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Sign inBoth works prove that every non-trivial union-closed family has an element in at least a fraction of its sets, above Liu's proved and his numerically predicted . Jiang's argument uses entropy bounds with global couplings and is computer-assisted; Costa–Sadhu reach and show that the inequality they use cannot exceed about . The entropy method as it stands cannot reach Frankl's .
Two concurrent preprints reach the same constant by related but separately implemented computations, which is corroborating. Neither is refereed and no external audit of the computations has been published.
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Jiang and Costa–Sadhu report the same value 0.38288525 to eight digits.
Re-execute the optimisation and confirm the final inequality is certified by interval or exact arithmetic, including the region near the optimum where the margin is about .
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Verify Jiang's global-coupling lemma and that the one-variable inequality being optimised implies the stated frequency bound, as in Gilmer's and Sawin's frameworks.
Neither preprint is published.