A nowhere-zero section annihilates generalized Euler classes ($s(x)\ne 0\text{ for all }x⟹e_h(\xi )=0$)

Normalize the section to the sphere bundle. Scaling it from zero to unit length gives a homotopy within the disk bundle from the zero section to this sphere-valued section. The image of the relative Thom class in absolute cohomology restricts to zero on the sphere bundle. Homotopy invariance therefore makes its zero-section pullback vanish.

// Target: geometry-and-topology.bigb