OpenAI and Anthropic Both Claim Their AI Refuted a 27-Year-Old Math "Law" — And the Math World Is Reeling

Two rival frontier labs announced within hours that their models had produced sub-exponential algorithms challenging the Exponential Time Hypothesis, a cornerstone of computational complexity theory first posed in 1999 — and the computer-science community has responded with a mixture of awe, disbelief, and outright irritation.

The Claim That Should Not Exist The Exponential Time Hypothesis, or ETH, is one of those quiet load-bearing walls of theoretical computer science. Published in 1999 by Russell Impagliazzo and Ramamohan Paturi at UC San Diego, it says that for a specific family of notoriously hard logic puzzles, even the fastest possible algorithm still needs time that grows exponentially with the number of variables. ETH is not a theorem; it is a widely believed assumption that props up a large chunk of what we think we know about hard problems, and it sits inside the broader, legendary P versus NP question.

Now OpenAI says its models found algorithms that crack those puzzles in "sub-exponential" time, which — if true — would topple ETH and could even open a road toward proving P equals NP, the holy grail of the field. Anthropic, not to be outdone, released its own refutation of ETH just hours later.

A 24-Hour Pissing Match Between the Frontier Labs This is where it gets genuinely fun. OpenAI did not simply post a paper. It staged three different releases tailored to three different disciplines: mathematicians got the work on GitHub, computer scientists were pulled into a working group and asked to fund a conference to debate the result, and physicists were handed their own working group to decide how the physics result should even be released.

That is less "we proved something" and more "we are reorganizing your entire academic social structure around our model output." Then Anthropic's mathematician Levent Alpöge publicly suggested OpenAI's solutions leaned heavily on pre-existing literature, and within hours Anthropic dropped its own independent ETH refutation, apparently to reclaim the spotlight. Two companies that are normally locked in a brutal commercial and talent war spent a single day racing to announce they had each broken one of the oldest assumptions in computer science.

Why This Is Exciting If the claims survive scrutiny, the implications are staggering. A refuted ETH means many complexity arguments built over the last two decades would need to be reopened, and a credible path toward P equals NP would reshape cryptography, optimization, and basically every field that secretly relies on problems being hard. Beyond the math itself, this is the clearest signal yet that frontier models are becoming genuine instruments of scientific discovery, not just chatbots.

MIT professor Ryan Williams, who previously called an earlier OpenAI proof dump a "DDOS attack on mathematicians," nonetheless agreed to help interpret the solution alongside the original authors of ETH. His verdict is the kind of quote that makes the whole episode feel historic: a few of the proof steps look "alien," and that alienness is exactly what convinced him the community must work on it together.

Why This Is Also Depressing Here is the uncomfortable part. The same episode that feels like a triumph also feels like a credibility stress test the field is not ready for. Williams himself does not regard the result as final and explicitly warned that OpenAI's algorithm is almost certainly not the best one possible.

The provenance questions are corrosive: when one lab implies another leaned on existing work, and when proofs arrive faster than any peer-review process can absorb them, the line between genuine breakthrough and elaborate pattern-matching artifact gets blurry. There is also a real cybersecurity scare buried in the story — Williams asked OpenAI whether the new algorithms break cryptographic systems that depend on certain key sizes, and the only reassurance on record is OpenAI's own internal check, relayed secondhand, that it is "reasonably confident" nothing breaks. For a claim this consequential, "we checked ourselves and we think it is fine" is not the level of assurance the world usually requires.

The episode lays bare how unprepared our norms of attribution, reproducibility, and verification are for a world where a frontier lab can generate a headline-grabbing technical claim overnight.

What Happens Next Williams, Impagliazzo, and Paturi have circulated an open letter asking interested scientists for feedback and proposing community norms for AI contributions. Working groups are forming. A conference is being discussed. In other words, the adult supervision is arriving — but it is arriving after the tweet, not before. Whether these contributions become durable advances or simply the most expensive screenshots in math history will be decided in the months ahead, in GitHub repos and seminar rooms rather than on company blogs.