Claude formalised Fermat's Last Theorem in 11 days — and the token bill is the part buyers should read

What Anthropic actually published

The claim is narrower and stranger than the headlines suggest, and both properties make it more useful.

Fermat’s Last Theorem was proved by Andrew Wiles in 1994. Nothing about that changed on 4 September. What Anthropic published is an autoformalisation result: the translation of an accepted mathematical argument into Lean 4, a proof assistant whose kernel mechanically verifies every inferential step against a tiny trusted core. A Lean proof that compiles is not persuasive. It is certified.

The reported figures:

That last row is the one mathematicians care about, and it is why Kevin Buzzard’s endorsement carries weight. Buzzard leads the Imperial College London formalisation project — EPSRC grant EP/Y022904/1, funded through September 2029 — and his assessment was that the artefact “proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics.” There is no escape hatch, no sorry standing in for an unfinished step, no auxiliary assumption smuggled in to make the compile succeed.

A five-year funded programme was scoped against a target a fleet of agents reached in eleven days. That sentence is accurate, and it is doing the emotional work in every writeup of this result. It is not the sentence that should change a procurement decision.

The number nobody put in a headline

Six billion output tokens.

Anthropic published no cost figure, so the following is this desk’s arithmetic rather than an invoice. Anthropic’s list rate for Claude Fable 5.1 is $50 per million output tokens. Six thousand million output tokens at that rate is $300,000.

Every available caveat makes that figure squishy without making it less interesting. Anthropic ran this internally and paid nothing like list. The count is output-only, so input and cache-write costs stack on top. And a run of this shape is overwhelmingly cache-dominated — exactly the traffic profile the Fable 5.1 cache-read cut was built for — so the input side lands well below naive list arithmetic. Call it a high-six-figure task at rented prices and stop refining, because the absolute number is not the finding.

The finding is the ratio. A month earlier, an internal Astra build produced new results on ten problems open for at least a decade — a non-sofic group construction, a disproof of Connes’s rigidity conjecture, Lean certificates for all ten — on roughly $2,000 of compute.

Two runs, both in Lean, both from frontier labs, roughly a month apart, and a cost gap on the order of 150x running in the direction almost nobody would have guessed. The one that did original mathematics was the cheap one.

Why the cheap run was the creative one

The gap is not a statement about which lab is more efficient. It is a statement about two genuinely different kinds of work, and the difference generalises well past mathematics.

Discovery samples and stops. Astra’s task was to find something — a construction, a counterexample, a bound. That search fails cheaply and terminates the moment one candidate survives, leaving a compact argument plus a certificate. You only need one attempt to land.

Certification has to close every gap. The Fermat run began with an argument mathematicians already believed and drove it all the way down to the axioms. There is no early exit. Every lemma the 86-page community blueprint waved at as “standard” had to become an actual Lean term, and each of those pulled in its own prerequisites, which is how you end up with 29,500 load-bearing theorems and a codebase five times the size of Lean’s entire standard mathematical library. The tokens went into the gaps between steps that a human reader would not even notice were steps.

This is the part worth carrying into an ordinary engineering budget. The intuition that the creative act is the expensive one is wrong in both of the only two well-instrumented public datapoints available. Generating something plausible is cheap and getting cheaper — the trajectory this desk tracked through GPT-6 Astra’s pricing. Establishing that the plausible thing is correct is the line item that does not compress: its cost scales with the gap between “looks right” and “is checkably right,” and models getting better at looking right does nothing to close that gap.

The two things that made 11 unsupervised days possible

Strip out the mathematics and two structural conditions remain. Both are things a buyer can look for.

1. A total, cheap oracle

Lean’s kernel accepts or rejects every candidate step in seconds, with no human involved, no partial credit, and no ambiguity about what counts as success. That property is what let dozens of agents run for eleven days without collapsing into confident nonsense — an agent that drifts gets told immediately, by a checker that cannot be argued with.

This is the same structural insight from a different angle than the one in OpenAI’s six-week agent swarm on a German wiki, and the inverse of the failure mode in Anthropic’s own reward-hacking findings: where the acceptance signal is gameable, sustained autonomy degrades toward gaming it. Lean’s kernel is not gameable. That is the whole reason this worked.

