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Source statement

Let $T$ be non-isosceles and admit a nonsquare, non-reptile tiling. The source claims the following classification of its nonsimilar tiles, up to similarity:

1. If $C=\pi/3$, $A<B$, and $\sqrt3\tan(A/4)\in\mathbb Q$, exactly two tile shapes occur, $(\alpha,\beta,2\pi/3)$ with $\alpha=A$ or $\alpha=A/2$ (in each case $\beta=\pi/3-\alpha$).
2. If $B=2A$ and both $\sin(A/2)$ and $\sqrt3\tan(A/2)$ are rational, exactly two shapes occur. With $\alpha=A$ and $\beta=\pi/3-A$, they are $(\alpha,\beta,2\pi/3)$ and $(\alpha,3\beta/2,\pi-\alpha-3\beta/2)$.
3. Otherwise precisely one row of Proposition 13 matches and the tile shape is uniquely determined by $T$.

Source and scope. Beeson–Laczkovich–Zhang, arXiv:2604.03609v3, Theorem 32, pp. 19–21. Statement and proof pointer with the gap below; this page does not claim a complete checked proof of the uniqueness assertions. The PDF prints $3\beta/2$ correctly; an extraction that reverses this fraction is not a source error.

Proof pointer and established part

Existence of the two tiles in case 1 follows from Propositions 26 and 31, because $\sqrt3\tan(A/4)\in\mathbb Q$ implies $\sqrt3\tan(A/2)\in\mathbb Q$ by the double-angle formula. Existence in case 2 follows from Propositions 27 and 28. Their Group 1 and Group 2 third angles differ, so these are distinct tile shapes.

For uniqueness in the first two cases, the source uses linear independence of incommensurable angles to rule out other row matches and checks angle permutations by determinants. For its final part, however, the printed p. 21 argument excludes simultaneous membership in the last two rows by subtracting $3\alpha+2\beta=\pi$ from $3\alpha+3\beta=\pi$. Those two rows may refer to different tiles and different angle labelings; the argument does not justify identifying their $\alpha,\beta$. It also does not explicitly settle all otherwise-only-one-row possibilities covered by part 3. Completing that exclusion requires an additional argument or a source clarification. No replacement proof is supplied here.

The solving Theorem 1 and the square-exception Theorem 3 do not use this uniqueness theorem. Their proofs remain separate from this lesser-result gap.

Bears on. Problem 633.