Unreleased Anthropic Model Advances Riemann Hypothesis Bound

An unreleased Anthropic model pushed the proven lower bound on a 150-year-old math problem from 41.6% to 67.2% in a 36-hour autonomous run. The model didn't solve the Riemann hypothesis — the $1 million bounty for a full proof stays unclaimed — but the result reopens a debate over whether AI systems can originate real mathematical ideas, not just remix training data.

A non-mathematician's prompt, then 36 hours alone

On Monday, August 10, Anthropic staff member Jarred Sumner — by his own description not a mathematician — asked the unreleased research model to "take a real stab" at the Riemann hypothesis, a conjecture about the distribution of prime numbers that has resisted proof since the 1850s. Sumner gave the initial prompt and occasional encouragement; the model, working inside Claude Code, took over from there. Over the next day and a half it tested 650 different ideas, ran 2,400 shell commands, wrote hundreds of Python scripts for numerical checks, and cross-referenced 54 arXiv papers to confirm its approach was novel.

Sixty subagents, divided labor

The work was coordinated across 60 subagents and consumed 31 million output tokens. A footnote to the resulting paper breaks down the division of labor: two subagents developed the key mathematical ideas, 13 contributed supporting ideas to them, 30 attempted but failed to develop new ideas, 13 served as validators checking the correctness of the arguments, and a final two helped write the paper. After deriving its result, the model volunteered to write up the finding and recommended human validation on its own — a structured research pipeline (generate, review, validate, publish) rather than a single long completion.

The specific advance: 41.6% to 67.2%

The Riemann hypothesis says all non-trivial zeros of the Riemann zeta function lie on one critical line — still out of reach to prove directly. A more tractable sub-problem asks what minimum percentage of those zeros are provably on that line; mathematicians had slowly pushed that lower bound to 41.6%. The Anthropic model combined prior work from mathematicians Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri in a novel, non-diagonal treatment of a quadratic form — what Anthropic's technical note calls having the "courage" to consider positive and negative definiteness together across the full space of functions — and raised the bound to 67.2%. That synthesis of existing frameworks into something new is, per Dev.to's writeup of the result, exactly the kind of conceptual leap that drives progress in mathematics, regardless of who or what performs it.

Verified, not just generated

Two of Anthropic's in-house mathematicians, Levent Alpöge and Ralph Furman, confirmed the work, and the result was formalized using the open-source Lean proof assistant. Anthropic gave no timeline for releasing the multi-agent research capability that produced it. The verification step matters for a specific reason: it's the difference between a model producing a plausible-sounding argument and one whose argument survives a machine-checkable proof, a distinction that has tripped up earlier AI math claims in other labs.

Why a failed attempt counts as a breakthrough

The model never solved the hypothesis — most of its 650 ideas were dead ends. What changed is the shape of the work: it moved from generating scattered attempts to running a multi-day project with division of labor, peer review among its own subagents, and formal verification before anyone claimed a result. That's a different capability than fluent chat or fast image generation, and it's the capability AI-engineering teams building multi-agent systems will likely study — a live example of a long-horizon agent task staying coherent, and checkable, across dozens of subagents and millions of tokens without a human in the loop for most of it.

Part of a wider pattern this year

The Riemann result lands in the middle of a string of AI-driven math results. OpenAI recently published 10 major results proved by its internal "Astra" model. A separate Anthropic effort disproved the long-standing Jacobian conjecture earlier this year, and a number of Erdős problems have already been solved by AI models in 2026. TechCrunch reports the growing tally has split mathematicians: in a public declaration signed in June, a group of prominent mathematicians warned that AI could undermine the field's standard that proofs be "attributable to specific authors who take credit for their discovery and assume responsibility for their correctness."

Not everyone reads the shift as a loss

Fields Medal winner Timothy Gowers pushed back on that framing in a blog post responding to the declaration. "If we arrive at a world where mathematical theorems are no longer associated with mathematicians," he wrote, "maybe that won't be any more problematic than the fact that stars aren't named after astronomers and most aren't named at all." Anthropic's own handling of the Riemann result — AI as discoverer, human mathematicians as validators, the process documented and formally verified — reads as one attempt at a middle path between those two positions. Watch for whether the next disputed result comes from a lab framing it the same way, or one that skips the human verification step entirely.