Claude Just Formalized Fermat in 11 Days on 6 Billion Tokens. The Harness Thesis Got Its Receipt.
Anthropic published the writeup this morning, Friday, September 5, 2026: dozens of Claude agents produced the first complete, machine-checked Lean 4 formalization of Fermat's Last Theorem. Eleven days of wall clock. Roughly 6 billion output tokens. About 13 million lines of Lean, roughly five times the size of Lean's main mathematics library. Around 29,500 intermediate theorems in the final graph. Independent verification by Lean itself, which either accepts the proof term or it does not.
Headline: the theorem is not the news. Andrew Wiles cleared Fermat in 1994 across 129 journal pages and seven years of solitary work. What is new is the shape of the compute that produced a mechanically checkable version of the same result. A general-purpose model every enterprise customer already has, wrapped in a graph of proof obligations, running in parallel across dozens of agents for eleven days, at a compute bill on the order of a seed check. The harness is the product, and this run is the first public receipt with a number attached to every column.
The Numbers
| Line item | Value | Notes |
|---|---|---|
| Wall clock | 11 days | Early to mid August 2026, launched by Anthropic + Columbia |
| Output tokens | ~6B | Across all agents, all rewrites, all failed branches |
| Lean lines produced | ~13M | Roughly 5x the size of Lean's standard mathlib |
| Intermediate theorems | ~29,500 | Nodes in the shared proof graph |
| Concurrency | Dozens | Agents working the graph in parallel via Prove2Me |
| Verifier | Lean 4 | Deterministic; the trust root sits outside the model |
Two things read straight off the table. First, this run consumed more output tokens than most public model evaluations combined. Six billion output tokens on a single research project is a compute footprint that sits in the neighborhood of a small pretraining fine-tune, not a benchmark. Second, the concurrency figure is the interesting one. Dozens of agents running for eleven days is not a serial reasoning trace, it is a parallel construction, which is exactly the shape a proof of this scale requires and exactly the shape a single-agent chatbot cannot express.
Why the Default Harness Failed
The important sentence in the writeup is the one about the first attempt. Anthropic tried the standard Claude Code multi-agent workflow, the same harness shape thousands of teams run in production every day, and it broke. Individual agents made local progress. Then they lost track of the overall project state and stopped coordinating. Long-horizon memory degradation is the canonical failure mode of the entire agent stack, and it fired on the hardest available test.
The fix was not a bigger model, a longer context window, or a smarter prompt. It was a different orchestrator. Tianyi Peng, an Anthropic researcher who initiated the project, built Prove2Me with collaborators at Columbia. Prove2Me maintains a directed acyclic graph of theorem statements. An agent picks an unfinished node, drafts a proof against the statement, runs Lean, and either commits a verified subtree or fails and hands the node back to the queue. The graph is the memory the individual context windows cannot hold. The graph is also what lets a dozen agents work at once without stepping on each other's proof state.
That is the whole insight, and it is a small one on paper. In practice it is the difference between an agent fleet that quietly stalls after three days and one that converges on a formal proof of the Modularity Theorem in eleven. The model contributed general reasoning. The harness contributed the ability to remember and to parallelize. Every commercial agent product in the market is going to have to answer the same design question inside the next year, because the shape that worked for Fermat is the shape a large software refactor needs, or a multi-week security audit, or a data-migration project.
What the Receipt Costs
Anthropic did not publish the compute bill, so build it from public pricing. Six billion output tokens at the Fable 5 rate of $50 per million is $300,000 in output alone. Agentic workloads at this shape run inputs roughly 20 to 50 times higher than outputs once cached context, tool traces, and Lean feedback are counted, and cached reads dominate the input column. Anchor an estimate at 200 billion input-equivalent tokens with 90 percent cached, price the cached portion at the pre-cut $1.00 per million that was in effect during the August run, and the bill lands somewhere in the low seven figures for the whole eleven days. Call it $500,000 to $1.5 million, all in, plus the researcher salaries that do not show up on the API invoice.
That is not zero. It is also not a scale that requires a new hyperscaler contract. It is the size of one team's quarterly project budget at a well-funded lab, or a single grant at a serious research university, or a rounding error against the $200 billion Google TPU commitment Anthropic signed in May. Formalizing a Millennium-adjacent result used to be a career. It is now a line item on a research budget with a bounded delivery window, which is a different thing entirely.
