Claims

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2026_05_20_alon_bloom_gowers_litt_sawin_shankar_tsimerman_wang_wood: Theorem 1.1 of Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Matchett Wood, arXiv:2605.20695, gives planar sets with a fixed-power excess of unit pairs; a human-digested proof of OpenAI's result, claimed.

2026_05_20_openai: Theorem 1.1 of OpenAI's report Planar Point Sets with Many Unit Distances gives n-point planar sets with n^(1+delta) unit pairs, fixed delta > 0, along infinitely many n; accepted on the companion's check and the curator's credit.

2026_05_20_sawin: Theorem 1 of Sawin's arXiv:2605.20579 gives, for arbitrarily large n, an n-point planar set with at least n^1.014114 over an absolute constant unit pairs; a second, quantitative disproof, claimed.

2026_05_26_anon: Theorem 1.1 of a 13-page manuscript with no author line gives, along an infinite set of n, u(n) >= n^(1 + c0 log log log n / log log n), which refutes the problem's bound with a gain tending to zero; claimed.

2026_06_02_emmerich: Proposition 2 of Emmerich's report re-optimizes the parameters of Sawin's criterion with Sawin's prime set and states u(n) > n^1.0152 for arbitrarily large n; claimed.

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