Forget quantum, forget light, your argument is getting obtuse. The point is whether mathematic proof is proof of truth and facts.
If the Pythagorean theorem is proved mathematically, is it therefore proven “true” - a “fact”? If it is mathematically proven false, is it therefore “false”. If it is mathematically proven unprovable, is it therefore a fact that it is “unprovable.”
If something is unprovable (in a given working model) it is simply unprovable. It is not "proven" unprovable.
Since we are on semantics, if it ismathematically proven false then it is mathematically false. That doesn't mean it is necessarily false as regards the real world.