Complete classification of sufficiently large equilateral tilings

Prove that, for every non-reptile and non-commensurate tiling of a triangle T by a triangle R=(a,b,c) having an angle equal to 2π/3, the possible tile counts are exactly the values m^2ab for integers m at least M=3⌈(c^2−a−b)/(ab)⌉.

Background

Theorem 3.1 constructs equilateral triangles with side lengths mab, and hence tilings with m2ab tiles, for all m above an explicit threshold M. The conjecture asserts that these constructed values exhaust the possible counts in the specified non-reptile, non-commensurable setting.

References

For all non-reptile and non-commensurate tilings of $T$ into $R = (a,b,c)$ with an angle equal to $2\pi/3$, the possible $N$ is the set $${m2 ab | m \geq M} $$ where $M$ is defined as in Theorem~(i).

— Tiling Triangles with $2π/3$ Angles  (2512.22696 - Zhang, 27 Dec 2025) in Conjecture 2, Section 3.1, Equilateral Triangles