In Today's Issue: What each lab claimed, what the refutation does and does not change, what Lean checked, how machine mathematics got here, a race with human costs, and who holds the capability behind the results. Records as of October 30, 2026.

Dear Readers,

On October 26, two AI labs posted papers claiming the same result. OpenAI's was titled "The Exponential Time Hypothesis is False". About six hours later Anthropic posted its own, attributed to "Claude / Anthropic". Both labs date the discovery to October 16, and both published formal versions checked by Lean, a program that verifies proofs step by step (OpenAI, 10/26/2026; Anthropic, 10/26/2026).

The conjecture they say they refuted concerns one of computing's oldest problems. In 3-SAT you get a long logical formula built from yes-or-no variables and must decide whether any combination of yeses and nos makes it true. With n variables there are 2^n combinations, a number that doubles with every variable you add. Clever algorithms have made the search faster, but every method known until now still slowed down at a fixed exponential rate: add a certain number of variables, and the running time doubles. The Exponential Time Hypothesis (ETH), proposed by Russell Impagliazzo and Ramamohan Paturi with Francis Zane, says that some fixed rate like this can never be escaped (OpenAI, 10/26/2026).

For decades computer scientists have used ETH to show that other problems are hard. Anthropic counts more than 800 papers that prove at least one result only under ETH or a stronger variant (Anthropic, 10/26/2026). Those results do not become false if ETH falls. They lose the premise they rest on, and with it what they say about how hard a problem really is.

The claims arrive while the field is still digesting September's OpenAI claim on a million-dollar Millennium Prize problem about fluid flow and more than 700 machine-written manuscripts from early October. The models behind these results are internal: only the labs can run them, and the labs decide when a result appears and who sees it first. Have the machines started producing genuinely new science, and if so, who controls it: the researchers who have to check it, or the two companies whose models produced it?

All the best,

Kim Isenberg

OpenAI announced three results that day: the ETH refutation; a proof in harmonic analysis, the mathematics of waves, which OpenAI says also settles the famous Kakeya conjecture; and an existence proof from particle physics (OpenAI, 10/26/2026). Anthropic announced one result, the ETH refutation, in a 49-page paper with its formal proof attached (Anthropic, 10/26/2026).

The two accounts of how the result came about differ. OpenAI says its AI found the disproof, written in ordinary mathematical language, on October 16, and that it was formalised in Lean the next day; its paper names no model and gives no compute figure (OpenAI, 10/26/2026). Anthropic writes that an internal Claude research model found the core algorithm, using 14 million output tokens and "no human input", with researchers working on the presentation afterwards (Anthropic, 10/26/2026). Neither account can be checked from outside.

The two proofs arrive at the same running time by different routes. That running time still grows enormously, but the growth keeps slowing: the larger the formula, the more variables it takes to double the running time, which is exactly what ETH rules out. The two main arguments share some of their machinery (OpenAI ETH paper, 10/25/2026; Anthropic, 10/26/2026), and Anthropic's paper already cites OpenAI's result as obtained "independently and concurrently". Two derivations that agree are a point in favour of the result. They also suggest that the ingredients were within reach once enough machine search was applied.