small finite-size corrections. Such
pairing occurs in the Ising/Majorana fermion chain and possibly in
parafermionic systems and strongly disordered many-body localized phases. The
proof here shows that the strong zero mode occurs in a clean interacting
system, and that it possesses some remarkable structure -- despite being a
rather elaborate operator, it squares to the identity. Eigenstate phase
transitions separate regions with different types of pairing.
