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Well, ANY physical concept is not more than a mental construct
Hmm. Is electrical charge a mental construct? Conservation of
charge is also a counterpart of a symmetry of Nature. Charge is,
somehow, a different sort of physical quantity than energy. For
example, the quantity of charge is independent of the frame from
which it is observed; the quantity of energy observed depends
upon the observer's frame. Charge is quantized; energy is not.
Charge is locally conserved; energy is not even localizable.
There are large qualitative differences between the concepts of
charge and energy. Charge is substantial; energy is insubstantial.
I believe that if your claim above is not trivial then it is not
correct, either.
(see Dewey's aftermail quote of Einstein).
I'm told Einstein spoke German. I don't recall a German quote
following Dewey's signature.
I know that if time invariance
then energy conservation; I ask if it also works the other way.
Is this an "if and only if"? I believe that it is.
You should look at the textbook by Eugene Hecht. There are many
gratuitous bows made to Noether's theorem, for no reason other
than Noether's sex, so far as I can tell. There is a big graphic
each time the subject arises. It is most distracting from the
physical thread!