Mixed Pentagon, octagon and Broadhurst duality equation

This paper is on elimination of defining equations of the cyclotomic analogues, introduced by the first author, of Drinfeld's scheme of associators. We show that the mixed pentagon equation implies the octagon equation for N=2 and the particular distribution relation. We also explain that Broadhurst duality is compatible with the torsor structure. We develop a formalism of infinitesimal module categories and use it for deriving a proof left implicit in the first named author's earlier work.