The short version: Anthropic left Claude running for eleven days to build a computer-checked proof of Fermat’s Last Theorem, with almost no human supervision. The reason it could be left alone that long was not that the model got smarter. It was that Lean, the proof checker, rejected every wrong step instantly and at no cost. That is the practical test for using AI in small business: the work you can safely leave running is the work where something other than you can tell, cheaply and immediately, that the output is wrong.
What Anthropic actually did
On 4 September, Anthropic published the results of a project led by researcher Tianyi Peng: a team of Claude agents produced the first complete, machine-checked formalization of Fermat’s Last Theorem, the problem Andrew Wiles proved in 1995 after three and a half centuries of failed attempts.
The numbers are unusual. The run generated 13 million lines of Lean code, proved 30,300 theorems, of which 29,500 were used in the final proof, and consumed roughly six billion output tokens. That last figure is the one with a price tag attached, and it is why the cache pricing Anthropic quietly cut last week matters more than the headline rate. The finished proof is more than five times the size of Mathlib, the community library of formalized mathematics it builds on. Human input, Anthropic says, was limited to occasional high-level instructions, one of which was simply “Jacobian as a scheme sounds high priority.”
Kevin Buzzard, the Imperial College London mathematician who has led the community effort to formalize this same theorem, reviewed the result. His assessment: “This extraordinary autoformalization achievement, which Anthropic researchers say only took 11 days, proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics.”
Two caveats belong up front, because most of the coverage has dropped them. This is verification, not discovery: Claude did not find a new proof, it translated an existing one into a form a computer can check line by line. And it did not start from nothing. Anthropic’s own repository credits 106 upstream files to Buzzard’s project and to Mathlib contributors. The eleven days sat on years of human groundwork.
Why could it run for eleven days with nobody watching?
Because Lean does not care how confident the model sounds.
Lean is a proof assistant, a language in which every claim either follows from the axioms and from previously verified results or fails to compile. There is no partial credit. A wrong turn was caught in seconds, by a machine, at a cost close to zero.
Think about what that made possible. Nobody read 13 million lines of code. Nobody could. Nobody had to, because the checking was never a human job in the first place.
Now hold that next to the way AI usually shows up in a small business. It drafts a customer email, and nothing rejects the draft for being subtly off. It suggests a price, and the error surfaces weeks later in your margin. The model is not obviously worse at those jobs than it was at Lean. The difference is that in Lean, being wrong had immediate consequences, and in your inbox it does not.
The first attempt failed, and not for the reason you would guess
This is the part worth sitting with. Anthropic’s early multi-agent runs did not fail because the model was not clever enough. They showed some early success, then, in Anthropic’s words, “quickly lost track of the project’s state and stopped collaborating effectively.”
What fixed it was Prove2Me, an open-source platform Anthropic did not build. It maintains a directed acyclic graph of theorem statements, so every agent can see what has been proved, what is still outstanding, and what earlier work it can reuse instead of redoing. It also speeds up compilation and lets agents search existing results in plain language.
So the breakthrough was bookkeeping. A shared, accurate, always-current record of the state of the work.
When an AI tool fails at a long task in your business, the instinct is to reach for a better model or a more expensive plan. The evidence from the most ambitious agent run yet published says reach for structure first: a clear record of what is done, what is pending, and where the last attempt stopped. The same pattern showed up in tools that automate work in applications with no API, where the failures cluster around losing track of state rather than around raw capability.
Which jobs in your business already have a checker?
This is the question that turns a mathematics story into a Monday morning decision. Go through your recurring work and sort it into three piles.
Already has a cheap, automatic checker. Bank reconciliation, where the balance matches or it does not. Data entry against a source document. Address and card validation. Inventory counts against your system of record. Appointment booking, where a double booking is caught by the calendar itself. These are the jobs where letting AI run with a light touch is defensible today.
Has no cheap checker. Whether a quote is priced right for this particular customer. Whether an email sounds like you. Whether advice fits a client’s situation. Hiring judgment. Anything whose error is found later by a person, often by the customer. Here checking costs about as much as doing, so you stay in the loop and the honest gain is a faster first draft, not an unattended process.
Could have a checker if you built one. This is the underrated middle pile. A quote generator with a rule that anything outside a set margin band comes to you for review is a checked process. A reply drafter that refuses to send anything containing a price, a date, or a promise is a checked process. The rule does not have to be clever. It has to be automatic, because a rule you apply from memory stops working on exactly the days you are busiest.
What does using AI in small business leave for the owner?
Not out of the picture. Buzzard’s project supplied the foundations, a human decided what mattered, and a human reviewed the result at the end. What moved was where the attention went: away from grinding through steps a machine can verify, toward choosing what was worth proving at all.
That trade is available at a much smaller scale, with one honest caution. When we looked at Xero handing bank reconciliation to an AI agent, the hours saved were real, but reconciliation had been the one recurring appointment that forced somebody to look at every dollar leaving the business. Automate the task and the looking goes with it unless you deliberately keep a moment for it. A checker proves the arithmetic is right. It cannot tell you the subscription you forgot about is still billing.
The eleven days are genuinely impressive. The transferable lesson is smaller and more useful: build the checker first, then decide how far to step back. Anthropic learned that in the order of failure, then fix, and they had a proof assistant waiting for them. Most businesses have to write the rule themselves, and it is usually one sentence long.
Frequently Asked Questions
Does this mean AI can now run my business unsupervised?
No. It means AI can run unsupervised on work that has an automatic checker, which is a far narrower category than “work” in general. Claude managed eleven days because Lean caught every error in seconds at no cost. Almost nothing in a normal business has a checker that strict, so the realistic move is to identify the few tasks that do, such as reconciliation or data validation, and treat the rest as work that still needs your eyes before it leaves the building.
What is Lean, and why should a non-technical owner care?
Lean is a programming language for writing mathematical proofs that a computer can verify. Every step either follows from the axioms and from earlier verified results, or it refuses to compile. It matters outside mathematics because it is the clearest demonstration yet of a principle you can apply anywhere: an AI system becomes trustworthy at scale when something automatic, rather than somebody tired, decides whether its output is correct.
Did Claude discover a new proof of Fermat’s Last Theorem?
No, and this distinction has been blurred in much of the coverage. Andrew Wiles proved the theorem in 1995. What Claude did was translate a simplified version of that proof into machine-checkable form, building on 106 upstream files credited to Kevin Buzzard’s formalization project and to Mathlib contributors. The achievement lies in the verification and the speed of it, not in the mathematics itself.
What is the cheapest way to add a checker to work I hand to AI?
Pick one recurring task and write down the single rule that would tell you the output is wrong, then enforce it automatically rather than by memory. For invoices it might be that the total must match the line items. For customer replies it might be that anything containing a price or a delivery date routes to you before sending. Most owners find they can write the rule in one sentence, and the value is that the rule then runs every time, including on the days nobody has time to look.
Which task in your business already has a checker you never think about, and would you actually be comfortable letting AI run it this week? We would like to hear where the line sits for you.
