Lebesgue's universal covering problem, posed in 1914, asks for the convex set of least area containing an isometric copy of every planar set of diameter 1. The known upper bounds descend through a ladder of constructions — the regular hexagon, Pál (1920), Sprague (1936), Hansen (1992), Baez–Bagdasaryan–Gibbs (2015), Gibbs (2018) — whose areas are quoted in the literature to at most twelve decimal places. So far as I can determine none had been evaluated with rigorous error control. This note certifies the whole ladder: hexagon √3/2 Pál 2 − 2/√3 Sprague 0.844137708435197570894066994044… Hansen 0.844137708397690336757182874586… BBG 2015 0.844115376859376746806104420762… (σ = 1.3) 0.844115297128419059214192192376… (Egan's σ) Gibbs 2018 a ≤ 0.8440935944, certified with margin 9.3×10⁻¹² The last line is, to my knowledge, the first verification of the record since its publication eight years ago; together with the certified lower bound of Xie (2026), the problem's bracket 0.833 ≤ a ≤ 0.8440935944 now rests on certified computation at both ends. Section 3 reconstructs Sprague's reduction and records a point of attribution: his 1936 paper proves only that Pál's cover is not minimal, and states no area. Section 4 describes the arithmetic. Section 5 confirms every digit of the Baez–Bagdasaryan–Gibbs table and settles a discrepancy with a figure circulated on the Azimuth blog. Sections 6 and 7 certify the 2015 cover and the 2018 record, the latter by running the record's five-step boundary map in second-order jet arithmetic over interval coefficients to bound the discretisation error rigorously. Section 8 reports a soundness defect in the kernel's first conversions — caught by a consistency gate between independently certified rungs — with its fix and its blast radius. Universality, that each set does cover every diameter-1 set, is taken from the literature throughout; the computational side conditions of the 2015 construction are certified here.
Gonzalez, V. (2026). Rigorous areas for the classical Lebesgue universal covering ladder. Zenodo. https://doi.org/10.5281/zenodo.21701519