> That there may be structures in nature that are 1-to-1 with any given (constructible) mathematical concept.

There may be, or there may not be. I don't think we can make a definitive argument either way without perfect knowledge of the structure of the physical world, or at least some part of it. What we have are mathematical models of physical systems that are valid to within some error margin within some range of parameters. The ultimate structure of the physical world is so far unknown, and may be forever unknown. Any actual physical situation is too complex for us to fully analyze, and we can only make our mathematical models work (to within some error) when we can simplify a physical system sufficiently.

I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it, i.e. even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?

That might be true for all I know. I'm not sure how to evaluate it (IANA mathematical physicist). It seems plausible though. There's the example of synthetic differential geometry, which has a different conceptual basis to the standard formulation of differential geometry, and at least suggests that you can't a priori rule out the possibility of alternate formalizations of any mathematical structure. I don't know enough about it to say whether or not it postulates something equivalent to unconstructible reals, it's just something that came to mind as maybe being along the lines of your point of view

https://ncatlab.org/nlab/show/synthetic+differential+geometr...

> I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it,

Yes, that is a good way to put it.

Unconstructible reals are a major and interesting "what if". What if there were numbers that had no finite relation to other numbers?

It is a great idea, from a mathematical boundary pushing way. So abstract we can never do anything not abstract with it!

> even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?

But unconstructible structures can never be reformulated as constructible by definition. That would mean they were constructible.

We can never define a specific unconstructible real.

But anyone who manages to create an interesting systems theory that uses them, with dynamics that constructible math can't match, would have created a major work of mathematical art!