Guide: Using OpenAI Models to Crack Discrete Geometry Conjectures

Problem

Researchers have long struggled with the unit distance problem, an 80‑year‑old question in discrete geometry that asks how many pairs of points at distance one can exist in a planar set. The problem sits at the heart of a broader conjecture that has guided countless papers and conferences. Until recently, no computational tool could settle the conjecture.

In late May 2026 an OpenAI model succeeded where human effort had stalled, delivering a proof that disproves the central conjecture. According to the OpenAI Blog, the model "solved the 80‑year‑old unit distance problem, disproving a major conjecture in discrete geometry" (https://openai.com/index/model-disproves-discrete-geometry-conjecture). This achievement shows that AI can move from assisting with language tasks to producing original mathematical arguments.

Prerequisites

• Access to an OpenAI model capable of reasoning over symbolic data. The exact architecture is not disclosed, but the model can ingest formal statements and output logical steps.
• Familiarity with the formal language used in discrete geometry. Researchers should be comfortable writing definitions, lemmas, and proof sketches in a machine‑readable format.
• A clean computational environment. Python 3.11, a recent version of the OpenAI API, and a notebook interface are sufficient.
• Version control for experiment tracking. Git or an equivalent system helps preserve prompt versions and model outputs.

Steps

1. Define the conjecture in a formal language

Start by translating the geometric statement into a set of axioms and a target theorem. Use a syntax that the OpenAI model accepts, such as Lean or a custom JSON schema. Keep each axiom atomic; the model performs best when the input is granular.

2. Prepare supporting data

Collect known results that the model can reference. This may include previously proven lemmas, counter‑examples, or computational bounds. Store them alongside the conjecture in the same file.

3. Craft a concise prompt

Write a prompt that tells the model what to do: "Given the axioms below, attempt to prove or disprove the theorem. Provide each logical step and cite any external result used." Include the formal definitions and any constraints.

4. Run the model

Submit the prompt via the OpenAI API. Set temperature to a low value (e.g., 0.2) to favor deterministic reasoning. Capture the full response, including intermediate steps.

5. Verify each step

Manually check the model’s output against the axioms. Use a proof assistant if possible; the assistant can confirm the logical validity of each line. Flag any gaps for further prompting.

6. Iterate with targeted prompts

If the model stalls, ask focused follow‑up questions: "Explain why Lemma 3 applies here" or "Show the missing case for configuration X". Iterative refinement often bridges the gap between a sketch and a full proof.

7. Document the proof

When the model produces a complete argument, format it for publication. Include a clear statement of the disproved conjecture, the model’s reasoning chain, and any human‑verified corrections.

Pro Tips

• Start small. Test the workflow on a known theorem before tackling an open conjecture.
• Use versioned prompts. Small changes in wording can alter the model’s reasoning path dramatically.
• Combine multiple models. A language‑focused model can generate the proof sketch, while a symbolic model can check correctness.
• Leverage community resources. Open‑source libraries for formal geometry can supply the axioms needed for the prompt.
• Track token usage. Complex proofs consume many tokens; budgeting helps avoid unexpected costs.

The breakthrough reported by OpenAI demonstrates that the workflow above is no longer hypothetical. By following these steps, researchers can harness the same capability that solved the unit distance problem, opening a new avenue for tackling long‑standing questions in discrete geometry.