Completeness of obtuse rational lattice triangles

Prove that every obtuse rational triangle with angles that are rational multiples of π whose unfolding yields a Veech (lattice) surface is either in one of the two infinite families with angles (π/n, π/n, (n−2)π/n) or (π/(2n), π/n, (2n−3)π/(2n)), or is the sporadic Hooper triangle with angles (π/12, π/3, 7π/12); equivalently, establish that the currently known list of obtuse rational lattice triangles is complete.

Background

The classification of lattice triangles asks which rational-angled Euclidean triangles unfold to translation surfaces with Veech (lattice) property. Acute and right-angled cases are classified, and two infinite families of obtuse examples and one sporadic example are known.

The paper focuses on the remaining difficulty in the obtuse scalene regime, especially the hard obtuse window where the largest angle lies in (π/2, 2π/3]. The stated conjecture asserts that no additional obtuse rational lattice triangles exist beyond the two known infinite families and Hooper’s sporadic example.