In mathematics that means we have three different ways to describe ’smoothness’. For example pure white noise is also scale invariant, so it is actually smooth.
There are various definitions of smoothness, but a standard definition is that for a function to be smooth it must be continuous and either infinitely differentiable and all the derivatives must be continuous or differentiable exactly n times and the nth derivative must be continuous. Notice that has nothing whatsoever to do with scale invariance. Here is a math stack overflow discussion which is more concrete and specific[1] and here is a nice treatment of the subject using animation etc[2]. Note that the sine function for example is not scale invariant but is smooth whereas white noise is an example which is scale invariant but not smooth because it is not continuous or differentiable.
I don't think anything you're doing here is mathematics at all let alone getting to the heart of mathematics.
Mathematics is defining axiomatic systems and proving results in those systems. What you're doing is making a physical analogy and then implying all sorts of things that are not rigorous and therefore are not provable or disprovable from that analogy.
Comments
There are various definitions of smoothness, but a standard definition is that for a function to be smooth it must be continuous and either infinitely differentiable and all the derivatives must be continuous or differentiable exactly n times and the nth derivative must be continuous. Notice that has nothing whatsoever to do with scale invariance. Here is a math stack overflow discussion which is more concrete and specific[1] and here is a nice treatment of the subject using animation etc[2]. Note that the sine function for example is not scale invariant but is smooth whereas white noise is an example which is scale invariant but not smooth because it is not continuous or differentiable.
I don't think anything you're doing here is mathematics at all let alone getting to the heart of mathematics.
Mathematics is defining axiomatic systems and proving results in those systems. What you're doing is making a physical analogy and then implying all sorts of things that are not rigorous and therefore are not provable or disprovable from that analogy.
[1] https://math.stackexchange.com/questions/12026/what-does-smo...
[2] https://www.youtube.com/watch?v=vD5g8aVscUI