Hyperbolic Staircases: Periodic Paths on 2g + 1–gons

The paper’s Theorem 3 explicitly states and justifies that for odd n the slopes of the diagonals of the staircase built from a double n‑gon are −v_{n−1}, …, −v_{(n+1)/2} with v_i = sin((1−i)π/n)/sin(iπ/n), via a stereographic-projection computation of the v_i together with the fact that the global skew acts by x ↦ −x on boundary slopes and takes the zig‑zag triangulation to right triangles and then rectangles (see the construction and theorem statements/proofs). The candidate solution reaches the same numerical formula but its key step is incorrect: it asserts that, under a single global affine skew, all “star” triangles Δ_k = conv{ζ^0, ζ^{k−1}, ζ^k} become right with legs aligned to the fixed horizontal/zig‑zag directions; an invertible linear map cannot simultaneously send the many distinct directions of the edges [0,k−1] and [0,k] to just two axis directions. The paper’s triangulation is the specific zig‑zag triangulation (not the star from vertex 0), and after the uniform skew it indeed yields right triangles whose hypotenuse pairings form the staircase rectangles; the model’s proof misidentifies these triangles and overuses the global skew. Hence the paper’s result is correct and justified, while the model’s argument contains a fatal geometric error, even though the final formula matches the paper’s theorem.