  We prove a rigidity theorem in the style of Urbano for the Allen-Cahn
equation on the three-sphere: the critical points with Morse index five are
symmetric functions that vanish on a Clifford torus. Moreover they realise the
fifth width of the min-max spectrum for the Allen-Cahn functional. We approach
this problem by analysing the nullity and symmetries of these critical points.
We then prove a suitable Frankel-type theorem for their nodal sets, generally
valid in manifolds with positive Ricci curvature. This plays a key role in
establishing the conclusion, and further allows us to derive ancillary rigidity
results in spheres with larger dimension.
