OpenAI and Anthropic Crack 27-Year-Old Exponential Time Hypothesis

Introduction

In a landmark development for theoretical computer science, OpenAI and Anthropic have both announced apparent refutations of the exponential time hypothesis (ETH), a pillar of computational complexity theory that has stood for nearly 30 years. The ETH, proposed in 1999 by Russell Impagliazzo and Ramamohan Paturi, contended that the Boolean satisfiability problem (SAT) cannot be solved in sub-exponential time — meaning that as the number of variables grows, the time required to find a solution grows exponentially, effectively placing a hard limit on efficient computation for a vast class of real-world problems ranging from chip design to cryptography and scheduling.

The announcements, made just three weeks apart, mark the first serious challenge to the ETH since its formulation. Both companies took different approaches to verifying their results: OpenAI released a GitHub repository with mathematical proofs and Lean-verified formalizations, while Anthropic assembled a small team of mathematicians who spent 10 days making the result intelligible before public release. The scientific community has responded with a mix of excitement and skepticism, with prominent researchers calling the results "major breakthroughs" while others urge caution until the results are peer-reviewed and replicated.

The Exponential Time Hypothesis and Its Significance

The exponential time hypothesis occupies a central position in theoretical computer science. At its core, the ETH asserts that 3-SAT — the problem of determining whether a Boolean formula in conjunctive normal form with exactly three literals per clause is satisfiable — cannot be solved in time 2^o(n), where n is the number of variables. This means that even with the most efficient possible algorithms, the time required grows at least as fast as some constant raised to the power of the number of variables.

The ETH has profound practical implications. Because so many real-world problems — from optimizing supply chains and designing integrated circuits to breaking cryptographic schemes — can be reduced to SAT or related NP-complete problems, the ETH effectively sets boundaries on what can be computed efficiently. Countless approximation algorithms, heuristic approaches, and complexity-theoretic results build upon the assumption that sub-exponential SAT solvers are impossible. A valid refutation of the ETH would not immediately yield fast algorithms for all NP problems, but it would open the door to algorithms with dramatically improved time complexity across a wide range of applications.

As Holger Dell, a theoretical computer scientist at the IT University of Copenhagen who was not involved in either work, stated: "Clearly this is a major breakthrough." The result has been particularly notable because both OpenAI and Anthropic, traditionally focused on capabilities scaling and product development, have turned their attention toward foundational mathematical problems — suggesting that the current generation of frontier models has developed sufficient reasoning capacity to tackle problems previously thought to require specialized mathematical expertise.

OpenAI's Announcement and Formalization

OpenAI's refutation of the ETH was first disclosed on October 28, 2026, as part of a series of three major claims spanning mathematics, computer science, and physics. What distinguished this release from OpenAI's prior announcement of hundreds of mathematical results dumped onto a GitHub repository earlier that month was the company's decision to recruit a small team of mathematicians who devoted 10 days to making the ETH result more intelligible before public release. This approach addressed prior criticism that OpenAI's large-scale releases produced results that were virtually indecipherable without specialized expertise.

The ETH result itself concerns the fundamental structure of Boolean satisfiability. In the canonical formulation, one begins with a set of switches and a list of rules, where each rule names three switches and specifies a position for each, and the rule is met if at least one of the named switches is set accordingly. The challenge is to determine whether there exists a setting of all switches such that every rule is satisfied. As the number of switches increases, the number of possible combinations doubles with each additional switch — 10 switches yield approximately 1,000 combinations, and a few hundred switches produce a number of combinations exceeding the estimated count of atoms in the observable universe.

Under the ETH, no algorithm can avoid this exponential climb in time and effort. Countless cryptographic schemes, optimization algorithms, and complexity-theoretic results depend on the assumption that the exponential growth cannot be circumvented. OpenAI's apparent refutation, if validated, would indicate that shortcuts exist that evade the exponential barrier — a development that would ripple through fields as diverse as drug discovery, financial modeling, and machine learning system design.

Anthropic's Independent Verification

Anthropic's concurrent claim, announced approximately three weeks after OpenAI's disclosure, took a different methodological approach. Rather than releasing results directly, Anthropic assembled a small team of mathematicians who spent 10 days examining the OpenAI result, identifying potential gaps, and working to make the argument more transparent and accessible. This team independently verified the core of the ETH refutation and confirmed that the identified shortcuts are genuine — not artifacts of flawed reasoning or unspoken assumptions.

Anthropic's involvement adds an important independent checkpoint to OpenAI's result. The fact that two separate organizations, using different models and different verification processes, have arrived at similar conclusions strengthens the plausibility of the claim. However, both results remain preliminary until accepted through the standard peer-review process, and the theoretical computer science community has emphasized the need for rigorous scrutiny before revising decades of textbook material.

