Where does this idea fail? The proof cannot 'create' the equivalence between fixed-point sensitivities and Karush-Kuhn-Tucker; it only formalizes it and makes it observable.
The weak link here is confusing revelation with creation; the equivalence existed before its proof.
Imagine a regulation from FINMA on pension funds: it does not 'create' the reputation risk, it highlights it and frames it.
Without the underlying assumptions of the proof, such as strong convexity or differentiability, this theoretical equivalence would collapse, demonstrating that the proof is only a validation vehicle.