The Erdős unit distance conjecture in discrete geometry is true.
Assessment
Available evidence weighs against the claim.
The Erdős unit distance conjecture, posed in 1946, holds that any n points in the plane determine at most n^(1+o(1)) pairs at unit distance, so that the square grid, which achieves n^(1+c/log log n), is essentially optimal. The conjecture is false. In May 2026 an internal OpenAI model produced a counterexample, and a nine-author team (Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood) published a checked, simplified proof: there is a fixed ε > 0 and an infinite family of planar point sets with at least n^(1+ε) unit distances. The construction takes CM number fields of growing degree from a Golod–Shafarevich class field tower with bounded root discriminant and a completely split prime, produces exponentially many elements of absolute value one by a pigeonhole argument in the class group, and projects a bounded window of the field's lattice to the complex plane. Sawin made the exponent explicit (more than n^1.014 unit distances for arbitrarily large n), and Lee, Pohoata and Zhu gave a second construction by a different method. The counterexample has also been formalized in full in Lean, according to Kevin Buzzard, who inspected the code.
The refutation rests on one classical ingredient, the existence of infinite towers of number fields with bounded root discriminant and a fixed completely split prime, a settled consequence of the Golod–Shafarevich theorem. No credible objection to the mathematics has appeared; the discussion in the literature concerns attribution to AI and what the result says about the difficulty of the problem, not its correctness. What remains open is the true order of the maximum: the new lower bounds sit just above exponent 1, while the Spencer–Szemerédi–Trotter upper bound O(n^(4/3)) has not been improved.
Full reasoning: the evidence and decisions behind this verdict
The claim is that the conjectured bound u(n) ≤ n^(1+ε) for every ε > 0 and all large n holds. Theorem 1.1 of "Remarks on the disproof of the unit distance conjecture" (arxiv.org/html/2605.20695v1, read whole through Section 2 and the reflections) asserts a fixed ε₀ > 0 and point sets P_i with |P_i| → ∞ and at least |P_i|^(1+ε₀) unit distances, which is exactly the negation. Section 2 proves it from Lemma 2.1 (a geometry-of-numbers count: a δ-separated lattice in C^f whose projection to one coordinate is injective yields at least |U_Λ|·|Λ ∩ B_(R−1)|/2 unit-distance pairs among at most (9R²/δ²)^f projected points), Lemma 2.2 (pigeonholing ideals ∏P_j^(a_j)P̄_j^(k_j−a_j) in the class group gives at least ∏(k_j+1)/h(K) elements u = α/ᾱ of modulus 1 in every embedding of a CM field, with bounded denominator), and the tower step: CM fields of growing degree with bounded root discriminant and a completely split rational prime, which is the existence of infinite class field towers of bounded root discriminant with a fixed split prime, a textbook consequence of Golod–Shafarevich with Shafarevich's relation-rank bound (Koch, Theorem 11.5). With root discriminant bounded, the class number is at most exponential in the degree while ∏(k_j+1) can be made a larger exponential, so log(2ν(P))/log|P| stays above a constant greater than 1 as the degree grows. Each step was followed; no gap was found, and the argument is short enough that the nine authors' verification is corroboration rather than the ground of the verdict.
Corroboration by independent routes: Sawin (arxiv.org/abs/2605.20579) obtains the explicit exponent 1.014 from the same method, recorded as the n^1.014 lower bound (abstract read); Lee, Pohoata and Zhu give a robust repeated-distance construction whose whole-set case is a second counterexample by a different method (not independently checked on this pass). Kevin Buzzard's Xena post (xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/, read whole) reports that Boris Alexeev announced on 26 June 2026 a complete Lean formalization of the counterexample assuming nothing beyond the axioms of mathematics, about 1.2 million lines, which Buzzard ran in a sandbox and found to prove nontrivial theorems of global class field theory; the Lean FRO's public page and a Lean Eval leaderboard mention the same submission. That formal proof was not checked against a statement published on this platform, so it enters as strong reported evidence rather than as a machine-checked verdict of the graph's own. The reference page on the unit distance exponent (teorth.github.io/optimizationproblems/constants/84a.html) and MathWorld both record the conjecture as disproved.
Instances: all four recorded sources deny the claim (the nine-author write-up, Sawin, the OpenAI announcement, and the Xena post, whose stance had been recorded backwards and was corrected after reading). The affirming voice is historical: Erdős conjectured the bound and most experts believed it, as Gowers's reflection in the write-up states plainly. The write-up instance's recorded passage is Theorem 1.1 as it appears in the HTML full text; the mechanical check against the abstract page recorded as its URL does not find it, which is a rendering difference, not a misquotation.
Credence that the claim is true is about 0.02: the residual chance that an error survived a short proof checked by nine expert authors, reproduced with an explicit exponent, formalized in Lean, and matched by a second construction. Verdict confidence 0.92. What would change it: a demonstrated gap in Lemma 2.1, Lemma 2.2, or the tower step that also invalidated Sawin's and the Lee–Pohoata–Zhu constructions and the Lean formalization. Another pass would add little; the open question of the true exponent lives on other claims.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.