OpenAI Proof Erases Landau-Siegel Zeros, Forcing a Reckoning Over P vs NP

On October 18, Stanford number theorist Jared Duker Lichtman wrote publicly that an eight-page AI-generated proof had just closed a number theory question that had resisted human mathematicians for roughly fifty years — and that the structure of the result strongly suggested P ≠ NP. He was not announcing a resolution to the most famous open problem in computer science. He was responding to something narrower and, in its way, more unsettling: the proof had been found by an AI model in a matter of hours, then verified by human experts over a matter of days. That asymmetry is precisely the intuition behind one of mathematics' deepest unsolved problems.

The proof in question is OpenAI's claimed resolution that so-called Landau-Siegel zeros do not exist — a result that emerged from the company's October 6 release of 722 mathematical manuscripts to a public GitHub repository. The release covered 372 families of previously open problems and was produced by an internal model that has not been publicly disclosed. By the company's own account, the model spent an average of roughly three hours of compute time per result, trying approximately 4,000 problems in total. According to Quanta Magazine, Lichtman later called it "the most significant AI-for-math achievement so far."

What Landau-Siegel Zeros Are—and Why Their Nonexistence Matters

Landau-Siegel zeros are hypothetical near-exceptional real zeros of a class of functions called Dirichlet L-functions — objects central to analytic number theory. L-functions encode arithmetic information about primes, particularly primes in arithmetic progressions (primes that land in sequences such as 1, 5, 9, 13, and so on). The existence of Landau-Siegel zeros close to the value 1 would indicate that standard control over how primes distribute themselves in these progressions breaks down in an unusually severe way. Their nonexistence gives number theorists tighter uniform control over those distributions — the kind of control that is a weaker but still substantial substitute for the Generalized Riemann Hypothesis (GRH).

OpenAI's model actually produced a larger result of which the Landau-Siegel result is a consequence: a claimed quasi-Riemann Hypothesis (QRH), a 199-page Lean-verified proof establishing that all nontrivial zeros of every Dirichlet L-function lie in a fixed strip, with the real part bounded strictly between 7/8 and 11/12. The proof is part of the openai/math GitHub repository. This does not reach the full Riemann Hypothesis, which asserts that all such zeros lie exactly on the line Re(s) = 1/2, but it represents a materially stronger zero-free region than anything previously proven in the literature. The independent confirmation of that result — through the Landau-Siegel simplification cascade described below — gave the mathematical community its most accessible entry point into the work.

From Eight Pages to Five: How Human Mathematicians Decoded the Proof

The compression of the Landau-Siegel proof by human mathematicians over the week following the October 6 release became one of the signal events in the broader story.

William Banks, a mathematician at the University of Missouri, was first. Five days after OpenAI's release, Banks posted a 19-page exposition reconstructing the Landau-Siegel argument with explicit constants — producing a human-readable rendering of the same result. Bryce Kerr then published a six-page version, achieving the compression by drawing on biquadratic field extensions and a classical result due to Masser. Ben Green, the Waynflete Professor of Pure Mathematics at Oxford, closed the rapid-fire simplification process with a five-page proof published around October 17–18.

The speed of this compression prompted Abhishek Saha, a professor at Queen Mary University of London, to pose the question now circulating in number theory circles: if the proof that no Landau-Siegel zeros exist requires only five pages, why did mathematicians fail to find it for half a century? The honest answer is that the AI did not follow the strategies humans expected to be productive. It identified a path through territory that the field had not systematically explored, possibly because standard heuristics pointed elsewhere. Human mathematicians, now reading the AI's approach through Banks's exposition and Green's compression, are working to understand not just whether the proof is correct but what the underlying strategy actually reveals about why it works. That question — what the AI saw that humans did not — may ultimately be more valuable than the result itself.

Why the Five-Page Proof Triggers the P vs NP Intuition

P versus NP, proposed formally by Stephen Cook in 1971 and carrying a $1 million Clay Millennium Prize, asks whether every problem whose answer can be verified quickly can also be solved quickly. The complexity class P contains problems solvable in polynomial time; NP contains problems verifiable in polynomial time. The question is whether these two classes are identical.

The overwhelming consensus among researchers is that P ≠ NP — that there exist verification problems for which no fast algorithm exists. No one has proven it. What makes the Landau-Siegel simplification interesting in this context is that it instantiates exactly the structure that motivates the P ≠ NP belief: a proof that took fifty years to find but five pages to verify once found. The search was computationally costly for the AI and even more so historically for human mathematics. The verification is cheap. That asymmetry is P ≠ NP in its most recognizable applied form.

This is not a proof of P ≠ NP. But it is a live demonstration that the gap between finding and verifying mathematical results exists and is large — which is what P ≠ NP asserts, in generalized form, about all computational problems. The implication circulating in the mathematical community is sharper: what researchers have long treated as genuinely difficult may look quite different from the vantage point of a system that evaluates mathematical structure rather than following established research heuristics.

What makes P vs NP genuinely hard to prove is not the belief in P ≠ NP but the proven impossibility of using most available techniques to establish it. Three barriers now constrain the field. The relativization barrier, established by Baker, Gill, and Solovay in 1975, showed that standard diagonalization methods cannot settle P vs NP. The natural proofs barrier, identified by Razborov and Rudich in a 1994 paper published in 1997, demonstrated that any proof strategy strong enough to prove circuit lower bounds would likely yield a construction breaking pseudorandom number generators — an apparent contradiction that rules out entire families of techniques. The algebrization barrier, formalized by Aaronson and Wigderson in 2008, extended the same impossibility to methods that use algebraic techniques over finite fields. Together, these barriers do not prevent a P vs NP proof, but they eliminate almost every method that researchers know how to use.

