PATCH FREQUENCIES IN RHOMBIC PENROSE TILINGS

The paper’s Proposition 4.1 explicitly proves that the Penrose rhombic tiling frequency module equals (1/10)Z[τ], by (i) showing any patch lies in a level-n vertex supertile whose frequency is τ^{-2n} times a vertex-type frequency and (ii) deriving two Z-linear relations among the eight vertex frequencies that yield 1/10 and τ^{-1}/10, hence 1/10 Z[τ] (see Proposition 4.1 and its proof, and the tabulated vertex frequencies) . The candidate solution reproduces the same structure: strict ergodicity for uniform frequencies, containment in vertex supertiles with τ^{-2n} scaling, and the same linear relations to generate 1/10 and (τ−1)/10. It adds a clear counting argument for pointed patches and an explicit minimal-denominator check; both are consistent with the paper. No substantive discrepancies found.