AI systems say they broke a 25-year-old math foundation. Here's the honest version.

If you've ever wondered why GPS apps can find you a fast route but can't guarantee it's the absolute best route — that question touches the same math that made headlines this week.

On October 26, OpenAI posted a 67-page paper on its website. Attributed to an internal AI model. The claim: the Exponential Time Hypothesis is false. Hours later, Anthropic published its own paper, attributed to Claude. Same conclusion. Different approach. Same week.

The math community's reaction, as best I can describe it: "This is extraordinary — now let's spend the next few weeks figuring out if it's actually true." That's not dismissal. That's how math works.

What the Exponential Time Hypothesis actually is

Here's the plain-language version, because you need it to understand why this matters.

Imagine a puzzle with thousands of interlocking rules — each one saying "if piece A is flipped, piece B must be too." You want to know if there's any arrangement that satisfies all the rules at once. This kind of puzzle, called 3-SAT (satisfiability), is the canonical "hard problem" in computer science.

The Exponential Time Hypothesis (ETH) is a 1999 conjecture by Russell Impagliazzo and Ramamohan Paturi, two mathematicians at UC San Diego. It says: there's no shortcut for these puzzles. Solving a 1,000-variable version takes at minimum exponentially many steps — roughly 2^n, where n is the number of variables. A 1,000-variable puzzle: 2^1000 steps. That's more steps than there are atoms in the observable universe. No algorithm, no matter how clever, can do it faster.

Why does anyone outside of academia care? Because ETH has been used as the foundation for hundreds of downstream proofs. Researchers working on logistics scheduling, computational biology, and algorithm design have used it to prove their problems can't be cracked faster than expected. It's less like a theorem and more like a load-bearing wall in a building. Disprove it, and a lot of adjacent structure needs to be inspected.

What happened this week

OpenAI's paper was formalized in Lean 4 — a proof assistant, meaning a computer program that can mechanically verify the logical steps of a mathematical argument. The 67-page paper was apparently demoed to US senators before being made public, which is a detail worth holding in mind.

Anthropic published a competing paper the same day. Different technical approach, same conclusion: a 2^o(n) algorithm for 3-SAT exists (that notation means "faster than exponential"). Two independent AI systems converging on the same refutation of the same conjecture in the same week.

The Lean 4 formalization is genuinely significant. When a proof is formalized in Lean 4, any mathematician with the right software can run the proof and verify that its logical steps are internally consistent. That's more transparent than most published math proofs. What it doesn't verify automatically is whether the theorem being proved is the theorem you meant to prove.

That scope question is exactly what the reviewers are working on right now.

What it doesn't mean

I want to be direct about this because the headlines have already been misleading.

This does not prove P equals NP. The famous "P vs NP" problem — which would mean every problem whose solution can be quickly verified can also be quickly solved — is a different, stronger question. Breaking most encryption would require P=NP. ETH is a narrower conjecture about specific problem classes. Disproving ETH gets you nowhere near collapsing the wall between easy and hard problems the way P=NP would.

Your passwords and bank account are not at risk. RSA encryption and elliptic curve cryptography — the math protecting your online banking, messaging apps, and email — rest on different mathematical foundations entirely (the difficulty of factoring large numbers and computing discrete logarithms). These papers don't touch that math.

The scope may be limited. Both claims appear to apply to specific encoded versions of 3-SAT, not necessarily every formulation of the problem. The original ETH authors are specifically checking this right now. A proof that works for fixed-width, canonically encoded instances isn't automatically the same as a proof that works for all 3-SAT instances. That distinction matters enormously for whether the 25 years of downstream proofs actually need revisiting.

Why everyone is waiting

The math community's caution here isn't AI-specific skepticism. It's the same standard applied to any extraordinary claim. Fermat's Last Theorem had a famous attempted proof in 1993 that took a year to fix a gap. A proof that would collapse 25 years of hardness results requires proportional scrutiny.

The reviewers are not random volunteers. Russell Impagliazzo and Ramamohan Paturi are the authors of the original ETH conjecture. MIT's Ryan Williams is one of the world's leading complexity theorists. These are the exact right people to evaluate these claims. And they're engaged.

The Lean 4 formalization helps. A machine-verified proof can't hide a logical error in its stated steps — it either checks or it doesn't. What it can hide is a gap between "what the theorem says" and "what we needed the theorem to say." That's the live question.

Source spread

• OpenAI — Building scientific understanding with academic communities — hype. The official announcement. Factually careful, but framed as a research milestone before peer review completes.
• Anthropic — ETH refutation paper — hype. Claude-attributed competing paper. Less press-release framing, more direct.
• Impagliazzo & Paturi — original ETH conjecture, 1999 — academic. The source document everything else this week is referencing.

What's real / What deserves a side-eye

What's real:

• Lean 4 formalization is publicly available and machine-checkable. That's a higher transparency bar than most math announcements.
• Two independent AI systems converging on the same refutation via different approaches is a meaningful signal — convergence is how math results get validated.
• The original ETH authors engaging in open review is the right process happening at the right speed.

What deserves a side-eye:

• Two competing AI labs publishing refutations of the same landmark conjecture on the same day is a remarkable coincidence. Science doesn't usually move like a product launch.
• "Formalized in Lean 4" and "proved the theorem you thought you proved" are different things. The Lean checker verifies the internal logic — it doesn't check whether the axioms match the real-world problem.
• Three separate millennium-level AI math claims in one week (ETH, Yang-Mills existence, Kakeya set conjecture) is not how math breakthroughs historically distribute. Something unusual is happening — either real progress or lab-driven coordination effects. Probably worth watching which of those it turns out to be.

What to do about it

This is one of those stories where the right practical answer is mostly "wait, and be skeptical of the people most excited to tell you what it means."

• Your encryption is fine. Nothing in these papers affects RSA, ECC, or the encryption protecting your passwords and banking. If you see a headline saying "AI breaks encryption," that headline is wrong.
• Don't update on the announcement. Update on the verification. The reviewers — the actual authors of the original conjecture — will have a public verdict within weeks. That's the signal. Press releases from competing labs on the same day are not the signal.
• Notice what "formal verification" actually does and doesn't do. Lean 4 checks the logic of the proof steps. It doesn't check whether the theorem being proved is the right theorem. That distinction is subtle and important, and it's the exact question the math community is currently working through.
• If you're using AI for technical research work, the concrete takeaway is that AI can now produce Lean 4-verified formal proofs at scale. That capability is real regardless of whether these specific results hold. What to do with that: TBD — the field is moving faster than the tooling guidance.

Further reading

• OpenAI — Building scientific understanding with academic communities — the announcement and paper links
• Anthropic — ETH refutation — Claude-attributed competing paper
• Impagliazzo & Paturi — original ETH paper, 1999 — the source material
• Samwise builders analysis of the ETH claims — technical coverage for those who want the complexity theory angle