Existence of a degree-one three-vertex algebra with profile (1,3,3)

Determine whether a ring-indecomposable, degree-one generated, three-dimensional twisted Calabi–Yau algebra on three vertices with odd-order Nakayama permutation and Gelfand–Kirillov dimension 3 can have local Smith profile (1,3,3), outside the superpotential-degree restriction of Hypothesis 1.4.

Background

For three-dimensional, three-vertex, ring-indecomposable twisted Calabi–Yau algebras with odd-order Nakayama permutation and Gelfand–Kirillov dimension 3, the parity constraints developed earlier restrict the possible local Smith profiles to (0,0,3), (1,1,3), (1,3,3), (2,2,3), and (3,3,3).

The classification of Gaddis–Rogalski under Hypothesis 1.4, which limits the superpotential degree to 3 or 4, realizes or accounts for the profiles (0,0,3), (1,1,3), and (2,2,3), while (3,3,3) is intrinsically excluded for ring-indecomposable degree-one algebras by Proposition 4.4. The profile (1,3,3) is therefore the remaining parity-compatible case not resolved by that classification. The paper notes that realizing it in the degree-one setting would require superpotential degree at least 5, and that weighted examples fall outside the cited classification.