Proving directional ergodicity for Solovay–Kitaev recursion

Prove an equidistribution or ergodicity theorem for the direction coordinate of the Solovay–Kitaev residual dynamics under Voronoi quantization by the specific synthesis net, with sufficient strength to establish the stationary limit law without assuming Hypothesis E.

Background

The paper proves contraction and stationarity properties for the residual scale in the Solovay–Kitaev recursion, but the block statistics also depend on the residual direction because direction determines the nearest-neighbour Voronoi cell and hence the emitted net word. The authors introduce Hypothesis E, which assumes uniform geometric ergodicity in the direction coordinate, and use it to derive the stationary compression limit law. They explicitly state that this ergodicity cannot currently be proved and identify an equidistribution theorem for the Voronoi quantization of the specific net construction as the principal open mathematical problem.