A machine-checked bound of 0.6934 for Besicovitch's 1/2-problem
Besicovitch's 1/2-conjecture asserts that a Borel set in the plane with finite one-dimensional Hausdorff measure whose lower density exceeds 1/2 almost everywhere is countably 1-rectifiable. Writing sigma_1(R^2) for the least threshold that forces countable 1-rectifiability, the conjecture is sigma_1(R^2) = 1/2. The previously published record is sigma_1(R^2) <= 7/10, obtained by De Lellis, Glaudo, Massaccesi and Vittone. This development proves, with a complete Lean 4 proof term, sigma_1(H) <= 6934/10000 = 0.6934 for every real Hilbert space H, without finite-dimensionality or separability…
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