No. Motion is absolute. Rest is relative. That is, rest is only an appearance. Since you are observing from earth everything you see is in motion since the earth is moving. So there is no absolute rest. And relative rest is just another name for motion.
Therefore, no object can have a property that exists only when the object is at rest. Because there is no absolute rest.
What is Galilean invariance? You are below deck in a ship moving uniformly on a smooth sea. No experiment can tell you if the ship is docked or if it has motion relative to the see. Assume that the ship is docked. In that case you say that the ship is at rest. And you attribute to the ship some properties because it is at rest. But is the ship at rest absolutely? Not at all. The ship is moving with the earth. So do you deny that the earth is moving? I assume not. If so how can you claim that the ship is at rest?
Motion is absolute means that all is in motion. This is an axiom. When you say motion is not absolute, can you give an example of an object which is at rest, which does not move with the earth and does not move with the galaxy? There is no such object.
> If so how can you claim that the ship is at rest?
The ship is at rest with respect to its own reference frame.
> So do you deny that the earth is moving? I assume not.
The earth is at rest with respect to its own reference frame.
It is moving with respect to other things, such as the sun, or the center of the galaxy. Motion and rest are relative. This is the key concept, and I don't understand why you're having such trouble with it.
Can you give an example of a system which is not based on axioms? There is none. There may be hidden assumptions but there will always be axioms. Axioms are simply definitions that are kept constant. In your case you make the assumption that absolute rest exists. The axiom of the universality of motion is a good one because it agrees with the observations.
> In your case you make the assumption that absolute rest exists.
This is a straw man. No one is making that assumption. Something is in motion or at rest with respect to a reference frame. For some reason, you keep misinterpreting this statement and ranting about "absolute motion" and "absolute rest".
Yeah, I agree, that sounds like word games. The whole point of it being relative is that rest and motion are indistinguishable without a point of reference. That point is the relative part.
But this is not word games. The statement is: "Nothing is absolutely at rest." Where is the word game in this statement? Or "all is in motion." These are clear statements. If people heeded Heraclitus who said this first thousand years ago astronomy would have advanced much faster. Astronomers built a whole geocentric system by assuming that the earth was stationary. They assumed stars were stationary. These turned out to be wrong. There is no absolute rest. The academic descendants of the same academics who believed in absolute rest now call themselves physicists and make the same mistake and assume that absolute rest exists.
> The statement is: "Nothing is absolutely at rest."
An object is at rest with respect to its own reference frame.
> The academic descendants of the same academics who believed in absolute rest now call themselves physicists and make the same mistake and assume that absolute rest exists.
In other words, you think all physicists are wrong and you're right?
Therefore, no object can have a property that exists only when the object is at rest. Because there is no absolute rest.