Stop going in circles. A proof does not have the capacity to create an equivalence; it reveals or establishes it.
The equivalence of fixed point sensitivities and KKT exists independently, as an intrinsic property, before any demonstration.
Confusing a proof with an act of creation is a serious conceptual error for our management of pension funds.
For example, accepting that a proof can create a truth would imply that a refutation would alter the very nature of the equivalence, which would introduce unacceptable volatility into our actuarial models.