Confluence relations, Hoffman basis over $\mathbb{Z}$_{(2)}, and the Deligne--Drinfeld conjecture

We express every admissible word as a ${\mathbb{Z}}_{(2)}$-linear combination of Hoffman words and explicit confluence relations. This implies that every multiple zeta value is a linear combination of elements of the Hoffman basis with coefficients having odd denominators, as conjectured by Hirose and Sato. Combined with Brown’s theorem and Furusho’s characterization of the Grothendieck–Teichmüller group $\mathrm{G}\mathrm{R}\mathrm{T}$ by confluence relations, our result identifies the de Rham motivic Galois group of mixed Tate motives over $\mathbb{Z}$ with $\mathrm{G}\mathrm{R}\mathrm{T}$ and gives an alternative proof of the Deligne–Drinfeld conjecture, for which a proof was previously presented in the OpenAI Math Release.