Interior transcendentality of the remaining rank-2 unit shapes

Prove that every unit shape of a unit-rank-2 number field with no nontrivial subfields and signature $(2,1)$, $(1,2)$, or $(0,3)$, and every unit shape arising from a totally imaginary sextic field whose Galois-closure group is $S_3^2$ or $S_3^2\rtimes C_2$, lies in the interior of $\mathcal{S}_2$ and is transcendental.

Background

The paper classifies most unit shapes associated with number fields of unit rank $2$ according to the Galois group of the Galois closure. The classification does not resolve fields with no nontrivial subfields or totally imaginary sextic fields with Galois groups S32S_3^2 and S32⋊C2S_3^2\rtimes C_2. Computational experiments using LMFDB data and PSLQ detected no low-degree algebraic relations for the examined examples.

The conjecture asserts that these unresolved families behave like the algebraically unconstrained interior families identified conditionally elsewhere in the paper: their shapes should be interior points of the rank-2 shape space and should represent transcendental complex numbers. A proof would complete the unconditional classification of rank-2 unit shapes.

References

The classification leaves open the cases of fields with no proper subfields and of $S_32$- and $S_32 \rtimes C_2$-sextic fields; computational evidence, including a PSLQ search that detects no algebraic relations, supports the conjecture that these too are transcendental and interior (\Cref{conj:remaining Galois groups}).

— On the Geometry and Shapes of Rank 2 Log Unit Lattices  (2608.24736 - Cruz et al., 25 Aug 2026) in Conjecture 4.6, Section 4; discussed in Section 1.1 and Section 4