Grant Sanderson (@3blue1brown) – AI disproved a famous math conjecture. Now what?

Summary

• Grant’s three-year-old call that IMO gold would be “another benchmark,” not AGI, aged well — and the pattern is fractal: within math itself, geometry has been brute-forced (“it just solves it in nineteen seconds since 2024”) while combinatorics holds out. The read-through for investors watching AI: the frontier stays spiky at every zoom level, so single-benchmark headlines are weak signals for economic automation.
• The load-bearing question is how an AI would prove the Riemann hypothesis: a Montgomery–Dyson-style “lightning bolt” between fields LLMs already know is LLM-native but implies little white-collar spillover; “mountain-building” — inventing new theory the way Fermat’s Last Theorem needed elliptic curves and modular forms — is “such a level of intelligence that it would be surprising if it didn’t permeate into other aspects of the economy.”
• The next milestone won’t be benchmarkable: “good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions.” You’ll detect it as a “tone shift” in how mathematicians talk about the models — and Dwarkesh’s kicker: what can’t be benchmarked can’t easily be trained for in the current paradigm either.
• Galois theory is the canonical counterexample to verifiable reward: a hundred-year verification loop from Lagrange’s inkling to Jordan’s modern group theory, with the “verifier function” (the academy) rejecting Galois repeatedly. Grant’s candidate fix: reward compression — “throw Kolmogorov complexity into your attempt to quantify what you mean by elegance.”
• Dwarkesh’s neglected driver of math/code progress: not just verifiability but grindability — deterministic, containerizable environments you can farm with parallel rollouts. Computer use is verifiable but not grindable (bot detectors, no replay); real-world business-building and trading are neither. The automation map follows this constraint, not raw model IQ.
• Lean is overrated as today’s engine, underrated as tomorrow’s: DeepMind dropped it for natural language at the IMO, and the unit-distance proof had none — but a self-extending Mathlib fork you could “press go… look away for ten years,” plus a green-checkmark correctness guarantee “every other field would kill for,” is unique to math.
• Grant changed his mind: AI will be good at explaining too, so mathematicians drift toward museum-curator and teaching roles — teaching being “one of the most stable post-AGI jobs” because it’s relational, not informational.
• Economic payoff of 10–100X math: expect incremental leakage via PDEs and simulation (one group’s insights “saved Boeing billions or something”), not step-changes — with the awkward tail risk that acceleration reveals pure math “has become wholly useless.”

Deep dive

1. IMO gold was “another benchmark” — and the spikiness is fractal

• Grant’s mechanism for why gold produced no “aha” moment: the AI frontier’s spikiness repeats at every zoom — “there’s a fractal nature to that spikiness.” Within the IMO’s four categories, geometry “just solves it in nineteen seconds since 2024 because it’s a brute force solver”; models would have taken gold in 2024 had the draw included fewer combinatorics problems — the “wild card… much more playful, puzzly-seeming problems” where they still struggle, and where “the last holdout of math for humanity” partisans locate creativity.
• Dwarkesh’s honest admission — “I’m totally moving the goalpost here”: he once pressed Dario on why models with superhuman breadth weren’t connecting known ideas into discoveries. OpenAI’s counterexample to the unit distance conjecture is exactly that, so the question resets: what’s the next thing that would be genuinely impressive?

2. How Riemann falls determines whether white-collar work falls

• Shape one — lightning bolts between existing fields: Hugh Montgomery writes down the pair-correlation of Riemann zeta zeros; Freeman Dyson recognizes the expression from eigenvalues of random Hermitian matrices — nuclear energy levels. LLMs “are experts at quantum physics… experts at analytic number theory. They should be able to see that similarity” without needing lunch at the IAS. Grant’s caveat: “that’s totally different from white-collar work” — your AI editor doesn’t fail for lack of lightning bolts.
• Shape two — mountain-building: Fermat’s Last Theorem is phrasable so simply (xⁿ + yⁿ = zⁿ), yet the proof required two mountains — centuries of elliptic curves plus modular forms — “before you can ask the right question that connects them.” An AI that builds correct new theory “is just such a level of intelligence that it would be surprising if it didn’t permeate into other aspects of the economy.”
• Shape three — raw hustle: a thousand-page proof “that no one’s really getting anything out of,” where what you’d then need is “the succinct, compressed versions of those ideas that would lend themselves to human understanding.”

3. The next milestone won’t fit a benchmark

• The framing both converge on, via a quote Grant cites from Polylog’s unit-distance video: “good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions.” Conjecture and definition generation is “the premium-tier mathematician” — but Grant “would be surprised if it ever took the form of looking like a benchmark.” Imagine the headline for GPT-5.4 coming up with a good conjecture: “‘We promise, everyone thinks it’s a good conjecture.’ It just doesn’t land the same way.”
• How you’ll actually know: “you’d feel a tone shift in conversations with mathematicians.” Grant’s forthcoming AI-and-math series has already captured one between mid-2025 and 2026 — “In the real world, that’s a very short amount of time. In the AI world, that’s eons.”
• Dwarkesh’s structural point: “there’s really no fundamental difference between a benchmark and a training environment” — what can’t be benchmarked can’t easily be trained for in the current paradigm. Both hedge that such dichotomies historically collapse: “it turns out you’re just thinking about it the wrong way, and actually it can do it pretty soon thereafter.”

