Ten to the power of ten thousand. This is the number of variables needed for a logical formula with five times more constraints than variables for the algorithm published on October 27 by Anthropic to beat the stupidest method, the one that tries all the combinations. Brute force already becomes impractical around a hundred variables. The algorithm will therefore be of no use in practice. If it is correct, it “will change an important part of theoretical computer science”, write the three researchers responsible for examining a twin result.
Because he doesn’t come alone. On October 26, OpenAI announced that one of its internal models had disproved the exponential time hypothesis, a conjecture formulated in 1999 on which hundreds of results are based. Less than six hours later, Anthropic published its own rebuttal, a 49-page article titled “A 2^o(n) Time Algorithm for 3-SAT” and signed “Claude.” The two laboratories date their discovery to the same day, October 16. At Anthropic, the central algorithm was found by an internal Claude research model, with 14 million tokens produced and without human intervention in the discovery itself, Anthropic mathematician Levent Alpöge told X, according to OfficeChai.
A conjecture on which more than 800 articles depend
The 3-SAT problem involves deciding whether variables can be assigned true or false in order to satisfy a list of logical constraints. Each added variable doubles the number of combinations to try. In 1999, Russell Impagliazzo and Ramamohan Paturi, of the University of California at San Diego, posited that no algorithm could escape this exponential growth other than by a constant. This is the exponential time hypothesis, or ETH.
Researchers use it as a benchmark. When they want to show that a planning, graphing or verification problem will never be solved quickly, they bring it back to 3-SAT and invoke ETH. More than 800 papers depend on it, ETH or a related hypothesis, according to the Anthropic article. If the refutation holds, their lower bounds lose their basis, and classical problems like the clique or vertex cover theoretically admit algorithms faster than expected.
Two proofs, the same terminal, no human proofreading completed
The two algorithms result in very close bounds, of the order of 2 to the power of n divided by the root of the logarithm of n, up to secondary factors. It’s barely less than exponential. But ETH prohibited precisely this “barely”.
The two laboratories accompanied their text with a proof formalized in Lean, a language where each step is controlled by a program. OpenAI says it completed its work on October 17. Anthropic had its verification verified by Lean Core and a second independent verifier, nanoda, according to its GitHub repository. Its appendix specifies that this development was written by AI models, proofread by other instances of the same models, and that at the time of writing no person was known to have verified it.
A formal proof guarantees that the stated theorem follows from the axioms. It does not guarantee that the formalized statement actually says what the authors claim. This is what humans must now establish. OpenAI asked Russell Impagliazzo and Ramamohan Paturi themselves, along with MIT’s Ryan Williams, to organize an open review. In their letter to the community, the three researchers warn that their participation does not constitute certification. They obtained from OpenAI the publication of the raw data and funding for a conference, according to The Next Web. Complexity theorist Lance Fortnow, who says he has not verified the proof himself, believes on October 28 on his blog that the hypothesis can now be considered dead.
Neither the encryption keys nor the solvers move
The first question a security manager asks has, for now, a reassuring answer. Ryan Williams asked Dan Roberts, OpenAI’s chief scientist, if any cryptographic systems were affected: he said OpenAI checked and was “reasonably confident” that none were broken, TechCrunch reported. RSA relies on integer factorization, which is not known to be as difficult as 3-SAT. Ryan Williams believes, however, that refinements of the algorithm could one day concern systems sized on precise key sizes.
Industrial SAT solvers, which check chips or plan supply chains, don’t gain anything either: they deal with millions of variables with heuristics that ETH has never been concerned with. The Anthropic article itself writes it, the result says nothing about solving the formulas in practice. P versus NP remains open. Barely subexponential time is still far from polynomial time.
The bottleneck moves from discovery to verification
This October 26, OpenAI also claimed proof of Christopher Sogge’s local smoothing conjecture in all dimensions, which would settle Kakeya’s conjecture, and proof of the existence of pure Yang-Mills theory in four dimensions, which only settles part of one of the Clay Institute’s seven millennium problems. For physics, the group led by Nobel Prize winner David Gross has chosen not to publish the material produced by AI for the moment.
At the beginning of October, OpenAI had already published 722 mathematics articles produced by its models at once. Ryan Williams called this salvo a “DDoS attack on mathematicians”. However, he agreed to organize the exam. He considers some parts of the evidence routine, but two or three elements very unusual.
Discovery is now measured in tokens. Verification is always measured in months of work by researchers, and they are the ones who are missing.
Quotes translated from English by ActuIA.
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