I guess what I'm wondering is how does the constant that lower bounds the diameter of a ball contained in $A(c)$ depend on the dimension $d$ and the Lipschitz constant $\alpha$.

@RyanBudney Why do you even bother answering questions if you're not going to explain what you mean in full? In the future please stop "answering" questions that I ask. Your answers are never helpful and all it does is stop other people from answering.

@RyanBudney Could you at least provide a source for that result?

Yes, I could assume that I have access to 4th derivatives. Could you elaborate further on how you turn that into a bound on the derivative of the gradient and how that gives a lower bound on the diameter of a ball contained in the stable manifold of a critical point?