A second gap is the wayward definition of a ridge on page 3 in Ref. . For example, at an SP of index one we have one negative eigenvalue of the Hessian but N-1 positive eigenvalues (if N is the dimension). If we go uphill from the SP perpendicular to the valley direction, we have positive curvature. This it is a (hyper)-ridge of dimension N-1. In Fig. below, in the region around the maximum, the region with two negative eigenvalues is inside the green line around the maximum.