Genady

We use GR. It is both. This how the metric of homogeneous and isotropic universe works. See FLRW metric.

A string is wound evenly around a circular rod exactly four times. What is the length of the string if the length of the rod is 12 cm, and its circumference is 4 cm?

Yes, relativistic Doppler effect includes time dilation. As I know of a solution to the twin paradox that does not include relativistic Doppler effect, could you refer to a solution that does?

PS. The mention of "always" in the OP means, "while the train moves forward."

That's the only part I know of. +1

Stew of Everything:

When a train moves forward there is always a part of it that moves backward relative to the ground. What part is it?

Yes, with a little subtlety. In democratic societies there are often several versions of history, accepted by different groups.

Perhaps you know that it is very difficult to modify constitution. Unfortunately, people go with it.

Constitution is a law that applies today. Unlike the historical facts.

Because people don't care?

No, not only. You use decimal system here, in your "essential step":

I think that if not for the continent underneath, the ice cover would be even more symmetrical.

Sorry @Mordred, I can't participate. I'm not trained on this level.

It is a Prime Law that all advanced civilizations use ten fingers for counting.

Oh, this is a good trick! +1

Let's go to the next plank under the top two. To stick out without falling these two can have their center of gravity on the edge of the plank under them: This position adds another 0.25m making x = 1 + 1/2 + 1/4 m. How far can we go if we continue like this?

And you are right. +1

OK.

I don't see anything here^^^ that you have hidden.

You haven't mentioned that the decimal system has a special significance in your theory. Does it?

Correct. +1

Hmm ... Except the twin paradox has nothing to do with the Doppler effect. It has to do with the time dilation. The latter is the same regardless of direction of the motion. It depends only on the relative speed between two frames of reference.