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I am sure I didn't say that altitude was a prerequisite for speed. I said that things at a higher altitude move at a faster speed relative to those things at a lower altitude.
The CERN is not relying on the spinning of the Earth to generate that speed.

the spinning of the earth is irrelevant.

Not if the subject is time distortion caused by relative speeds. Think about a wheel or a gear or a merry go round or anything that spins. If you are inside the merry go round you don't have to run very fast to jump on, if you are on the outside of the merry go round you have to run a lot faster. Both gravity and speed affect the passage of time, even standing a single foot higher than something else changes the rate of time. They have taken a pair of matched clocks and put one one foot higher and measured the difference to confirm this, as one of the many many experiments confirming this. They have done the same by putting one clock on a train and the other not on a train. They have to regularly recalibrate the GPS satellite clocks to correct for the time distortion.

you are confusing 'thought experiments' with actual empirical studies.
the difference in time in the 'experiments' that you are describing would be so miniscual that no clock yet devised could measure it.

Nope, the putting a one clock thing a foot above another thing and putting clocks on a train and putting them on an airplane have all been done in real life with atomic clocks and confirmed Einstein's thought experiments. In fact we have devised such clocks. And the GPS satellites are subject to the effect so strongly that if they didn't take it into account they wouldn't work right.

Clocks on planes:

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/airtim.html

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