[Math] Why the quarter in the $\frac{1}{4}$-pinched sphere theorem

Question

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!

Answer 1

The new proofs add a layer of complication that is unnecessary for your question. From Cheeger and Ebin, Comparison Theorems in Riemannian Geometry first paragraph of Chapter 6:

The symmetric spaces of positive curvature are known to admit metrics such that $1 \geq K_M \geq > \frac{1}{4}$; see Example 3.38. In fact, we will prove that any riemannian manifold with $1 \geq K_M > \geq \frac{1}{4},$ which is not a sphere, is isometric to one of these spaces.

Back in chapter 3, page 73,

A complete classification of symmetric spaces is available (Helgason [1962]). In particular, the only simply connected symmetric spaces having positive curvature are the spheres of constant curvatures, complex and quaternionic projective spaces, and the Cayley plane. These are sometimes referred to as the rank one symmetric spaces, and except for the spheres they have canonical metrics varying between $\frac{1}{4}$ and $1.$ As an example, we will compute the curvature of complex projective space. The calculations for the other rank one spaces are similar.

Then they begin the example, which is numbered Example 3.38. The punchline is that $\alpha$ and $\beta$ are orthogonal unit vectors, $J(\alpha)$ and $\beta$ are unit real vectors, so their real inner product lies between $-1$ and $1$ by Cauchy-Schwarz, followed by an identity about the sectional curvature that I shall write as $$ K_{\alpha \beta} = \frac{1}{4} + \frac{3}{4} \langle J(\alpha), \beta \rangle^2. $$