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on Erdős minimum overlap problem (AlphaEvolve problem 5)

The statement

Let $C$ be the largest constant for which

\sup_{x \in [-2,2]} \int_{-1}^1 f(t) g(x+t) dt\geq C

for all non-negative $f,g: [-1,1] \to [0,1]$ with $f+g=1$ on $[-1,1]$ and $\int_\mathbb{R} f = 1$ , where we extend $f,g$ by zero outside of $[-1,1]$ . What is $C$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.