You mention that you accept caesium decays are slowed at high speeds. I wonder if the problem here is that we're using different definitions of the word, "time"?
Indeed. The math in special relativity works just fine. I've got no problem with that. But the explanation is completely upside down because it's definition of "time" and "space" ... and worse.... "Spacetime" (the word itself is a bastardization of reality.)
Objects -in- space and time alter. But no matter where or when something happens... you can't change space. Einstein admitted he based much of his theory on the M&M failure... they failed to prove an aether, therefore he assumed it didn't exist... then wrote a theory stating "If you go real fast, you can bend the "spacetime" you're traveling through...
which inherently assumes space is the aether which he assumed did not exist!? <-this is bold because it represents me yelling in a quite upset tone.... with a hint of disappointment.
... just... how? How do you disagree with an aether, then write both special relativity and general relativity? How does a person base a theory on the idea that an aether doesn't exist... when the theory itself describes an aether existing? I think he did it just to annoy me. Like the person who put all those silent letters in the french language. Just to annoy me, personally. (I enjoy my dillusions of grandeur... do not take them away from me

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Perhaps you are imagining time as a sort of constant beat through the universe that objects conform to, more fundamental than clocks.
Constant? I don't like the implication. "Not there. At all. Figment of your imagination." More fundamental than clocks... yes.
Tell me... do clocks define time? Or are clocks a way to describe time? If the power goes out on my alarm clock... did time stop? Or did my explanation of time fail to match the reality? If I spend an hour spinning in my chair with a rotational speed of 97% of C... what does it matter what if it only seemed like a minute to me? How does that change the fact that the earth rotated by 15 degrees?
What time dilation means is that events in moving reference frames happen more closely together (in time) than events in stationary ones (by "stationary frame" I mean "the frame of the observer").
If events happened more closely together while moving, time would be speeding up, not slowing down due to speed.
You could, I suppose, think of this not as time slowing down but as events slowing down
THAT is perfectly sane.

I agree that at high speed, events (including atomic events, such as muon decay) slow down. But saying that changes time itself is like saying that buses and trains ACTUALLY make a lower-pitch sound whenever they pass by you... the Doppler effect couldn't simply be an auditory illusion.
So the clock on the plane does slow down, but so does the heartbeat of the pilot, so does the combustion of the engine fuel, and so do the thoughts of the passengers. Any experiment performed on the plane seems like it is going more slowly than usual to the people on the ground (the reverse is true for the people on the plane: the people on the ground seem to be going more slowly, until the plane slows down and the reference frames realign).
INDEED! A very important point! The clock on the ground was going JUST as fast relative to the clock in the jet as the clock on the jet was going relative tot he clock on the ground. Were SR legitimate... each clock would see the other one as behind itself equally, until they stopped moving relative to each other. So, why is it, when the clocks in the experiment "realigned" and were brought back to the same frame of reference, the clock on each jet read a different time than the clock on the ground?
Explain that one, if ya don't mind.
Also, if you have the time (HA! I made a joke!), tell me why, in the HafeleKeating experiment, the clock on the plane traveling eastward LOST 59 nanoseconds, while the clock on the plane traveling westward GAINED 273 nanoseconds?
The experiment claimed to account for the earth's axial rotation, since one aircraft was moving with the rotation of the earth, and the other was moving against it... However, the rotation of the earth doesn't affect the relative speed between the aircraft and the ground (and therefore the clocks on the aircraft and ground) any more than the earth's orbit around the sun affects it... since the clock on the ground is moving with the earth's rotation and orbit just as much as the clocks in the jet.