Nonexistence of lattice triangles in the hard obtuse window

Determine whether any rational lattice triangle exists whose largest interior angle lies in the interval (π/2, 2π/3]; specifically, prove the conjecture that no such lattice triangles exist in this hard obtuse window.

Background

The hard obtuse window refers to rational triangles whose largest angle is between π/2 and 2π/3. This regime is described as the most mysterious and is central to the unresolved part of the lattice triangle classification.

The paper develops number-theoretic obstructions ruling out a density 1 subset of candidates in this window, supporting the conjectural nonexistence of lattice examples there.