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!