Sturmian lattices and Aperiodic tile sets

The paper formalizes the projective simplex Δ of tile ratios, the Veronoi curve TVC(α) = [(1−α)^2:2α(1−α):α^2], and the SAB matching rule, showing that any realizable tiling must lie in TA ∩ TVC where TA is the convex hull of finitely many rational points arising from patch-tiles . It proves the trichotomy: (1) rational α are enforced by a singleton A = {T}, (2) quadratic α by two tiles via a rational line orthogonal to n(α) (constructed from the minimal polynomial), and (3) no finite A enforces non-quadratic α; the argument reduces to intersections of a rational polygon with the conic TVC and the fact that a line meets a conic in at most two points, yielding only rational or quadratic parameters on edges and an arc with rational points in interiors . The candidate solution mirrors this geometry: Δ, TVC as a conic, enforcement as TA ∩ TVC = {TVC(α)}, the rational line cut for quadratic α using the minimal polynomial, and the impossibility for non-quadratic α via tangency considerations. It adds a standard (but nonessential) remark about an inflation coming from an SL(2,ℤ) matrix with eigenvalue a quadratic unit. Net: same core proof idea; the paper supplies full hypotheses (matching rule, density computation) and a constructive orthogonality criterion n(α), while the model provides essentially the same proof in slightly different language. The conclusions match exactly.