Phases in the cyclic quantum dilogarithm: correction to "Cyclic quantum Teichmüller theory"

We first correct ${\mathit{\mu }}_N$-phase issues in the basic definition around the cyclic quantum dilogarithm operators in [Ish26]. In particular, the notion of global coefficients must be extended by incorporating additional lifting data forming a ${\mathit{\mu }}_N$-cover, and the pentagon relation generally acquires an extra ${\mathit{\mu }}_N$-factor. Nevertheless, we prove that every point in a regular locus of the partially compactified cluster Poisson variety lifts to a \emph{strict} extended global coefficient, for which this extra ${\mathit{\mu }}_N$-factor is trivial. Consequently, the original statements of [Ish26] hold for such lifts and are therefore restored.