OpenAI Model Disproves Erdős's 1946 Conjecture

An OpenAI Model Disproved a Central Conjecture in Discrete Geometry

Paul Erdős posed the unit distance problem in 1946. In May 2026, an internal OpenAI reasoning model finally broke it — not by proving the conjecture, but by knocking it down.

For nearly eighty years the working assumption in discrete geometry was that the maximum number of unit-distance pairs you can squeeze out of n points in the plane grows only slightly faster than linearly. The classic square-grid construction looked essentially optimal. Erdős himself conjectured an upper bound of n^(1+o(1)) — the number of unit-distance pairs is n to a power that barely edges past 1.

A general-purpose reasoning model from OpenAI found an infinite family of counterexamples: point sets with at least n^(1+δ) unit-distance pairs for a fixed exponent δ > 0. The original AI's exponent was microscopic, around 10^(-38), but Princeton's Will Sawin later pushed it to δ = 0.014 in a refinement (OpenAI, Scott Aaronson).

That is not a cosmetic tweak. Going from "barely faster than linear" to "a genuine polynomial excess" is a qualitative jump, and the proof survived contact with some of the strongest minds in the field.

The Problem Anyone Can State

Place n points anywhere on a flat plane. Count pairs of points that are exactly one unit apart. Call that number U(n). The question: how big can U(n) get?

It is the kind of thing you can explain to a non-mathematician in thirty seconds, and then think about for decades.

Here is a brute-force way to count unit pairs for a small set — the boring, developer-approved version of the problem:

import itertools
import math

def unit_pairs(points):
    """Count pairs separated by exactly one unit of distance."""
    count = 0
    for a, b in itertools.combinations(points, 2):
        if math.isclose(
            (b[0] - a[0]) ** 2 + (b[1] - a[1]) ** 2,
            1.0,
            rel_tol=1e-9,
        ):
            count += 1
    return count

# A 4x4 square grid: a decent number of unit pairs, but is it optimal?
grid = [(i, j) for i in range(4) for j in range(4)]
print(unit_pairs(grid))

The grid construction Erdős used in 1946 gives n^(1+Ω(1/log log n)) unit pairs — growth only slightly faster than linear. For decades the best upper bound known was O(n^(4/3)), due to Spencer, Szemerédi, and Trotter in 1984, and the conjecture that the grid was essentially optimal stood firm.

What the Model Actually Did

This is the part people keep getting wrong, so let's be precise.

The result is not an AI regurgitating a known proof from training data, and not a brute-force computer search over a bounded configuration space. It is a model independently assembling a non-obvious connection between two branches of mathematics and constructing an original argument using techniques from algebraic number theory — the study of fields that extend the integers and rationals (arXiv:2605.20695, AI Beat).

The construction replaces the Gaussian integers — the engine behind Erdős's original square grid — with more complicated algebraic number fields that carry richer symmetries. Those fields create many more unit-length differences than the humble grid can.

The key machinery: infinite class field towers and Golod–Shafarevich theory. The existence of these towers was established back in the 1960s, and they were well known to algebraic number theorists. That they have anything to do with a simple geometric question about points at unit distance in the plane came as a genuine surprise.

Why Mathematicians Are Actually Convinced

A skeptic's first question is always verification. This is where the result differs sharply from earlier AI math claims.

A companion paper, Remarks on the disproof of the unit distance conjecture, was signed by nine working mathematicians (affiliations per the arXiv listing):

• Noga Alon (Princeton)
• Thomas F. Bloom (University of Manchester)
• W. T. Gowers (Cambridge, Fields Medalist)
• Daniel Litt (Toronto)
• Will Sawin (Princeton)
• Arul Shankar (Toronto)
• Jacob Tsimerman (Toronto)
• Victor Wang (Academia Sinica)
• Melanie Matchett Wood (Harvard)

The paper is transparent about its provenance: it presents "a human-digested, somewhat simplified, and somewhat generalized version of the AI proof." Per AI Beat's reporting on the announcement, Gowers said he would recommend the proof for publication without hesitation, describing it as the first AI-produced mathematics he has seen that contains a genuinely original idea rather than a well-structured search through existing technique space.

