Strictness of the flat autoconvolution inequality

Determine whether the autoconvolution constraint forces the strict inequality \(C_{6.3}<1\) for the supremum of the normalized flatness functional over nonnegative integrable square-integrable functions.

Background

The flat autoconvolution problem asks how closely the self-convolution of a nonnegative function can approximate a flat-topped function. Hölder’s inequality gives the general upper bound C_{6.3}≤1.

The unresolved issue is whether equality can be approached arbitrarily closely under the additional requirement that the output be an autoconvolution. The paper proves that binary step functions suffice for approaching the unrestricted supremum but does not settle whether the supremum is strictly below one.

References

Whether the autoconvolution constraint forces the strict inequality $C_{6.3}<1$ remains open.

— Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment  (2608.23691 - Chung et al., 24 Aug 2026) in Section 3, subsection “Flat autoconvolution”