Establish the general index-discrepancy pattern for orthogonal and symplectic theories

Establish the observed pattern that the first discrepancy between the pure-gaugino classical indices of \(SO(2r+1)\) and \(Sp(2r)\) occurs at order \(t^{2r+2}\), with leading expansion \(\Delta_r(t)=t^{2r+2}+2t^{2r+3}+O(t^{2r+4})\), for all ranks \(r\geq3\).

Background

The paper computes the pure-gaugino index difference for ranks three through six and finds the same leading pattern, with the discrepancy interpreted through the non-Coulomb class $\mathcal S^{r+1}$ present in $SO(2r+1)$ but absent in $Sp(2r)$.

The authors explicitly note that the observed rank-dependent pattern has not been proved in general. Establishing it would clarify whether the first index discrepancy is universally controlled by the first non-Coulomb power of the gaugino bilinear.