4. Galois: a hundred-year verification loop RLVR can’t close

• The chain as Grant tells it: Lagrange plants the seed that symmetries of roots are the right lens on the quintic — no result, “he just asked it,” and it wasn’t even important math at the time. Abel proves impossibility and dies at twenty-six of tuberculosis. Galois, a teenager writing from prison, is rejected by the academy — “the verifier function… is rejecting what he wrote,” and “frankly, it was not very coherent” — then dies amid the romantic story of a duel. “If you’re a young genius, don’t work on the quintic.”
• Even after death the loop stays open: twenty years until Liouville sees something in the notes, another twenty until Jordan produces a modern treatment of group theory, and the twentieth century before Gell-Mann predicts quarks from a purely group-theoretic question. “You could easily imagine history turning differently… Galois could have been forgotten if he was a less florid character.”
• The reward-design question this forces: what metric captures “the instinct inside Galois’s mind when he says, ‘I think there’s something here’”? Grant’s candidate, from his current “compression is intelligence” series: “maybe you throw Kolmogorov complexity into your attempt to quantify what you mean by elegance” — reward the smallness of the concepts required, not just the solve.

5. Proof isn’t explanation — and Grant no longer thinks explaining is safe

• His favorite paper opening, from Timothy Chow on forcing (the method behind the continuum hypothesis’s yes-and-no, axiom-dependent answer): “everyone knows the idea of an unsolved research problem. I want to propose the idea of an unsolved expository problem.” Grant: “this is my whole life… There is a difference between proof and explanation.”
• The nightmare is abc-conjecture-shaped: an otherwise reputable mathematician in Japan builds “inter-universal geometry,” an alien mountain that takes years to even parse — “people work for years to go up the mountain, and they’re like, ‘Dang it. This just isn’t right.’” David Bessis’s “The Fall of the Theorem Economy” names the deeper break: theorem-proving “gets all the credit, but it’s really a parasite on the coming-up-with-the-definition stuff” — fine when one person did both, broken when AI automates the theorem half.
• Grant’s change of mind, flagged as such: “I used to think that AIs would become these automated theorem provers” and mathematicians would shift to his job, explaining. But the people with genuinely novel insights — Einstein, Claude Shannon, Feynman — are also lucid expositors, so the same faculty likely transfers: “digesting and explaining… is probably actually not what’s left for mathematicians. That’s a way my beliefs have changed.”
• What’s left is the art-museum-curator role, grounded in relationship: “they’re trusting you as a curator” — the same reason human musicians survive objectively better MP3s. Dwarkesh’s gallows summary of doing this until he dies: “You build a man a fire, and he’s warm for one night. But set a man on fire, and he’s warm for the rest of his life. So that’s where I am with AI.”

6. Langlands, the autoregressive box, and engineered entropy

• Most mathematicians aren’t ticking off problems; the Langlands program is “not even a field of math so much as it is a research ethos” — preemptively mapping connections across the landscape. Grant’s five-year call: “that’s my guess on what most of the useful progress from these models will look like in the next five years… filling in that landscape of connections.” The Erdős problem 1196 example had this shape — Markov chain, bottom-up probabilistic argument, von Mangoldt function — “if you say that to someone in the know, they’d know how to run with it.”
• Why breadth hasn’t already produced bolts — Dwarkesh’s box thought experiment: locked in a box predicting slips of paper with your memory wiped each time, then shown the result — “This is awful. That’s not the essay that I would’ve written.” Autoregression makes you “a slave to your context,” and the valuable cross-field connection is by nature the unlikely next token.
• Grant’s response runs through data, not architecture: diffusion text models produce nothing “of a wholly different character”; agents learned “let’s step back and assess my mistake” because environments rewarded it. Build environments that require cross-field connections — e.g., frontier math-like problems, partially synthetic — “and then it doesn’t really end up mattering what the loss function is.”
• The counterintuitive digital advantage may be the opposite of pooling knowledge: systematically erasing context. The “troll” IMO problem — which Terry Tao also failed — rewards escaping “the context of being in the IMO”; a street brain-teaser framing cracks it. So spawn agents with deliberately different contexts and biases, while remembering biases aren’t neutral: “you want to make sure you don’t accidentally have all your LLMs be Einstein, because you might halt progress on quantum mechanics.” And per Dwarkesh, with labs throwing billions at math for PR, “quantity has a quality all of its own.”

