Is there any hope of a high-level explanation of why the fraction $\frac{1}{4}$ plays such a prominent role as a sectional curvature bound in Riemannian geometry? My (dim) understanding is that the idea is that if the sectional curvature of a manifold is constrained to be close to 1, then the manifold must be topologically a (homeomorphic to the) sphere $S^n$. Conjectured by Hopf and Rauch, proved by Berger and Klingenberg, and strengthened by Brendel and Schoen to establish diffeomorphism to $S^n$. Here is the definition of "local" $\frac{1}{4}$-pinched from Brendel and Schoen's paper "Manifolds with $1/4$-pinched curvature are space forms" (J. Amer. Math. Soc., 22(1): 287-307, January 2009; PDF):
We will say that a manifold $M$ has pointwise $1/4$-pinched sectional curvatures if $M$ has positive sectional curvature and for every point $p \in M$ the ratio of the maximum to the minimum sectional curvature at that point is less than 4.
I know the "$\frac{1}{4}$" in the $\frac{1}{4}$-pinched sphere theorem is optimal, and perhaps that is the answer to my question: $\frac{1}{4}$ appears because the theorem is false otherwise—punkt! But I am wondering if there is a high-level intelligble reason for the appearance of $\frac{1}{4}$, rather than, say, $\frac{3}{8}$, or $\frac{e}{\pi^2}$ for that matter?
I am aware this is a "fishing expedition," and a fair response is: Study the Brendel-Schoen proof closely, and enlightenment will follow!