Well, duh. It was two different examples. You mentioned real numbers and "theoretical integers, i.e. x,,y,z, etc." (whatever that is.) So I gave examples of numbers and variables .
Quantum mechanics by definition cannot be proven by reality, because the reality "breaks down" in that model.
No, gross mistaken over-simplification. Theory is tested and proved by "reality." In experiments. Bell's theorem for example, the wave/particle duality of light, photons, sub-atomic particle cloud chambers, the Large Hadron Collider, etc. (For the record, these are different examples.)
So, then, math can be used to describe real events but it can also be used as fiction, to describe imaginary world…
Certainly, there is abstract math - though it follows the real world rules of all math - but the examples I gave are not abstract, they are set theory, polynomials, and basic math. I hope you're not next going to exclude calculus as not "real world."
Your initial reply requested specificity in "facts, truth, proof." Math is the most specific, most precise language. Unfortunately for the argument that "provable" and "true" are identical sets, math proves it wrong.
Bell's theorem is just that, a theorem, not a fact. A theorem simplpy means something that is proposed on the basis of previously agreed upon rules or axiomatic assumptions, not necessarily on facts.
The problem with experiments is that some of them are not real. For example, an optical flat can be certified only to the degree limited by the interferometric device used and the ability to control all other parameters. Our measurements are as good as the precision (and errors) of our instruments. If we can't measure something then it is an approximation, not a fact.
the wave/particle duality of light, photons
The wave particle issue has to do with our inadequate mathematical models. Just as a quadratic equation can have two distinct solutions but never one. The problem is in our math, which is inadequate to provide answers as to the nature of a "particle' of radiant energy.
Ancient math was unable to do what calculus was bale to achieve, but ti took Leibniz and Newton to discover that mathematical model.
Quantum theory is still theory, not a fact. Even is something is mathematically provable does not mean that it reflects or describes the real world. Ptolemaic navigational systems comes to mind.
Let's set your sophism aside for a moment. Facts must be provable or else they are not facts. They are baseless claims.
Without reality to corroborate it, math is just numerical fictional prose. Math does not make reality "real" but rather the observed reality makes math real.