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Remarks on the Disproof of the Unit Distance Conjecture

Source: arXiv:2605.20695
Authors: Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, Melanie Matchett Wood
Date: May 2026

TL;DR

An OpenAI general-purpose reasoning model autonomously disproved the Erdős Unit Distance Conjecture (1946) — that the maximum number of unit distances among n points in the plane is roughly n — by constructing point sets achieving a provably larger exponent. Nine leading mathematicians verified the proof and published an expository paper. Thomas Bloom called it "a $500 Erdős problem solved." Jacob Tsimerman said he would "accept it for any journal without hesitation."

Historical Context

  • The Problem (Erdős 1946): What is the maximum number of unit distances among n points in the plane?
  • Previous Bounds: Erdős gave n^(1+c/log log n) using √n × √n grids. The best upper bound was O(n^(4/3)). The consensus belief was that the true answer was n^(1+o(1)) — that is, nearly linear.
  • The Conjecture: That n^(1+o(1)) is the correct order of growth.

The AI's Breakthrough

The key insight came when the AI shifted perspective: rather than varying the point structure within a fixed number field (as everyone had tried), it fixed the primes and varied the number field — a paradigm shift no human mathematician had considered.

The construction uses:

  1. CM fields — algebraic number fields where having absolute value 1 in one embedding implies the same in all embeddings
  2. Golod-Shafarevich infinite class field towers — to build fields of increasing degree with bounded root discriminant
  3. A geometry-of-numbers lemma — projecting lattice points from high dimensions onto a single coordinate

The resulting exponent is roughly 1 + 6.24 × 10⁻³⁸ — tiny but definitively > 0, disproving the conjecture.

Why It Matters

  • First AI-disproved prominent open conjecture in mathematics (not just AI-assisted, but AI-originated)
  • Kolmogorov complexity modulo experts (Gowers' phrase): the proof has low complexity but required a non-obvious paradigm shift — precisely where AI excels
  • Methodological lesson: The AI wasn't substituting for human reasoning — it was exploring combinatorially larger search spaces for the right conceptual framing