AIArticle Math's AI moment shows capability lands wherever verification is cheap, public, and machine-checkable.
Mariana Souza In late May, an unreleased OpenAI reasoning model produced a construction that disproved Paul Erdős's 1946 unit distance conjecture — an 80-year-old question about how many pairs of points in the plane can sit exactly distance one apart. Erdős believed the square grid was essentially optimal; the model found an algebraic-number-theory construction that beats it by a polynomial margin, and Will Sawin quickly made the exponent explicit (n^1.014, since pushed to n^1.0318). A verification paper followed from mathematicians with no stake in the announcement — Noga Alon, Tim Gowers, Sawin, and Mark Sellke among them. Gowers wrote he'd recommend the result to the Annals of Mathematics "without any hesitation." Gil Kalai compared the moment to the computer-assisted four-color theorem of 1976.
Seven months earlier, the same company was a punchline. In October 2025, OpenAI staff claimed GPT-5 had "solved" ten open Erdős problems; within days it emerged the model had merely located existing papers that Thomas Bloom, who maintains the problem database, hadn't yet cataloged. Demis Hassabis called the episode embarrassing, and he was right.
The distance between those two events is the real story, and it's not primarily about model IQ. Erdős problems are falling because mathematics accidentally built the one thing that reliably converts raw model capability into results: a full verification pipeline. Developers should study this, because it's the cleanest natural experiment yet in what happens to a knowledge field when that pipeline exists.
An accidental benchmark #
Bloom started erdosproblems.com as a personal reference project: a catalog of the thousand-plus conjectures Erdős scattered across six decades of papers, each one short, self-contained, and machine-readable in spirit. Last August he bolted on a comment section. That single feature turned an archive into a public review layer — and, unintentionally, into the AI industry's favorite eval. The ledger now stands at 565 problems solved and 652 open, out of more than 1,200 cataloged.
Terence Tao's team ran a companion wiki tracking every AI contribution, sorted with taxonomic care: fully autonomous solutions, AI rediscoveries of forgotten literature, AI extensions of known work, human–AI collaborations. By the time the wiki stopped updating at the end of June, it listed over 400 contributions across more than 200 problems, roughly 60 of them autonomous. Read that number again: the tracking infrastructure, staffed by some of the best mathematicians alive, couldn't keep pace with the output.
The stack, not the model #
Look at what actually got assembled between October's fiasco and May's landmark. Generation got stronger, sure — GPT-5.2 Pro cracked problem #728 in early January, a 23-year-old with no advanced math training one-shotted the 60-year-old problem #1196 with GPT-5.4 in April, and Google DeepMind ran Gemini-based agents against a set of 353 problems formalized in logic, resolving nine. But every result that survived did so because of what sat downstream of generation.
First, mechanical certification. The #728 proof was formalized by Aristotle, the prover built by the startup Harmonic, and checked in Lean — meaning its correctness is a compiler question, not a social one. The #1196 solution got the same treatment. Second, expert review where formalization wasn't enough. Jared Duker Lichtman, who'd wrestled with #1196 in his own doctoral work, put it plainly: "The raw output of ChatGPT's proof was actually quite poor." He and Tao sifted it, simplified it, and shortened it. The model supplied the key move — deploying the von Mangoldt function, a classical prime-number tool nobody had thought to aim at this problem — and humans turned that move into mathematics.
And third, fast public falsification. On Christmas morning, a claimed first-ever autonomous solve lasted only hours before commenters surfaced a 1977 paper settling the question — written by Erdős himself. The system caught it because the review layer is public and adversarial. Compare that with Bloom's current nightmare: AI-generated papers running 100 to 200 pages where, as he puts it, "no human has read it, and no human is going to read it." The pipeline holds exactly as far as verification reaches, and not one theorem further.
The economics are the part that should make you sit up. DeepMind reported per-problem costs of a few hundred dollars. These are questions that consumed careers.
If you build software, this rhymes #
Every failure mode above has a name in your issue tracker. The October fiasco is the AI pull request that "fixes" a bug by vendoring someone else's patch without attribution. The unread 200-page paper is the plausible-looking generated diff nobody reviews. The Lichtman workflow — cheap generation, expensive expert sifting — is exactly where serious engineering teams have landed with coding agents: the scarce resource moved from writing to reviewing.
Which points at the transferable lesson. LLM capability doesn't land evenly; it lands where three things coexist: a corpus of well-posed problems, a mechanical verifier, and a public review culture. Erdős problems had all three by late 2025 — short statements, Lean, and Bloom's comment threads. Software has all three too, which is why coding fell early. If you work in a domain that lacks a mechanical verifier — infrastructure migrations, data pipelines, security posture — the highest-leverage thing you can build right now isn't a better prompt. It's the verifier: the property-based test suite, the formal spec, the reproducible check that turns "looks right" into "is right." The moat was never the model.
A real shift, extrapolated carefully #
The human signals say this is no hype cycle. Alon, who spent decades solving Erdős problems by hand, told Quanta that "once AI started to solve them, there is no point anymore." Jacob Tsimerman won a Fields Medal on July 23 and announced days later he's taking leave from Toronto to join OpenAI, saying the mathematical career as we know it won't exist in its current form. When the people winning the game start leaving the table, believe them.
But keep the asterisk. Erdős problems are unusually AI-shaped: self-contained, provable in a sitting, frequently vulnerable to a wide-enough bag of known techniques — breadth over depth, which is precisely the axis where models beat any individual human. The multi-hundred-page theory-building programs that win Fields Medals live in a different regime, one with no benchmark database and no one-file Lean target. The unit distance disproof shows machines can now produce Annals-grade objects; it doesn't yet show they can decide which objects are worth producing.
The durable takeaway isn't "AI does math now." It's that eighteen months of infrastructure — a database, a comment box, and a proof checker — did more to unlock frontier-model capability than any single training run. That's a reproducible recipe, and mathematics just published it.
Sources & further reading #
Why the Legendary Erdős Problems Are Falling to AI— quantamagazine.org - Amazing: Erdős' Unit Distance Problem was Disproved! It was achieved by AI!— gilkalai.wordpress.com -
[Amateur armed with ChatGPT 'vibe maths' a 60-year-old problem](https://www.scientificamerican.com/article/amateur-armed-with-chatgpt-vibe-maths-a-60-year-old-problem/)— scientificamerican.com -
[AI contributions to Erdős problems](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems)— github.com -
[Remarks on the disproof of the unit distance conjecture](https://arxiv.org/abs/2605.20695)— arxiv.org -
[Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof](https://arxiv.org/abs/2601.07421)— arxiv.org -
Jacob Tsimerman Wins 2026 Fields Medal for André-Oort Conjecture Proof— quantamagazine.org
Mariana Souza· Senior Editor Mariana covers the fast-moving world of machine learning and generative AI, with a particular focus on how these technologies are reshaping development workflows. When she isn't stress-testing the latest foundation models, she's usually at a local hackathon.
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