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In the reference frame of the moving rod, its length is unchanged. In the reference frame of the observer it is much shorter.

. No. I didn't read the rest of your post (I suffer from MEGO).

Your example is a particular case. It does not address the general problem, where the number of sets is uncountable.

The problem with infinite sets sets is that to get the choice set, you need an infinite amount of operations. For non-countable sets, it seems impossible, so we have an axiom.

The axiom of choice refers to infinite sets. For finite sets, there is no problem in choosing an item. For infinite sets, the question is forming a set made up of one item from each of the infinite sets.

How local do you mean? For example, when you boil water, the water in the pot has more energy, but outside you were putting it in.

https://en.wikipedia.org/wiki/Neutrino_oscillation This should answer your question - individual neutrinoes ooscilate.

Photons are electromagnetic waves. They can collide, but otherwise they can interact as waves.

The picture of the standard model is a little misleading. Lining up the bosons with the fermions hints at relationships which don't exist.

My impression - problem with solar power. The sun is not out all the time. Therefore you need long lasting battery storage (or some other storage) or an always available power source, such as gas generated.

Tell your partner that natural gas is better than coal. It is cheaper and cleaner.

You can compute the distance before he starts slowing down and after he stops. These depend only on the relative speed at that time. What happens in between doesn't matter.

In general, when talking about angle between intersecting lines (curves) the angle is between tangents at the point of intersection. Radius is perpendicular to tangent, therefore perpendicular to circumference at that point.

The deceleration time is irrelevant. The point is before C stops he measures L' and after C stops, he measures L. If he is continuously measuring while he is stopping, the measured distance will change continuously.