We have claimed that a randomly generated point lies on the equator of the sphere  independent of where we pick the North Pole.  To test this claim randomly generate ten  vectors in 128 dimensions whose coordinates have value ±1. Think of these ten vectors  as ten choices for the North Pole.  Then generate some additional random vectors with ±1 coordinates.  For each of the new vectors determine how many of the original vectors they are close to being perpendicular to.  That is, they lie close to the equator.