Harmonic’s Bet: Teach AI Math Directly, and the Riemann Hypothesis Falls by 2029

Last Tenev checked, Metaculus put a 43 percent chance on the next Millennium Prize Problem being solved by AI, or by a human with significant AI assist. He thinks that is too low. Larry Guth, he notes, might already be on a path toward the Riemann hypothesis with classical tools, which would be wonderful. But if the next prize is still taken by a human, it would probably have to happen in the very near future. The one after that, he says, will probably be solved with substantial AI involvement.

Sequoia’s Sonya Huang and Pat Grady welcome Tenev and Achim onto *Training Data* with a biography most listeners only half know. Tenev is founder and CEO of Robinhood, and also, the hosts stress, an enormously talented mathematician. The two have teamed up to create Harmonic, an AI research lab whose goal is to push the frontiers of human knowledge. Specifically they hope to create mathematical superintelligence, on the thesis that understanding math lets a system reason about much of the broader world.

The conversation is framed around a bet that is, the hosts say, somewhat contrarian. A lot of foundation-model labs hope mathematical skill will emerge as a property of scale. Harmonic’s core belief is that math *is* reasoning, and that the best way to train a model to perform well at it is to teach it math directly. The rest of the episode is the argument for that choice, including the ingredients they think it takes: synthetic data, reinforcement learning, self-play, and a proof assistant named Lean.

Achim’s case for starting with math is almost architectural. Look around at the fields of science and engineering, he says, and math sits at the foundation of almost all of them. It is how people model phenomena from black holes to atoms, and how they design things. Two properties make that possible. The universe happens to be explainable by a fairly compact set of symbols. And math is a way to build shared cognitive theories that are objective, clear, and transparent: you write down a set of deductive rules, agree on the ground rules of whatever you are modeling, and derive conclusions from there.

With humans, he argues, people who become very good at math tend to be good at other quantitative work in science and engineering. Harmonic’s bet is that you will see the same transfer in a machine. It might not immediately write the world’s best history essays. Ask it to do something scientific or something in engineering, though, and it should be really, really good. That is why they started with math. Where the outer edge of that skill sits, between a homework helper and a history essay, is a question the hosts press and the founders do not pretend to have closed.

Tenev offers himself as the non-AI exhibit. He was good at math as a kid, and he remembers the seventh graders who raised their hands at side-angle-side and demanded to know when they would ever use it. The teacher would mumble that they probably would not use it soon, but that it would make them good at other things. The other kids were skeptical. He bought in. He majored in math at Stanford, went on toward a math PhD, and spent ten hours a week banging his head against abstract-algebra problem sets on the theory that business problems would be easy by comparison.

He thinks it basically ended up correct. He took maybe one intro programming class at Stanford. Five years later, as an entrepreneur, he found it easy to pick up code and even to read contracts: the logical underpinnings felt simple next to analysis. For humans, at least, he considers himself an example of math transferring to “other very monetizable things,” and his intuition is that AI should work the same way. Achim adds the industry evidence already on the table. It is an open secret, he says, that training on a lot of code data leads to much better performance on reasoning benchmarks. Imagine, he says, what that looks like with math datasets that encompass more general types of reasoning.

Huang notes that Tenev studied, briefly, under Terence Tao, often called the world’s greatest living mathematician, and still has lunch with him when he is in Los Angeles. Do they talk stock tips? No. One of the unfortunate things about being a public-company CEO in finance, Tenev says, is that he has lots of stock tips and cannot share any of them, cannot even use them, and basically cannot trade anymore. He loves trading. That part is just gone.

What amazes him about Tao, a professor at UCLA, is breadth. Most mathematicians go very deep in a pretty narrow domain. Tao can go very deep across dozens: number theory, combinatorics, harmonic analysis, applied math. He is now one of the leading Lean contributors, Tenev says, formalizing his own papers, hopping on the community Zulip to engage with students, and keeping a popular blog. The way he has been able to do this is that he is smarter than 99.999 percent of people, “probably even more than that.” From a very early age it was clear he was on another plane.

Tenev’s undergraduate honors thesis was under Larry Guth, who recently posted what Tenev calls a groundbreaking non-AI result related to the Riemann hypothesis. Guth suggested he look at UCLA. He went, studied first-year graduate analysis with Tao, and then dropped out after one year, a status he is happy to claim because it is a *graduate* dropout. During some reading with Tao, the professor lent him a book and signed it, apparently to make sure it came back. Signing it guaranteed the opposite. Tenev brings it up every time they meet: the book sits on his shelf next to his other autographed first editions, and Tao is not getting it back.

