For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least \(c_dn^{2/d}\) distinct distances, with \(c_d\gt 0\) depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
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It bears on Erdős problem #1083, as listed by Thomas Bloom, erdosproblems.com.
The paper states
In The higher-dimensional Erdős distinct-distances conjecture, 23 September 2026
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For every integer d\ge3, there is a constant c_d>0 such that every finite set P\subset\mathbb R^d of n\ge2 distinct points satisfies
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The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000900 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: No.
Yao Xu, with AI assistance, cadamcat/distinct-distances-lean4: Lean 4 formalisation of the higher-dimensional Erdős distinct distances bound (JSP-000900), 10 October 2026. Source
Independent Lean formalisation of the higher-dimensional distinct distances bound.
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Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 1083 (distinct distances in higher dimensions), names Tidor, Yu and Zakharov, and says it is not formalised, heavily algebraic and 103 pages long.
mo271, ErdosProblems/1083: erdos_1083 is true (issue 6905), 7 October 2026. Source
The release’s higher-dimensional distinct-distances manuscript gives at least c_d n^(2/d) distinct distances for d >= 3, which settles Erdős 1083.
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Why this passage is shown
Theorem 1.1
In The higher-dimensional Erdős distinct-distances conjecture, 23 September 2026