Operations on spectral partition Lie algebras and TAQ cohomology

We determine all natural operations and their relations on the homotopy groups of spectral partition Lie algebras, which coincide with $\mathbb{F}_p$-linear topological Andr\'{e}-Quillen cohomology operations at any prime. We construct unary operations and a shifted restricted Lie algebra structure on the homotopy groups of spectral partition Lie algebras. Then we prove a composition law for the unary operations, as well as a compatibility condition between unary operations and the shifted Lie bracket with restriction up to a unit for the restriction. Comparing with Brantner-Mathew's result on the ranks of the homotopy groups of free spectral partition Lie algebras, we deduce that these generate all natural operations, thereby also recovering unpublished results of Kriz and Basterra-Mandell on $\mathbb{F}_p$-linear TAQ cohomology operations. As a corollary, we determine the structure of natural operations on mod $p$ $\mathbb{S}$-linear TAQ cohomology.