What OpenAI Solved in Theoretical Computer Science—and What It Did Not

Within the 722 manuscripts, 40 families fell under theoretical computer science. Several landed on principal routes toward P vs NP.

The most consequential is likely the claimed proof of the Unique Games Conjecture (UGC), released with Lean formalization. The OpenAI UGC manuscript addresses a conjecture proposed by Subhash Khot in 2002, asserting that approximating certain constraint-satisfaction problems beyond a specific threshold is NP-hard. The conjecture has become a central organizing principle in approximation complexity; nearly twenty years of hardness results for approximation algorithms rest on it. The OpenAI proof claims to establish UGC unconditionally under standard P ≠ NP — without assuming the conjecture itself as an axiom — by first constructing a Unique Games instance with near-perfect completeness and arbitrarily small soundness, then propagating that gap to Basic-SDP threshold problems via the dictator test framework.

The correctness of that proof is still being reviewed. Dana Moshkovitz — a leading UGC expert who has spent much of her career working on the conjecture — described the AI manuscript in texts published by Scott Aaronson as feeling "like something written by someone who's on psychedelics," a manuscript she said was nearly impossible to read without AI assistance. On the night of the release, her nine-year-old son reportedly taunted her about a machine having solved the problem she had spent her entire academic career pursuing. The anecdote spread widely through the mathematical community.

Other theoretical computer science results include claimed circuit lower bounds at depth-3, a proof that the complexity classes L, RL, and BPL are equal (meaning randomness does not help for space-bounded computation), and a bound on the matrix multiplication exponent of approximately 9/4 — described in a preprint in the openai/math repository — a meaningful step toward the conjectured optimal value of 2, though still at a substantial distance from it.

Despite the density of these results, Lance Fortnow's Computational Complexity blog, which has tracked the field for decades, concluded the day after the release that none of the results brought researchers closer to settling P vs NP, and that the model had resolved fewer than 10 percent of the problems it had been given. Scott Aaronson, who characterized October 6 as one of the most significant days in mathematical history, also noted that the problem list included explicit attempts at P ≠ NP and at settling P = BPP — whether randomized polynomial-time algorithms can be fully derandomized — and that OpenAI's model failed on both.

Four Gaps Between the GitHub Release and a P vs NP Proof

The distance between OpenAI's October 6 release and a P vs NP proof is structural rather than merely quantitative.

First, only roughly 42 percent of the manuscripts included Lean formalizations. The remaining manuscripts rely on human review, which has barely begun. Aaronson noted that almost no humans have read the full text of any of the longer proofs. The UGC manuscript — the most consequential theoretical computer science result — is acknowledged by its principal community expert to be nearly unreadable without AI assistance.

Second, the race to publish had already begun before OpenAI's release. According to Quanta Magazine, the MIT research group of Dor Minzer, having heard rumors of OpenAI's work around September 11, rushed to publish an independent variant proof before OpenAI could release. That human mathematicians raced to establish independent priority suggests the community is recalibrating its sense of which open problems are reachable, even as the specific proofs remain under review.

Third, Lean formal verification guarantees only logical validity given the formal axioms — it does not guarantee that the formal statement precisely captures the intended mathematical object. A Lean proof that a formally defined structure has certain properties is correct within its formal system, but the correspondence between that formal definition and the intuitive problem is itself a human judgment that the community must still make.

Fourth, and most fundamentally, the missing piece for a P vs NP proof is not any of the results OpenAI released. It is a general Boolean circuit lower bound — a proof that no polynomial-size, unbounded-depth Boolean circuit can solve a problem in NP. The theoretical computer science breakthroughs in depth-3 circuits and algebraic circuits do not reach this general regime. Number-theoretic results, however striking, cannot substitute for it.

The Deepest Irony in Theoretical Computer Science

On October 26, OpenAI announced a separate result: a claimed refutation of the Exponential Time Hypothesis (ETH), which asserts that Boolean satisfiability cannot be solved in subexponential time. On the same day, Anthropic independently published its own ETH refutation, titled "A 2^o(n) Time Algorithm for 3-SAT" and attributed to its Claude model — marking the first occasion on which two AI companies independently announced resolutions to the same major theoretical conjecture on the same day. Both organizations were explicit that these results do not imply P = NP. If the refutations are confirmed, they would eliminate a conjecture that underpins much of fine-grained complexity theory.

The cumulative effect of these developments has changed how researchers think about the landscape. Some questions considered genuinely open and expected to remain so for another generation have been closed, or credibly approached, in a matter of weeks. The peripheral territory around P vs NP — approximation hardness, circuit lower bounds at restricted depth, derandomization, arithmetic complexity — has been substantially reorganized. What has not changed is the core: a proof that no polynomial-time algorithm can solve NP-hard problems in the general Boolean circuit model. That gap remains. The natural proofs barrier specifically suggests that any technique strong enough to yield such a proof would have consequences that seem inconsistent with what researchers believe to be true about pseudorandomness — a constraint that applies to AI-generated proofs exactly as it applies to human-generated ones.

The philosophical footnote to all of this is genuine. AI systems consume enormous compute to generate mathematical proofs. Humans, once given those proofs, verify them quickly. The asymmetry between generation and verification is not incidental to AI as a technology; it is structurally embedded in how current large-scale models work. If a future AI model does eventually prove P ≠ NP, it will do so by expending vast computational resources to find a proof that any graduate student can then read and verify in an afternoon. That would be, among other things, the most self-referential theorem proof in the history of mathematics. The next milestone to watch is not another number theory result. It is whether any AI system — or any human working with AI tools — can produce a general Boolean circuit lower bound that navigates around all three known barriers. The October 6 release changed what researchers expect to be possible in the near term. It did not answer the question.