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on An autocorrelation problem with functions that can take negative values (AlphaEvolve problem 4)

The statement

Let $C$ be the best constant for which one has

\max_{-1/2 \leq t \leq 1/2}\left |\int_\mathbb{R} f(t-x) f(x) dx\right| \geq C \left(\int_{-1/4}^{1/4} f(x) dx\right)^2

for all $f \colon [-1/4,1/4] \to \mathbb{R}$ (note $f$ can take negative values). What is $C$ ?

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