Yes I know that. That's why I said if the mass were "somehow actually retained". I'm positing that the some of the extra energy somehow becomes real mass. Yes, this is not how special relativity works. That's why I said "wacky theory". But is there a simple and obvious way to falsify this?
If all of the kilograms were undergoing radioactive decay, then the ones that traveled back and forth would be younger, and thus heavier, because of the twin paradox. You'd probably need a new undetectable form of radioactive decay and relativistic trains to make this plausible.
Sure, you will only need two approximately equal masses, a high precision scale and a round-trip airplane ticket. Seems easy enough for me. Strictly speaking this seems like the most obvious way to test your theory. Of course it would be mighty odd if you would find any effect, but who knows?
