Strike a drum and you hear its lowest tone, and above it the next. How far apart can those two tones be? Measured by the ratio of their squared frequencies, the answer for drums of every shape has been known since 1992: the round drum is best, with a ratio of about two point five four. If the drum must be a polygon with a fixed number of sides, the regular polygon is believed to be best. For triangles this was completed in 2022: the equilateral triangle wins, with a ratio of seven thirds.

This release from Evidence Press settles the next case. It proves that among all four-sided drums, convex or dented, the square has the largest ratio, exactly five halves, and that no other quadrilateral reaches it.

The proof combines short mathematical arguments with a very large number of rigorous computations. Nearly triangular shapes and very thin shapes are handled by hand. Near the square, a careful expansion shows that every small deformation lowers the ratio. Everywhere else, the shapes are divided into hundreds of thousands of small families. For each family, a certificate computed in interval arithmetic, which keeps track of every rounding error, proves the inequality for every shape in it at once. Every certificate was checked again by replay programs that stop at the first discrepancy, and the frozen archive was replayed once more from a fresh copy.

The result is an unrefereed candidate. It has not been reproduced by an independent team, formally verified or peer reviewed.

The paper, its supplement, the code and every certificate log are linked from this page, and archived on Zenodo as version zero point one point zero candidate, released in October 2026. This summary uses an AI-generated voice. It is a communication aid, not additional mathematical evidence.