@A.T. are you trying to say that there is no angular momentum in the upward direction?
the spinning wheel has mass and it is rotating about the pivot with some angular velocity in that direction due to precission doesn't that mean it has an angular momentum.
Once the gyroscope starts to spin about the pivot it acquires an upward angular momentum. But since it did not have any angular momentum in this direction before, and the only torque is due to the gravitational force, how does this new angular momentum suddenly arise. Isn't this a violation of...