The Veech Discrete 2-Circle Problem and Non-Integrable Flat Systems

The paper proves Theorem 3.2 for flat finite polysquare-b-rational translation surfaces by constructing the interval exchange transformation of the α-flow, establishing ergodicity under the separation hypothesis for badly approximable α (Lemma 5.1), and then upgrading from almost-every starting point to every non-pathological starting point via a Furstenberg-style argument (Lemma 6.1), thereby obtaining uniform distribution of every half-infinite α-geodesic (Theorem 3.2) . The candidate solution incorrectly treats these b-rational surfaces as square-tiled/origami (hence lattice) surfaces and invokes Veech dichotomy; however, introducing b-rational gates fundamentally alters the structure (e.g., a 2-square-b surface with irrational b is not a polysquare surface and exhibits behavior that violates the classical uniform/periodic dichotomy) . The model also discards the separation hypothesis, which is essential to the paper’s ergodicity step and cannot be dropped in general . Moreover, the model’s projection-to-torus and “uniquely ergodic for all irrational α” claims contradict known counterexamples on 2-square-b surfaces for non–badly approximable α .