7. Grindability, not just verifiability, is the hidden driver

• Outside the labs, Dwarkesh’s self-described “totally naive theory” starts from everyone crediting verifiability, while computer use is highly verifiable (“Is my Etsy package coming?”) and still slow. Grant’s answer is that it lacks grindability — bot detectors and compute costs make a thousand parallel rollouts of the same Amazon checkout flow impossible. “You’ll get shut down by Andy Jassy. Him personally. He presses the red X on Dwarkesh button.”
• The mechanism: sample efficiency is unsolved — “sucking supervision through a straw, as Karpathy says” — so learning needs containerized, deterministic replays where the diff between a successful and failed rollout solves credit assignment. Code and math are the exceptions; building a business or “trading in the markets for a day” changes daily and “you can’t keep replaying and grinding and farming the simulator.”

8. Lean: overrated as today’s engine, underrated as tomorrow’s

• Dwarkesh: “I feel like Lean just doesn’t matter that much for the current level of progress.” Evidence: the unit-distance chain of thought contained no Lean; DeepMind went all-Lean at the IMO one year, all natural language the next; and DeepSeek’s DeepSeek Math paper showed natural-language verification works when a meta-verifier trains the verifier.
• Grant’s first rebuttal — autonomy: a fork of Mathlib plus an endlessly running prover is “a very unique thing that math has that nothing else has, where you could press go and just pour compute at it, look away for ten years, and then come back and say, ‘What do you have?’” — AlphaGo-style, no human check-ins. Terry Tao’s project exhaustively searching the space of possible algebras is the template: mostly “trash,” but occasionally “this little island of a completely different type of axiom system” rich in theorems, whose motivation you might retrofit the way group axioms turned out to be about symmetry.
• Second — trust at scale: Alex Kontorovich’s point applies if AI mathematicians generate ten papers a day with any error rate — “even if 99 out of 100 are right,” locating the error is brutal. Auto-formalization gives the green checkmark: “every other field would kill for that.” Grant: Lean is “maybe overrated regarding its importance as a VR environment… but I definitely wouldn’t write it out of the story.”

9. Why writing lags: it can’t be slop, and models lack face muscles

• Dwarkesh’s two-part theory: judges get “totally derailed by B*” — the shitty essay that hits every bell an A essay is supposed to hit, pure reward hacking — and writing isn’t modular. With code, “if it works, it works”; in writing “the end product is directly the thing the AI is producing… It can’t be slop in the way that code can be slop and still produce the outcome you want.”
• Grant’s twist: distillation is already quasi-superhuman — Dwarkesh confesses a revealed preference for pasting expert writing into an LLM (“the explanation will be better than the thing produced by the human”) — but real writing is knowing “exactly the correct point when you want to make an unpredictable move.” What generated the book worth distilling was never an LLM.
• The theory-of-mind gap: Andy Matuschak and a collaborator tried RL on open models, chain of thought, big prompts to the best closed model — and couldn’t make LLMs write good spaced-repetition flashcards, which require “projecting somebody’s mind in three months.” Grant’s vaguely remembered Botox experiment: people who had freshly gotten Botox were reportedly worse at reading emotions — understanding is embodied mimicry. “They don’t have face muscles… It’s like an alien trying to empathize. How could it have theory of mind?”

10. Learning, careers, and the payoff question

• Grant’s learning heuristic: “who matters more than what.” LLM explanations feel “a lot like Wikipedia — which is to say, amazing,” but Wikipedia is a crowdsourced “local minimum where every sentence has to be correct,” versus a Princeton Companion article deliberately crafted by one author. So he uses the LLM as “a very souped-up version of Google” — “Who should I read?” — even after Claude gaslit him by misattributing someone else’s semiconductor video to 3Blue1Brown.
• Dwarkesh’s setup: Strogatz’s Nonlinear Dynamics and Chaos lecture, textbook, and LLM each on a third of the screen — the human curates the order and motivation, the LLM prunes around the branch. What neither gets from a model: the A+ teacher’s ability to “jujitsu your way of thinking” and reframe a bad question — LLMs are “a little too placating. ‘Oh, what an insightful question.’”
• Career advice, heavily hedged (“I wouldn’t trust any advice that I give… I’m a YouTuber”): students chase the next hoop instead of asking “where the money is coming from, what value you’re actually adding, and the connection between those two.” Teaching is “one of the most stable post-AGI jobs” because it’s relational — where abundant parents spend. And curating where to point “this behemoth of new math” becomes “a much more levered move than it had been previously.”
• On whether 10–100X math matters economically: super spiky. Algebraic number theory unlocking something “feels unlikely,” but a PDE/dynamics group’s simulation insights were recalled as having “saved Boeing billions or something” — expect “meaningful leakage” at the fringes, not massive economic step changes from solving Navier–Stokes. The awkward tail: acceleration may reveal math “has just become wholly useless” — “every time we wrote those grant proposals and said, ‘Trust us, the elliptic curve progress is going to help with cryptography,’ it shines a light on the fact that maybe it doesn’t.”