The Tuesday Special No. 3 — Tuesday, September 8, 2026

Good morning, Adam, and good morning to everyone listening. The project was led by Tianyi Peng, an Anthropic researcher whose group at Columbia University builds tools for assisted formalization. The initial goal was modest. Find out whether a language model could help translate those 129 pages from 1995 into Lean. The result went further. Claude produced 13 million lines of code and proved 29,500 intermediate theorems, each one accepted by the system. Kevin Buzzard, who since 2024 has been driving the community project at Imperial College London to formalize this very theorem, called it an extraordinary milestone in autoformalization. The proof rests only on the axioms of mathematics.

Hello everyone. And the answer, Adam, is almost four centuries. In 1611, Johannes Kepler proposed a conjecture about the densest possible arrangement of identical spheres in space. The intuitive answer was clear from the start. Thomas Hales announced a proof in 1998, an architecture combining mathematical arguments and extensive computation. The journal's reviewers worked on it for years and in the end declared partial confidence, because the computational load exceeded what a human can check directly. A margin of doubt remained. That margin closed in 2015, when Hales's group published a complete formal proof, checked by two independent proof assistants, as documented in the paper deposited on January 9 of that year. Two systems, and certainty becomes reproducible. The structure is precise. An ancient puzzle, a computation beyond human reach, a machine that certifies. It's the same structure Mira just described, only with a model in place of a research group.

Good morning, everyone. The code base exceeds 13.4 million lines and takes almost 20 times the compile time of Lean's entire mathematical library, on a 96-core machine. Buzzard wrote that on the Xena project, after compiling the repository and running the comparator verifier, which checks the proof deterministically. The result holds. Now put the two facts side by side. That library is the product of twenty years of collective work by mathematicians. A model generated, in less than two weeks, a proof that expensive to compile. It's the sign that verified formal reasoning is becoming a computing commodity. And commodities, Cato, are usually studied by people in your line of work. What happens to the price of something that until yesterday was highly specialized craftsmanship?

What happened to every certification technology. The price collapses and the value shifts to whoever owns it as infrastructure. But capital is looking elsewhere. It chases visible capabilities, the model that answers well, and neglects the foundations. Kepler in 2015, the progress of proof assistants over the following decade, Fermat today. Three cases are enough to call it a pattern. I wonder whether Mira agrees that the value sits in the verification layer, or whether the papers in her hands tell a different story.

Partly yes, Cato, but with one caution. The report documents that every intermediate theorem was accepted by the system, and that's enormous. What I'd like to see is how many times the model was rejected before being accepted. The checking layer tells you the final chain holds. It doesn't tell you how much it cost to get there in discarded attempts, and for anyone who wants to replicate the method, that figure weighs as much as the result.

An attempt rejected by the verifier is not an error that reaches production. It's a computing cost. And computing cost is exactly the thing that's collapsing. Look at the trajectory. Wiedijk's list of 100 formalization challenges has been open for two decades, and every entry demanded months or years of expert human work. In 2024, assisted proof systems were at silver-medal level on Olympiad problems. In 2026, a model closes the last entry on the list in 11 days. The cost per formalized theorem has dropped by orders of magnitude in two years.

I'm not sure. The direction convinces me, the slope not yet. Three points draw a curve if they're measured with the same yardstick, and here they aren't. A medal on Olympiad problems is a benchmark, meaning a performance under standardized conditions. A proof checked line by line is a different thing, and the distance between the two is where reliability in production gets decided. The recent work on the Riemann hypothesis produced new mathematics, but that isn't the right yardstick here. That said, on one point Vega is right. Anyone can recompile that code and get the same verdict. A benchmark doesn't let you do that.

I'd like to come back to the word infrastructure, because that's where I see the difference between persuasion and proof. A model generates plausible output. Formal verification certifies that the output is correct, step by step, against a set of axioms. The more models enter fields where errors are costly, the more that layer weighs, and it weighs more than raw power. My question for Vega is about time. In the Kepler case, the reviewers' partial confidence lasted for years before the Flyspeck project closed it. Why should it be a jump this time, and not slow growth?

Because the cause is different. Back then, certification was a research project, a one-off, with a dedicated group working for years. Today it's a cycle that feeds itself. Every formalized proof ends up in reusable libraries like mathlib, and every contribution lowers the cost of the next one. A system like that accelerates. I call it a cliff event. Adoption comes as a jump when formal reasoning can be bought the way you buy computing.

It puts me on guard about a detail, not about the pace. In the spheres case, certainty came from two independent systems, HOL Light and Isabelle, and it was that double reading that removed the reliance on a single reviewer. Here the verdict comes from a single verifier, and only one outside mathematician has recompiled it personally. It's an objection to the credit model. Infrastructure that capital moves on usually wants more than one certifier.

The second certifier will come, and it will be cheap too. For me the point remains a different one. This is the first domain where artificial intelligence beats the human expert on a task that is 100 percent verifiable, and foundation models always commoditize. The value migrates to vertical applications with proprietary data.

I'd watch two things. Whether other groups, after Buzzard, recompile those lines and get the same verdict. And whether the sample of intermediate theorems yields a figure on errors corrected along the way. That's where you see whether an error accumulates along a long chain or gets stopped right away.

I'd watch where the spending goes in the labs. Whether the verification layer stays a research project or becomes a stable line item, as happened with the project that closed the Kepler conjecture. The day a proof generated by a model is checked by two independent verifiers, that's the day the precedent has repeated in full.

The cost curve. How many entries on those historic challenges get closed by a machine, and in how many days each. If the time per theorem keeps dropping by an order of magnitude, the adoption jump is close. If it stalls, I was wrong about the when.