Fermat Formalized in 11 Days: A Regime Change

An ordinary Tuesday, mathematics changes regime

An internal Anthropic model has formalized the complete proof of Fermat's Last Theorem in the Lean language, and this special Tuesday in research marks a regime change. The fact is documented, verified line by line, made public on September 4, 2026.

The consensus reads the event as an academic curiosity. The consensus has the wrong frame.

The proof develops Fontaine's theory and part of Mazur's work on the Eisenstein ideal. It follows the Darmon-Diamond-Taylor exposition from 1995, rather than the modern proof. The result remains complete and verifiable.

The numbers that matter

Here are the numbers that count. The codebase exceeds 13.4 million lines and requires nearly 20 times the compilation time of Lean's mathematical library, on a machine with 96 cores, as mathematician Kevin Buzzard reports on the Xena project^[1].

Buzzard compiled the repository and ran the comparator verifier: the result holds.

The technical account adds the most significant data: the model completed in 11 days work that took humans years, according to TechTimes^[2]. Anthropic describes the process in its own research report^[3].

The comparison withstands scrutiny against the industry's heaviest infrastructure. Twenty years of collective work by mathematicians built Lean's library. One model generated a proof twenty times more expensive to compile in less than two weeks.

The cost curve tells who is right

The consensus looks at the prestige of the theorem. The data that predicts the future is the cost curve of verified formal reasoning.

Three points trace the trajectory. Wiedijk's list of 100 formalization challenges has been open for about twenty years, and each entry required months or years of expert human work. In 2024, assisted proof systems reached silver-medal performance on Olympic problems. In 2026, one model closes the final entry on the list in 11 days.

Observe the causal mechanism, beyond the correlation. Every formalized proof feeds reusable libraries like mathlib. Every contribution lowers the cost of the next contribution. The system accelerates its own growth.

The direction is clear: the cost per formalized theorem collapses by orders of magnitude in two years. The cost curve tells who is right about the pace.

The cliff event: formal reasoning as a commodity

Adoption of formal verification follows a jump, rather than linear growth. The cliff event has a precise cause: formal reasoning becomes a computational commodity.

This desk has long held a thesis: foundation models become commoditized, and value migrates to vertical applications with proprietary data. Mathematical formalization is the first domain where AI beats the human expert on a 100% verifiable task.

When the machine produces a proof, comparator checks it deterministically. The error has zero probability. This property makes the domain explosive: total trust, plummeting marginal cost.

The distinction matters: this is a prediction about technology, with high confidence. The exact timing of market adoption carries medium confidence, and I declare this openly.

What this work represents, and what it leaves open

The work closes a benchmark, yet leaves distinct human tasks open. Buzzard remains funded by the EPSRC to formalize the modern proof, the one based on ideas from Khare and Taylor.

Anthropic's proof applies to exponents greater than or equal to 5. The case of regular prime exponents was already formalized by Best, Birkbeck, Brasca, Rodriguez, van der Velde and Yang, and the smallest irregular prime is 37. The set thus covers the entire theorem.

Anthropic produced a static formalization. The dynamic document that allows humans to explore the modern proof is missing, and this part remains open work for the mathematical community.

Three categories that change form by 2028

Three categories change form by 2028:

• Critical software verification: aerospace, medical devices, automotive.
• Cryptography and security protocols, where formal proof replaces manual audit.
• Chip design, where formal verification of silicon becomes continuous.

The companies positioned on this curve have concrete names. Harmonic builds models dedicated to mathematical reasoning. DeepMind has pushed AlphaProof toward Olympic problems. Anthropic now demonstrates the industrial scale of the method.

A concrete example makes the stakes clear. A cryptographic protocol formally verified eliminates entire categories of vulnerabilities before deployment. The cost of this guarantee today is prohibitive, and collapses along the same curve.

The formal software verification sector, today slow and expensive, becomes economical and pervasive. Every line of critical code acquires an attached proof. The procurement signing multi-year contracts today on manual audit buys technology in the process of becoming obsolete.

My position, and what would falsify it

My position is clear: mathematical formalization is the first frontier where AI surpasses the human expert on an entirely verifiable task, and this commoditizes formal reasoning within 24 months.

Value leaves the model layer and concentrates in vertical applications: integrated verifiers, proprietary libraries, specialized training data. Who buys generic capacity buys what becomes commodity.

The reasoning extends beyond pure mathematics. Every domain with explicit formal rules inherits the same dynamic: law, quantitative finance, systems engineering.

What would change my idea? A slowdown in the curve. When in the next two years only Anthropic remains capable of complete formalizations, the thesis of regime change loses force. Diffusion across multiple actors is the signal that confirms the trajectory.

The forecast, with horizon and kill signal

Here is the explicit forecast, with horizon and kill signal.

By December 31, 2027, at least one laboratory besides Anthropic will publish a complete, verified formalization in Lean of an entry on Wiedijk's list or an open research result. Confidence: 70%.

Kill signal: zero complete, verified formalizations from laboratories other than Anthropic by December 31, 2027. This data would falsify the thesis of rapid diffusion.

What this means for decision makers

For the CTO and Chief Innovation Officer: reevaluate the verification stack now, before it becomes obvious. Formal proof tools enter the standard development cycle.

For venture capital: the bet on vertical applications of formal reasoning seems premature, yet the data supports it. The application layer captures the margin of the next decade.

For the Chief Strategy Officer and procurement: a three-year plan built on manual audit assumes a world that is disappearing. 90% of analysts are right about the present, and wrong about the pace of change.