Asked what the math community thinks of AI math, Tenev says it is split, and not in a surprising way. Younger mathematicians tend to be pro-AI, pro-verification, and pro tools like Lean. Older folks are more skeptical. His mental model is chess. First there will be a lengthy period of humans plus AI assist, and that will produce a lot of good results. Over time the AI will get better. In chess today, if a human were assisting the engine, the engine would be annoyed: it would want to delete all of the input, because the input would only make the results worse. He is not sure mathematics gets all the way there. Humans will still need to guide the algorithms. The definition of what a mathematician does will still fundamentally change.

He asked a young mathematician friend at MIT whether he was worried about being in a field that was about to be rewritten. The answer was that the field of math has always changed. In the 1800s, mathematicians sat in royal courts as glorified calculators, solving quadratic equations by hand, and they worried that computers and calculators would erase the job. Mathematicians get to define what math is. At some point, Tenev expects, the work will be prompting and guiding these AIs. Even if an AI solves the Riemann hypothesis, “a human will always be in the loop because the humans kind of pose the Riemann hypothesis to begin with.” Achim puts a sharper job description on that loop. In the future a lot of compute will be dedicated to math. The human question becomes by which procedure you decide where to direct all that reasoning firepower: what to work on, how to interpret results, how to interpret failures to find answers.

Can an AI math system actually solve Riemann? Tenev thinks it should be able to, or prove that the statement is undecidable, which would also be an interesting result. A great mathematician like Tao synthesizes lots of papers and frontier results, finds connections other top humans miss, and uses them to create new and more complex theories. That, he says, is exactly how the system they are engineering works, and it is what computers and AI models are already great at: synthesizing large amounts of information, finding patterns, recursively self-improving. He comes back to the 43 percent Metaculus line and repeats that he thinks it is an underestimate. Guth might still get there the old-fashioned way. After that, AI will be in the room.

Grady restates the architecture as a closed loop with no humans inside it. Lean lets you encapsulate math in code. Formal verification tells you objectively whether something is right or wrong, which is an objective reward function. Self-play and reinforcement learning can then run on very fast cycle times. Achim accepts the picture and then draws the contrast that excites him. Recursive self-improvement already works in perfectly observed zero-sum games such as AlphaGo, but those games hit an optimal strategy and then stop. Math has no upper bound. You keep putting compute in and it keeps getting better, with no end. Millennium prizes are human milestones. The real question is whether it ever stops. He thinks the systems will end up solving problems much harder than the Riemann hypothesis, problems we have not even conceived of yet because we cannot write them down. Tenev’s analogy is a video of an AI beating Minecraft in 20 seconds: you know what the game is, and you cannot grok what is happening on the feed.

Under the hood, Achim says, Harmonic is a collection of agents trying to solve problems. Because they use Lean, they can check whether answers are correct and derive a variety of training signals from that check. He is careful about what the check is not. Lean lets you verify things. It does not tell you whether you are getting closer to the answer, or whether you are getting smarter. It only tells you whether a candidate is correct. Making the system get better quickly is still a pile of open scientific challenges.

Lean itself is “just another programming language,” Achim says, a really great one created by Leonardo de Moura. Mathematical statements are encoded in the type system. Functions have input types that correspond to the assumptions of a theorem and an output type that is the conclusion. If you write a program that implements that function and it compiles, you have derived the output type from the input type, which means you can conclude the conclusion from the assumptions. Tenev, sitting next to him, calls it the best programming language, and jokes that we might all be writing Lean, or that the AI might just be writing Lean in the future.

The historical twist is that de Moura, now at Amazon AWS, is not a mathematician. He wrote Lean as a software-verification tool. His belief was that software in the future would be verified, and that existing tools such as Coq and Isabelle, multi-decade-old systems with a poor developer experience, were frankly unusable. Build something better, more people use it, software gets better. What he did not fully anticipate was the math community. Proving that software has certain properties is the same kind of work as proving theorems, and thousands of mathematicians and students started building an organic library called mathlib, which Tenev describes as the largest open-source math textbook, sitting on GitHub and growing at a fast clip. Usage of Lean for math, he thinks, may already have surpassed usage for verified software.

Reinforcement learning has existed for a long time, and math for much longer. Why start Harmonic now? Achim gives two reasons. First, the AI systems got better in a way that matters for theorems. In 2015 and 2016 he was already talking with a close friend about RL for theorem proving, and the blocker was that nobody had a great model for an infinite action space. In Go you place a black or white stone somewhere. In math the next step can be anything. Autoregressive language models made that tractable. Second, Lean itself got good. Twenty years ago, telling a mathematician that a large fraction of the field would be excited about formal methods might have sounded crazy; the successes lived in logic and corners of graph theory, with the Four Color Theorem as the famous example. Mathlib now gives you a body of knowledge to build on, and Lean 4 was only officially released in September 2023. Those two things happening together, he says, made it the right time to attack the problem.

