Tiling spaces are covering spaces over irrational tori

The paper defines p̃: Ω → Tv = ℝd/Rv, proves it is a smooth surjection (Lemma 4), and shows its structure groupoid is fibrating, hence p̃ is a diffeological fibration (Proposition 7). It then states Theorem A—Ω is a covering over Tv—but the proof appeals only to the dictum that a covering in diffeology is a fiber bundle with discrete fiber, without establishing that the fiber of p̃ is discrete or providing the needed local trivializations along plots . By contrast, the model gives an explicit, well-defined construction of p̃ via a vertex-patch coordinate, builds local trivializations using local lifts for the quotient diffeology, and proves the fiber is diffeologically discrete (plots are locally constant) using that the return module Rv is countable. These steps fill the paper’s missing ingredient and are compatible with the paper’s diffeological setup (e.g., the diffeology on Ω and the quotient diffeology on Tv that guarantee local lifts) . Hence the model completes and corrects the argument.