[Pinegulf]

>Uncertainty principles are not formally defined

In physics it seem to be pretty straight forward. For instance https://www.wolframalpha.com/input/?i=uncertainty+principle

[diffeomorphism]

Not really. You defined exactly one, the Heisenberg uncertainty principle. There are many, many more.

For instance, a function cannot both be compactly supported and have a compactly supported Fourier transform. Or you can have other transforms, other operators than position and momentum etc.