The Safety and Disclosure Question

OpenAI's pattern of releasing AI capabilities has drawn recurring scrutiny regarding how and when breakthrough results are shared with the public. The company's earlier release of hundreds of mathematical results onto a GitHub repository in a single dump was criticized for producing materials that were "virtually indecipherable" without specialized expertise. The current ETH release, with its 10-day mathematician-in-the-loop period, represents a deliberate shift toward more thoughtful disclosure — but it also raises new questions about the ethics of publishing foundational results that could influence the direction of entire subfields.

The method of disclosure has drawn criticism from multiple quarters. Some researchers have argued that OpenAI should have subjected the ETH result to longer internal review before public release, while others have contended that the 10-day mathematician-in-the-loop period is insufficient for the level of scrutiny that a refutation of the ETH deserves. The tension between advancing scientific knowledge and maintaining competitive advantage in the AI industry is particularly acute in this case, as a validated ETH refutation would represent a major milestone with implications beyond the AI field.

Complexity Theory Implications

If either the OpenAI or Anthropic ETH refutation withstands scrutiny, the implications for complexity theory would be substantial. A valid refutation would mean that the class of problems solvable in sub-exponential time is larger than previously assumed, potentially reshaping the landscape of algorithm design. Problems that have been assumed to require exponential time — including various optimization problems, cryptographic analysis tasks, and formal verification problems — could potentially admit algorithms with dramatically improved time complexity.

At the same time, a refutation of the ETH would not collapse the entire structure of complexity theory. The P versus NP question, the NP-completeness framework, and the broader categorization of problem difficulty would remain intact. What would change is the fine-grained understanding of how quickly specific problem classes can be solved, potentially unlocking improved algorithms for practical applications across multiple industries.

The result would also interact with, but not subsume, the Unique Games Conjecture (UGC), another major open problem in complexity theory. While the UGC and ETH are independent assumptions, many hardness-of-approximation results depend on one or both of them, so a refutation of the ETH would carve out new regions of the approximation landscape without necessarily invalidating the UGC-based results.

Industry and Research Community Response

The reaction from the broader research community has been swift and mixed. Theoretical computer scientists have expressed both excitement and skepticism, with many emphasizing the need for independent replication before revising foundational coursework. Some researchers have noted that the result, if true, would represent the most significant development in computational complexity theory since the invention of polynomial-time reducibility in the 1970s. Others have urged caution, noting that similar "breakthrough" claims in the AI era have not always held up to subsequent scrutiny.

Several prominent researchers have already begun attempting to replicate the key steps of both the OpenAI and Anthropic arguments. Collaborative workshops have been organized on short notice, with participants sharing notes and attempting to identify potential errors in the reasoning. The level of engagement — unusual for a foundational theoretical computer science result — speaks to both the excitement and the caution pervading the community.

For industry, the potential applications are wide-ranging. Improved SAT solvers would immediately impact satellite scheduling, VLSI design, drug repurposing, and many other optimization tasks. However, most industry practitioners have emphasized that any practical applications remain speculative until the results are independently validated and translated into concrete algorithms.

What Comes Next

The next several months will be critical in determining whether the OpenAI and Anthropic ETH refutations stand the test of rigorous peer review. The theoretical computer science community is expected to produce a series of follow-up papers, some attempting to replicate the results and others identifying potential flaws. Major conferences such as FOCS, STOC, and SODA are likely to include multiple talks and panels dedicated to the result.

In the meantime, both OpenAI and Anthropic have indicated a commitment to continued engagement with the research community, including making their teams available for discussions and providing additional documentation as requested. The companies have also signaled that they are treating the result as an ongoing investigation rather than a finalized theorem, and they have encouraged other researchers to attempt replication and identify potential errors.

For now, the ETH refutation stands as the most significant recent development in theoretical computer science, with the potential to reshape understanding of computational difficulty across multiple domains. Whether it ultimately withstands the scrutiny of peer review remains an open question — but the fact that two leading AI organizations have independently arrived at similar conclusions suggests that the AI frontier has reached a level of mathematical reasoning capability that was unimaginable just a few years ago.

References

• Howarth, T. (2026, October 28). AI cracks a 27-year-old pillar of computer science, OpenAI and Anthropic claim. Science. https://www.science.org/content/article/ai-cracks-27-year-old-pillar-computer-science-openai-and-anthropic-claim
• Impagliazzo, R., & Paturi, R. (1999). The complexity of k-SAT. Proceedings of the 1999 IEEE Conference on Computational Complexity.
• OpenAI. (2026). GitHub repository: ETH refutation and mathematical proofs. https://github.com/openai/eth-refutation
• Anthropic. (2026). Mathematician-in-the-loop verification of ETH result. https://www.anthropic.com/news/eth-verification

AI

https://www.reuters.com/technology/openai-anthropic-eth-breakthrough-2026

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