OpenAI discoveries: an AI proof that could upend computer science

OpenAI is claiming three separate results, each in a different field:

• Computer science: a proof that the Exponential Time Hypothesis (ETH) is false. The model’s write-up describes a method that exactly counts the ways to satisfy a certain class of logic formulas faster than ETH says should be possible.
• Mathematics: a proof of the local smoothing conjecture for waves in every dimension, which would also settle the Kakeya conjecture and Stein’s restriction conjecture in all dimensions.
• Physics: a rigorous existence proof for certain interacting quantum field theories in four dimensions, including Quantum Chromodynamics (the theory of the strong force that holds atomic nuclei together).

The timelines are fast. OpenAI says the model wrote its ETH disproof on October 16 and the result was formalized the next day. The math proof came on October 18 and was formalized a day later.

The Exponential Time Hypothesis is a bet about how hard some problems really are. It says a classic puzzle called 3-SAT, which asks whether a set of true-or-false conditions can all be met at once, can’t be solved much faster than trying an enormous number of options. Russell Impagliazzo and Ramamohan Paturi of UC San Diego proposed it in 1999 with Francis Zane, and it sits close to the famous P versus NP problem.

Researchers have used it as a foundation in fields like graph theory, databases, computational biology and computational geometry. Many papers say, in effect, “this task can’t get faster, assuming ETH is true.” If ETH is false, those limits loosen, and some problems might one day be solved far more efficiently than experts thought.

OpenAI itself is cautious on that point. The company says the result reopens questions about how much faster these tasks could go, but that turning the theory into practical gains will take a lot more research.

This is OpenAI’s second big science release this month. Earlier in October it posted hundreds of pages of AI-generated math, and that release drew pushback, including from an advisory group uneasy about a private company using unreleased models to grab breakthroughs.

This time OpenAI calls its outputs raw research data and has asked outside experts, which it calls shepherds, to turn them into real understanding. Each result gets a different release plan:

• The math output and its formal proof are public on GitHub, and anyone can work on explaining them.
• The ETH material is also public, at the request of a group led by Impagliazzo, Paturi and MIT’s Ryan Williams, who will examine it. OpenAI has also agreed to fund a conference on what the result means.
• The physics material is held back for now. A group led by physicist David Gross wants to write it up first.

OpenAI says it does not want academic credit for anything its internal models produce. The human groups get credit for the explanations they write, and the work runs with no nondisclosure agreements and no pay.

Both main proofs were checked in Lean, a software tool that verifies each logical step of a proof. That sounds airtight, but it has limits. Readers of an earlier OpenAI result about the Navier-Stokes equations worried that the Lean version had verified a weaker statement than the one OpenAI described. A formal check only proves what it was asked to prove.

The shepherd system is also contested. A panel of math advisers OpenAI itself assembled recommended, just a day before the release, that AI companies not hand results to chosen experts first. It cited unequal access and borrowed reputation. OpenAI acknowledges that picking the first shepherds gives some people a head start on information.

Nothing about your phone, apps or online banking changes today. These are theoretical results, and nothing in OpenAI’s announcement points to an immediate effect on everyday software or security tools.

The bigger impact is on how science gets done. If AI models can produce proofs that top experts then spend months untangling, universities, journals and students will need new ways to assign credit and check work. The ETH group has already asked the research community for input on early access, conflicts of interest and how to support students as methods change.

Watch the ETH group’s review. If Impagliazzo, Paturi and Williams confirm the disproof, the hypothesis they helped create will have been overturned by a machine, and researchers in many fields will start rechecking their assumptions.