OpenAI Model Disproves Planar Unit Distance Conjecture

An OpenAI general-purpose reasoning model has autonomously disproved a longstanding conjecture in discrete geometry regarding the planar unit distance problem. This result is a milestone in AI research, marking the first time a prominent open problem in a mathematical subfield has been solved autonomously by an AI system.

Resolution of the Planar Unit Distance Problem

The planar unit distance problem, posed by Paul Erdős in 1946, asks for the maximum number of pairs of points $u(n)$ that can be exactly distance 1 apart when $n$ points are placed in a plane. For decades, the prevailing mathematical belief was that rescaled square grid constructions were optimal, yielding a growth rate of $n^{1 + C / \log \log n}$. Erdős conjectured an upper bound of $n^{1 + o(1)}$, where the additional term tends to zero as $n$ increases.

An OpenAI model has disproved this conjecture by providing an infinite family of examples that achieve a polynomial improvement. Specifically, the model constructed configurations of $n$ points with at least $n^{1 + \delta}$ unit-distance pairs for some fixed exponent $\delta > 0$. While the original AI-generated proof did not specify the value of $\delta$, a subsequent refinement by Princeton professor Will Sawin established that $\delta = 0.014$.

Technical Approach: Integrating Algebraic Number Theory

The model's solution was achieved by applying sophisticated concepts from algebraic number theory to an elementary geometric problem, a connection that was previously unexpected to mathematicians.

From Gaussian Integers to Algebraic Number Fields

Erdős's original lower bound relied on Gaussian integers (numbers of the form $a + bi$). The AI model expanded this approach by replacing Gaussian integers with more complex generalizations known as algebraic number fields. These fields possess richer symmetries that allow for the creation of significantly more unit-length differences.

Advanced Mathematical Tools

To prove that the required number fields actually exist, the model utilized high-level tools including:

• Infinite class field towers
• Golod–Shafarevich theory

Significance for AI and Mathematics

This breakthrough demonstrates that advanced reasoning models can move beyond acting as assistants to human mathematicians and are capable of generating original, ingenious ideas.

Impact on Mathematical Research

The result does not merely settle a specific conjecture but creates a bridge between algebraic number theory and discrete geometry. As noted by Thomas Bloom in the companion paper, the discovery suggests that "number theoretic constructions have to say about these sorts of questions than we suspected" and that the required number theory can be "very deep."

Capabilities of the Reasoning Model

Unlike systems trained specifically for mathematics or those scaffolded for proof searching, this result was produced by a general-purpose reasoning model. This indicates a capacity for deep, coherent reasoning and the the ability to connect distant areas of knowledge—capabilities that OpenAI suggests are applicable to other scientific fields such as biology, physics, and materials science.

Expert Perspectives

External mathematicians have verified the proof and written a companion paper to provide context. Key insights from the experts include:

"In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition"

— Arul Shankar, leading number theorist

"a milestone in AI mathematics"

— Tim Gowers, Fields medalist