It gets cheaper next quarter. Fable 5.1 shipped on September 1 with cache reads at $0.25 per million, a 75 percent cut on the line that dominates this workload, per the cache-read repricing we covered yesterday. The next attempt at a project of this shape on the same tier of model runs at roughly half the price, without a single change to the harness or the prompt.
The Harness Thesis, With a Receipt
We wrote up the shape of this bet twice already. In July, when Anthropic launched Claude Science as a workbench of coordinating agents and shipped no new model, the harness-is-the-product piece argued that the lab was selling the workflow and letting the frontier model ride underneath it. In April, when a wave of coding harnesses started opening the gap between model capability and delivered capability, the harness-gap essay argued that the axis of competition had already moved. Both were forward reads on a curve that had no killer public receipt attached.
Prove2Me plus Fermat is the receipt. It is the first public run at frontier scale where the same model produced a qualitatively new capability strictly because someone wrapped a better graph around it. The DAG is 200 lines of Python and a scheduler. The agents are the Claude every paying customer already has. The Lean verifier is open source. Put those three pieces together with a research question that decomposes into 29,500 provable statements and the model clears a formal proof no single agent could plan. Take any of the three away and it stalls. That is a stack claim, not a model claim.
What the Receipt Does Not Say
Three things worth naming, because the temperature on this announcement is going to run hot for a week. First, the project formalizes an existing proof; it does not discover a new one. Wiles and Taylor did the mathematics. Kevin Buzzard's team at Imperial spent years planning the Lean blueprint that Claude filled in. flt-regular gave the agents a working Kummer proof to lean on. The result is scale, not insight, and every honest headline says so. Second, the token efficiency is dreadful by human standards. A working mathematician does not need six billion output tokens to reproduce the argument; the whole textbook of algebraic number theory fits in maybe fifty million tokens. The receipt is about parallel construction, not about efficient reasoning. Third, none of this generalizes automatically. A DAG of theorem statements exists for Fermat because Buzzard spent years writing it. There is no equivalent graph for the Riemann hypothesis, or the Hodge conjecture, or the Navier-Stokes existence problem. The bottleneck has moved from the proof to the plan, and the plan is still a human artifact.
Those caveats are real. They are also exactly the shape of caveats that get quietly smaller quarter by quarter, because a general model plus a general graph plus a deterministic verifier is a research pattern that can grind through any problem someone bothers to plan. The next public run does not need to be Riemann. It only needs to be a formalization someone has been waiting a decade to finish.
Our Take
The interesting sentence in the writeup is the one about the failed first attempt. Anthropic buried it, and the industry press ran the 13-million-line number instead, but the failure of the default Claude Code harness on the hardest available task is the disclosure that reprices the agent stack. If the flagship in-house harness cannot hold project state across an eleven-day run without a purpose-built graph on top of it, then every commercial agent product on the market is running on the same wrong end of the same scaling curve, and the fix is orchestration, not tokens. That is a positioning problem for the model layer and an opportunity for whoever ships the second Prove2Me.
Practical read for builders on the API. If your agent product hits a ceiling on long-horizon tasks, the ceiling is almost certainly not the model. It is the shape of the memory your agents share. Prove2Me is 200 lines and a DAG, and it beat the default multi-agent harness on a task nobody thought a general model could clear. Whatever shared state your agents rely on today, that state is the axis to iterate on, not the prompt and not the tier. The receipt Anthropic just published is a very expensive proof that the orchestrator wins.
Three signposts for the next 90 days. Whether a second lab publishes a comparable formal proof of a hard formalization target (the Poincare conjecture in Lean, the Odd Order theorem in a modern prover, or one of the still-open Millennium problems reduced to a formalizable subresult), which is the direct test of whether Prove2Me is a pattern or a one-off. Whether the DAG-orchestrator shape shows up inside a shipping commercial agent product (Cursor, Cognition's Devin, Claude Code itself, a new entrant), which is the direct test of whether Anthropic ports its own research finding into the customer stack before someone else does. And whether Anthropic's next revenue disclosure breaks out research or science workloads as a separate line, because at $500K to $1.5M per formal proof and dropping fast on the cache-read curve, this is a segment that can be sold to hundreds of research groups on the same infrastructure that runs the coding harnesses. Two of the three fire and the harness thesis stops being a TF read and becomes a category on the pricing page.