J.C.MacSwell

I assumed Johnny meant the Earth station frame, but I think his statement is true (contradictory but true given my assumptions) for any choice of inertial frame.

I think you can claim mathematically that they are (at rest wrt each other), although some laws of physics may require corrections/adjustments to allow for your frame/perspective.(which I think is a very useful frame for cosmology) To claim that they are also "absolutely" at rest is a bit of a "stretch" if you will excuse the pun. I think the experiment, which requires instant transmission of results and therefore is excluded as a possibility in SRT, would be expected to produce a null result in that no rest point on any inertial frame would be expected to have a faster clock than any other.

OK, what is wrong with this in principle: Two clocks go by the Earth at near c in opposite directions. As they go by they are synchronized by a signal from Earth. Each assumes the other clock to be much much slower. They go "straight" around the Universe/hypersphere. When they meet again Earth is long gone. As they pass by they prepare to resynchronize via an Earth station our descendants have built and positioned for the occasion. How do their watches compare?

Does such a point exist? I'm questioning your perspective, not your ideas which I think are interesting.

Then you would not be at the center of mass. Your first answer (8x) is correct.

Check again. Center of mass is linear and gravitation is inverse squared. Hint: You are closer to the bigger mass!

Picture yourself in space at the center of a system that includes you, a solar mass 200 million miles away to your right and a 2 solar mass 100 million miles away to your left. You are at the center of mass. Which way would you be pulled by gravity?

If you had a closed system, and chose an inertial frame based on the center of mass being at rest, then the center of mass would stay at rest. The gravitational net force at this point would not necessarily be zero, nor would it necessarily stay zero if it was.

The pressure is highest at the center. The net gravitational force is lowest, goes to zero, at the center. The center of an all one material sphere would be generally denser at the center as you describe The density gradient depends on the bulk/volume modulus of the material, temperature gradients and bulk coefficient of expansion with temperature and any phase changes involved. The Earth is considerably denser at the center as the denser materials gravitate toward the center.

In which direction do you think it should be pulled away?

If entangled communication could work that and the experiment would yield consistent results (disregarding gravity strength anomalies) and if the results produced a "fastest clock" in a particular direction and speed then that would distinguish that reference frame. It's an interesting bit of thinking even though we would expect a null result due to the second if (I think that was Swansont's point) and in fact probably no results at all due to the first if.

Equivalently? Who said equivalently? I said you raised the Earth. You close the distance (sit down) 99.9999999999???????? % and the Earth will close (raise up) the remainder...small as it is. If you want a sense of it being equivalent, then the total mass (mass of the Earth) times distance the Earth moves will be equal and opposite to the mass (your mass) times distance that you move. (all other influences notwithstanding). This is basic conservation of momentum.

So 144 pounds / square feet would convert to 1 pound / square inch?

When you turn on a detecting device the interference pattern disappears, whether it's electrons or photons etc. I think they all "self interfere", the different probabilities of the individual probability wave interfering, and not "with each other", but I am not sure.