With all due respect, that sounds like straight-up circular logic to me. You're arguing that 1/2 = 2/4 because "1/2 [...] is [...] 2/4". You're just restating the conclusion as an explanation of itself, which is completely unsatisfactory.
Another way of putting it:
> 2 slices of four is the same fraction as 1 slice of 2.
The phrase "is the same fraction as" is obviously inappropriate and explicitly circular and self-referential in any definition of "fraction". I don't think you've thought about this as deeply as you think you have.
Another way of putting it:
> 2 slices of four is the same fraction as 1 slice of 2.
The phrase "is the same fraction as" is obviously inappropriate and explicitly circular and self-referential in any definition of "fraction". I don't think you've thought about this as deeply as you think you have.