A rational triangle has angles that are rational multiples of π. Unfold it — reflect repeatedly across its sides — and the unfolding tiles a translation surface. If that surface has a closed orbit in moduli space (a Teichmüller curve), the triangle is called a lattice triangle. Lattice triangles have extraordinary symmetry: their billiard dynamics are completely understood.
The paper on the paucity of lattice triangles (arXiv: 2603.23928) attacks the “hard obtuse window” — triangles with largest angle between π/2 and 2π/3 — where it is conjectured that no lattice triangles exist.
The method uses the Mirzakhani–Wright rank obstruction, reformulated arithmetically. The rank of the GL₂(ℝ)-orbit closure of the translation surface constrains which triangles can be lattice. The arithmetic reformulation converts this into a number-theoretic condition that can be checked computationally. The result: all but a density-zero subset of triangles in the hard obtuse window are ruled out.
The paper notes that the main engine was autoformalized in Lean by AxiomProver — a formal verification tool checked the combinatorial argument.
The through-claim: lattice triangles are rare because the arithmetic conditions are overdetermined. Being a lattice triangle requires the unfolding to have maximal symmetry (rank 1 orbit closure), which imposes algebraic conditions on the angles. In the obtuse window, these conditions are generically unsatisfiable — almost all triangles fail the arithmetic test.
2603.23928. Dynamical systems / translation surfaces / lattice triangles / Teichmüller curves / arithmetic constraints.