In that post are you considering a mirrored rim or mirrored sides of a wheel?

Rim.

Those circles really confused the direction of that thread.

Those circles were Tach's, as presented in the debate thread. Perhaps you should consider the significance of that, if the evidence Tach presented to support his case confused the direction of the discussion of his case.

Also, what do you mean by "simultaneously moving toward and away from the light source"?

Exactly what it sounds like.

Consider the scenario where the light source us to the right (or in the 3'oclock position relative to) of a stationary wheel (for simplicities sake) and the wheel is rotating clockwise about its axel. In the frame of the light source (or in the rest frame) the top of the wheel is moving towards the light source, and the bottom of the wheel is moving away from it, with the 3 oclock position neither moving towards or away from the observer.

Now consider exactly the same setup, only this time the axel has a translational velocity towards the light source such that $$v<r\omega$$. This instance there will be some point on the rim of the wheel that appears to be stationary, because the instaneous velocity is at a sufficient angle that its horizontal component is equal, but opposite to the motion of the wheel.

In the case that $$v=r\omega$$ (at which point we're dealing with what is physically equivalent to a wheel rolling without slip), that part of the wheel is located at the 6 o'clock position, and obviously for $$v>r\omega$$ all points on the wheel appear to have some velocity in the direction of the motion of the axel.

My contention was/is that irrespective of whether we're talking about specular reflection, diffuse reflection, or emission, then in the special case of a wheel rolling without slip, the part of the wheel that is in contact with the surface it is rolling along shows no doppler shift, but the rest of it shows red shift, or blue shift, depending on the relative direction of motion of the wheel.

I understand Tach's claim that he has made, and I even understand the physical principles of Tach's claim, however, I have yet to convince myself of the universality of it within specular reflection, let alone beyond that. And I will

*not* be bullied into accepting it.