Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge

What the math gives to ML

The paper gives an explicit arithmetic construction of balanced binary words and a unique Stern-Brocot/Farey recursion for generating them from smaller coprime fractions. The transferable asset is not the tiling geometry itself, but the deterministic distribution property of the carry word: exactly m of k positions are selected, while every prefix tracks the ideal density m/k with rounding error below one. This can parameterize structured sparse attention or conditional-compute masks whose connectivity is evenly distributed rather than clustered, with only a coprime slope and a small recursive description. The cleanest first experiment is a circulant Christoffel-word attention mask at long context, compared against local, random, and strided sparse attention at equal edge count.