Exact E6 wall-period ratio and cyclotomic-field membership

Determine the exact value of the ratio \(\lambda_{E_6}=P_2/P_1\) of the six simple-wall periods for the restricted \(E_6\) arrangement, and establish whether \(\lambda_{E_6}\) belongs to the maximal real subfield \(\mathbf Q(\zeta_{35})^+\).

Background

For the six simple-wall periods P1,…,P6P_1,\ldots,P_6 associated with the E6E_6 restricted arrangement, the six-orbit Varchenko block supplies four independent linear relations. These relations reduce the unresolved information to the single ratio λE6=P2/P1\lambda_{E_6}=P_2/P_1, while leaving its common scale undetermined.

Direct numerical quadrature gives λE6=1.6329738485…\lambda_{E_6}=1.6329738485\ldots, but the paper does not provide an exact reduction or a sufficiently precise numerical representation to decide whether this value lies in Q(ζ35)+\mathbf Q(\zeta_{35})^+.

References

The associated six-orbit block has rank $4$ and nullity $2$, leaving the single ratio $\lambda_{E_6}=P_2/P_1$ undetermined. Direct quadrature supports eleven significant decimal digits, \lambda_{E_6}=1.6329738485\ldots, so membership in $\mathbf Q(\zeta_{35})+$ remains unresolved.

— Second-pole wall periods for Witten zeta functions in the classical families  (2608.17815 - Matuzas, 18 Aug 2026) in Section 7, “Further questions,” subsection “The remaining E6 period ratio,” Proposition 7.1 (numbering inferred from the section structure)