Cutting sequence and Sturmian sequence in billiard

The paper states the key facts needed here — that the shear [1 0; −1 1] shortens every A-block by one for slopes >1 (Theorem 5.1), and the flip [0 1; 1 0] swaps A/B and inverts the slope; it also presents the shear/flip algorithm corresponding to continued fractions and asserts that, for slope p/q in lowest terms, a period contains p A’s and q B’s — but gives only example-based or sketch justifications and some statements are imprecise (e.g., the membership test Theorem 5.2 erroneously rejects patterns like …BBBABBB… that do occur in valid cutting sequences for 0<m<1). By contrast, the model supplies a complete, rigorous line-by-line proof of the shear rule via a floor-count argument, correctly ties the sequence-level moves to the Euclidean algorithm/continued fractions, and proves the p/q letter count and minimality. Hence the model is correct while the paper is incomplete and contains at least one flawed claim .