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You seem to think that I am just changing for example 1 Joule to 2 half-Joules. That is *not* what I am doing. I am changing 1 Joule to 2 Joules. Noether's theorem just gives an expression ##h## (with a physical dimension!) that is conserved and doesn't say whether the correct energy to use is ##E=h## or ##E=2h##.As @vanhees71 said, the four momentum is a vector, which is a single unchanging geometrical object. What you are doing when you say ##E’=2E## is simply to represent that single geometric object in a different basis. I.e. as a larger multiple of smaller vectors. This change of size of your basis is a change of units. It doesn’t change the geometrical vector, just the numbers you use to represent it.