Realizability of the remaining portions of the excluded rays

Determine whether every modulus on the portions of the rectangular and rhombic rays not covered by the constructions of Doyle and Schwartz, the present paper, and its corollary can be realized by an eight-vertex paper torus.

Background

Doyle and Schwartz realized every modulus outside two excluded rays in the moduli space of flat tori: the rectangular ray and the rhombic ray. The present paper realizes the square and hexagonal endpoints and an initial segment of each ray. The unresolved problem is whether the rest of those two rays is likewise realizable with eight vertices. The paper notes that two planar families sweep these rays, while embeddedness along the families is deferred to a sequel.

References

The first question is what remains of the two rays. Doyle and Schwartz supply every modulus off them , and Theorem \ref{t:main} with Corollary \ref{c:rays} supplies an initial segment of each, so what is open is the rest.

— The two most symmetric flat tori as eight-vertex paper tori  (2609.10703 - Lander, 9 Sep 2026) in Section 'Open questions' (Section 6), first paragraph

The last, with degree sequence $(4,5,5,6,7,7,7,7)$, resisted every numerical search. We found no paper torus on it at any modulus, and we suspect that none exists.

— The two most symmetric flat tori as eight-vertex paper tori  (2609.10703 - Lander, 9 Sep 2026) in Section 'Open questions' (Section 6), third paragraph

A single triangulation carrying every flat torus exists, the universal one of with $2434$ faces, but we do not know whether one of the $8$-vertex triangulations realizes every flat torus.

— The two most symmetric flat tori as eight-vertex paper tori  (2609.10703 - Lander, 9 Sep 2026) in Section 'Open questions' (Section 6), third paragraph