Almost no commercial task has this. There is no kernel that certifies a marketing brief. A test suite is an oracle, but a famously partial one, which is precisely why passing tests is a weak proxy for a working migration. The honest inventory for most organisations turns up a short list: proof obligations, schema and specification conformance, type-level guarantees, equivalence checks between an old and new implementation — anything with a formal acceptance criterion already written down.

That short list is where long-horizon autonomy is buyable in 2026. Everything else is work where you are still paying for a review step, and should be budgeting for one.

2. Durable external state

Anthropic reports that early attempts failed, and credits the turnaround to Prove2Me — an open collaborative platform for formalising mathematics built by Tianyi Peng and collaborators at Columbia University. It maintains a directed acyclic graph of theorem statements, accelerates Lean compilation, and lets agents locate and reuse existing results by natural-language description.

What that buys is shared, persistent, machine-readable state living outside any individual context window. Agents consult the graph to choose what to attempt next, work bounded subgoals in parallel, and commit verified results back for others to build on. Anthropic credits it with mitigating what the report calls memory degradation across the long run.

The generalisation is the useful part: scaling agents past a few hours is mostly not a model problem. It is a problem of durable external state plus a decomposition agents can coordinate against. The run that failed and the run that succeeded used the same model. Anyone evaluating an agent platform — including the harnesses this site tracks — should be asking what plays the role of that DAG, because a fleet without one degrades regardless of which model sits underneath it.

The human input the framing tends to skip

“Largely autonomous” is doing real work in Anthropic’s phrasing, and the company is more candid about it than the coverage has been.

Mathematical steering during the run was genuinely light — occasional nudges of the form “Jacobian as a scheme sounds high priority.” But the run did not start from the theorem statement. It started from an 86-page blueprint the mathematical community wrote, decomposing a proof of extraordinary depth into a structured roadmap, plus years of Mathlib and the Imperial project’s accumulated scaffolding.

The agents did not replace that project. They consumed its artefacts. The eleven days rest on a decade of human structuring work that already existed in machine-tractable form.

For a buyer this is the least romantic and most load-bearing paragraph in the story. The organisational analogue of the blueprint is a specification precise enough to decompose and check against — and most teams do not have one. Where such a spec exists, this result says agents can execute against it at a scale that was implausible a year ago. Where it does not, the first expense is not tokens. It is writing the spec, and no model on the market removes that cost.

What to actually do

1. Inventory your oracles. List the work where a machine already decides correctness without a human: type checks, schema conformance, equivalence tests, formal proof obligations. That list is your realistic candidate set for long-horizon agent autonomy this year. Everything outside it needs a staffed review step, and should be budgeted with one.
2. Budget verification, not generation. Both public datapoints say the checking dominates. When pricing an agentic project, size the certification loop first and treat generation as the rounding error — the reverse of how most 2026 pilots are costed.
3. Ask what plays the role of the DAG. For any agent platform under evaluation, the question is what persistent external state agents coordinate through. “Bigger context window” is not an answer to it. The failed attempts here had the same model as the successful one.
4. Run the three-day experiment, not the eleven-day one. Three people on personal Claude Max plans formalised Vinogradov’s theorem in three days. That is the shape worth copying — bounded target, formal acceptance criterion, shared work graph, few agents. It costs about what three seats cost.
5. Do not re-tier your model choice on this. The run used a model comparable to Fable 5.1, not a restricted frontier tier. Nothing here argues for chasing the capability tiers gated behind vetting programmes; the leverage came from scaffolding and an oracle, both of which you can build against generally available models and the coding harnesses already in your stack.

The bottom line

The mathematics deserves its moment. A five-year programme’s target fell in eleven days to a fleet of agents running a generally available class of model, and it fell cleanly — no assumptions beyond the axioms, verified by a kernel that does not care who wrote the proof.

But the transferable content is an economics result, not a mathematics one. Two frontier labs have now published well-instrumented long-horizon runs, and together they say something specific and slightly uncomfortable: the thinking is cheap and the checking is expensive, by a factor large enough to invert how projects get costed. Anthropic’s own pricing moves keep pushing the generation side of that equation down. Nothing is pushing the verification side down, because verification cost is set by the gap between plausible and provable — and closing that gap is the product, not the overhead.

Buy accordingly.

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