Synthetic data is the fuel. Mathlib is an amazing resource, but it is human-written Lean in a general, compact style aimed at proving advanced theorems, not at problem solving. As a result, Achim says, almost the only data you can use is synthetic data you generate yourself. It is a data-poor regime compared with most of AI. The primary way to get signal is the process he already described: agents try to solve problems and thereby generate training signals, starting with really simple facts and amplifying themselves recursively. Huang’s aside lands the scarcity: there is not as much math data on the internet as there are cat videos.

Tenev splits the famous data wall into two walls. One is the exhaustion of internet text that foundation-model labs are already hitting. The other is quality. Cat videos and nice Wikipedia content set a ceiling on how smart those models can get; it is an open problem how to become significantly smarter than the corpus you trained on. To surpass human mathematicians you need a self-reinforcing, self-play regime, the AlphaGo-to-AlphaZero path, in which the models create the vast majority of their own data and the data increases in complexity as they iterate. Math has a simple meter for that: you can measure the difficulty of a problem by how many lines of Lean it takes to solve. Hard problems get broken into smaller chunks with fewer lines. Get good at the chunks, then at assembling them, and the gears turn toward incrementally harder work. It mirrors pen-and-paper math. Fermat’s Last Theorem ran about 200 pages. The Riemann hypothesis might take hundreds of pages if not more. Math, Tenev says, is to some extent “the original synthetic data”: start from simple axioms and build giant complex structures.

What determines the rate of improvement? At the highest level, Achim says, energy. More energy means more attempts in parallel, which means you generate data faster. There are plenty of rate-limiting steps, but no fundamental constraint on how fast it gets better other than how much compute you put in. Tenev adds the unfinished inventory. Mathlib is incomplete: there is very little geometry, and the library is very theoretical and abstract. At some point the models have to create new theories, expand to other domains, and get really good at formalizing things humans have not formalized. He imagines pointing a system at something as simple as a baseball team throwing balls back and forth and having it auto-formalize the scene into Lean on the fly. That is not reliable yet. When it is, he thinks it will be a really big unlock, and that it will happen within the next several years.

If everything goes right, what does Harmonic become? Achim’s mission statement is to explore the frontiers of human knowledge, with the constraint that what they produce is correct and useful to humans. In the best case they build a tool mathematicians use to close all the Millennium Prize Problems and go far beyond. The commercial twin is software. The dream is to check that code is correct, which requires a very good model of how code and libraries work and what they promise. He can imagine a future where safety-critical software, and then general software, is proven correct, and the way software engineering is done changes with it.

Tenev thinks the discipline is already moving. People are doing crazy things with Cursor and Claude 3.5. In the future, he argues, software engineering will be less about reviewing and collaborating on code as an artifact and more about collaborating on specs: what do we want the code to do, and can we be more rigorous about that. Verification becomes a bigger deal because as the cost of software goes to zero, the cost of verified software goes to zero with it. Look out five or ten years, assuming the capability curve holds, and “the vast majority of software written will be verified and provably correct.” On the more theoretical side he wants to accelerate frontier physics, which is essentially math, and help figure out experiments to test why the universe is the way it is.

Are they staying in math-adjacent work, or is anything that involves reasoning in scope? They are still a small company, Tenev says, so they try to be focused. Over the long term, if you believe math is reasoning, and they do, then computer science is a natural analog and anything is in scope for those models, even the history essay. Achim would rather wait and see; he does not want to touch history essays. Tenev, who enjoyed writing them even when his parents told him to ignore language arts, insists his math skills led him to write great history essays too. “Hopefully Aristotle will be no different one day.” Huang calls him a polymath. He cites the *Poetics*.

The lightning round forces dates. When will they win the IMO? Achim says “soon.” Tenev offers maybe 2024. A Millennium Prize? Tenev guesses 2029, or 2028 for an AI-human hybrid; Navier–Stokes he would give 2026, Riemann 2029. Huang notes that today’s LLMs still cannot reliably do arithmetic. Superhuman reasoning more broadly, something that would give Tao a run for his money, Tenev puts a couple of years out, with models reaching the 99.99th percentile on math problems within the next year, a call he thinks Tao would agree with. Asked whom they admire, Tenev picks John von Neumann, the mathematician whose businessman father tried to talk him out of a field that was not monetizable, and who then pioneered the architecture of modern computers, contributed to the Manhattan Project, and wrote a canonical text in game theory. Achim picks Leibniz, who besides competing with Newton on calculus sketched, in the late 1600s, a “universal characteristic”: a deductive language, an automated procedure for deduction, and an encyclopedia of work you can build on. That, Achim says, is essentially a prediction of Lean plus AI, three and a half centuries early. The only missing pieces were computers and models that were good enough.