Stop going in circles. A mathematical proof does not create an equivalence; it reveals or demonstrates it, the relationship is inherent.
The equivalence between sensitivities based on fixed points and KKT conditions is an intrinsic property that existed before any formalization.
We must decide to recognize that the mathematical demonstration validates a pre-existing property; it does not generate it.
For example, the law of gravity did not create gravity; the formula just explained it.
The robustness of our models depends on the validity of their mathematical foundations, regardless of our ability or inability to prove them at a given moment.