The old manual still applies because the act of proving an equivalence does not create it; it only reveals or confirms it, just as fundamental economic principles are not invented by the models that describe them.
Historically, scientific discovery has always consisted of unveiling pre-existing truths, not creating them.
It is confusing to suggest that a proof could 'create' the equivalence between sensitivities based on fixed points and KKT; this relationship already existed.
Consider the law of gravity: Newton did not 'create' gravity, he simply formulated its proof, making the phenomenon understandable, but the force was already at work.