I'm looking for a source for the following result.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset of the critical points of $M$. Consider the negative of the gradient vector field $V$, that is at each point we consider the $(x, f(x)) \in M$ we assign the vector $V(x,f(x))=(−\nabla f(x),−|\nabla f|^2)$. This generates a flow $F: M \times \mathbb{R} \rightarrow M$ on $M$ and a set of integral curves that travel along the manifold toward the critical points of $M$. Consider a critical point $c \in M$ and define the attractor set $A(c) = \{p \in M: \exists t \in \mathbb{R}, F(p, t) = c\}$ of $c$.
Suppose that we have a bound on the fourth derivative of $f$ at the critical points of $f$. I've been told that given a bound on the fourth derivative of $f$ you can convert that into a lipschitz bound on the derivative of the gradient, $\nabla f$. This gives you a lower bound on the diameter of a ball contained in the $A(c)$ for a critical point $c$ of $f$.
I'm looking for a reference and official statement of this result.