That endorsement carries extra weight in context. Thomas Bloom is the same mathematician who publicly debunked OpenAI's October 2025 claim that GPT-5 had solved ten previously unsolved Erdős problems — every "solution" turned out to be already in the literature. He did not take OpenAI's word the second time around.

The October 2025 Contrast Matters

In October 2025, an OpenAI executive claimed GPT-5 had solved a batch of Erdős problems. The claim collapsed within forty-eight hours when Bloom pointed out that every solution was a known result. The post was deleted.

This May result is the opposite shape. It landed with a verified, co-authored paper signed by nine mathematicians whose reputations do not survive endorsing a flawed proof. The verification chain is what makes the claim load-bearing — and it is now the standard against which every future AI math announcement gets judged.

Scott Aaronson described how it came about: his former student Lijie Chen "simply gave GPT the problem," and after thinking for a while the model "output a several-page argument that, on analysis by human experts, turned out to be correct" (Shtetl-Optimized).

No scaffolding, no fine-tuning on mathematics, no model purpose-built for theorem proving. OpenAI says it was evaluated on a collection of Erdős problems as part of a broader effort to test whether advanced models can contribute to frontier research, and this one produced a counterexample (OpenAI).

How to Read an AI-Math Claim

This article applies an implicit standard to the May result. Made explicit, it is a checklist any reader can reuse the next time a lab announces a machine-made proof:

1. Named signatories with reputations at stake. Nine mathematicians co-authored the companion paper — anonymous "internal evals" do not count.
2. Human-digested proof text, not a model transcript. The argument survived expert analysis, not just a benchmark score.
3. A track record of retracting false alarms. The October 2025 claim was deleted within forty-eight hours once the solutions turned out to be known results — a lab that corrects publicly earns sharper attention next time.
4. Attribution kept honest. "No scaffolding, no fine-tuning" is OpenAI's self-report, relayed through Aaronson's anecdote — read it as the lab's claim, not an independent measurement.

The Meta-Lesson for Developers

There are two ways to read this story.

The first: an AI system just did something a roomful of brilliant humans hadn't managed in eighty years. That is the headline.

The second is more interesting, and it is the one that should matter to people who build with these models. The model did not "understand geometry better" than the experts. It did something subtler: it was willing to try paths the community had quietly written off, and — crucially — it was not emotionally invested in proving Erdős right. The community spent most of its effort trying to prove the conjecture true, because that was the shared belief. The model spent most of its chain-of-thought looking for a counterexample to the widely believed upper bound, then imported machinery from a domain most discrete geometers never touch.

That combination — a willingness to explore dismissed paths plus an encyclopedic reach across subfields — is a genuinely different research profile from the typical human specialist. It does not mean AI replaces mathematicians. The verification, digestion, and generalization of the result were deeply human jobs. What it does mean is that "medium-hanging fruit that experts missed for contingent reasons" is now being picked. The same alien-cognition theme runs through the slowdown argument from OpenAI's own chief scientist — the trace is useful precisely because the thinking behind it is not human-shaped.

The wall that stood for eighty years has a hole in it. The next eighty years of unit-distance research start from a different floor — and the AI-assisted workflow that made this one possible is now part of the toolkit.

This is not a one-off, either. Shortly after, the same line of attack inspired a disproof of the Erdős–Szemerédi strong sumset conjecture over the reals — this time with humans, including Sawin, doing almost all of the work, directly inspired by GPT's refutation (Shtetl-Optimized).

If you are still skeptical of what reasoning models contribute to research, this is the cleanest demonstration yet that the answer to "what can they do autonomously?" is no longer "nothing."

For more context on an OpenAI math milestone heading into the same storm — the Navier–Stokes result, the 10,000-agent swarm, and the credit controversy around it — see our earlier deep dive.

Sources

• OpenAI: An OpenAI model has disproved a central conjecture in discrete geometry
• arXiv: Remarks on the disproof of the unit distance conjecture (arXiv:2605.20695)
• Shtetl-Optimized (Scott Aaronson): Dispatches from the possibly last days of human relevance
• AI Beat: Eighty Years, One Model, One New Idea
• Gary Marcus (Substack): Checking the math behind OpenAI and Anthropic's latest announcements