Possible vanishing of the non-degeneracy determinant

Determine whether the determinant condition $\det(F+D_R)\ne0$ can fail for a kernel-vector Bethe class constructed from the trigonometric formula, and characterize all such failures if they occur.

Background

Theorem \ref{thm:Hall} computes the handle-gluing operator for kernel-vector classes under the additional condition that det⁡(F+DR)≠0\det(F+D_R)\ne0 on every RR-run. The condition is satisfied in all cases computed, but the theorem is conditional because the determinant has not been shown to be nonzero in general.

A vanishing determinant would obstruct the Schur-complement/Hill-formula argument and could correspond to additional degeneracy phenomena among the Bethe classes. Establishing universal non-vanishing, or identifying precisely when it fails, is therefore necessary for a complete general classification.

References

It did not fail on the $49$ words of length $3$ to $12$ examined: at all $4236$ classes of Theorem~\ref{thm:allwords} in both sectors, the smallest absolute value was $6\cdot10{-5}$, at $L2RL2RLR3LR$. For $b\ge2$ the determinant takes negative values, and smaller absolute values occur on other words, for instance $4\cdot10{-6}$ at $L3R6L2R$. Whether it can vanish at a kernel vector is not known.

— Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua  (2610.18512 - Evans, 13 Oct 2026) in Section 5.3, immediately after Theorem \ref{thm:Hall}