Anthropic Claude mathematical research

Table of Contents

Anthropic’s research-math program points unreleased Claude research models at open mathematics through Claude Code, with subagent fleets doing the work and Lean plus named human experts as the trust gates.

The frontier research loop is now ordinary engineering tooling: one staff member, a coding agent, about sixty subagents, thousands of shell commands, and a compiler as the judge.

What it is
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A series of Anthropic Science results produced by Claude models working in a Claude Code multi-agent harness, validated and published with artifacts. On August 10, 2026, Anthropic reported that an unreleased research version of Claude, asked to take a real stab at the Riemann hypothesis, failed at the hypothesis itself but raised the lower bound for the share of zeta zeros on the critical line from 41.6 to 67.2 percent. The run took two sessions and 31 million output tokens: after 650 failed ideas, Claude spent a day and a half coordinating about sixty subagents that ran 2,400 shell commands and hundreds of Python scripts, checked against known zeta zeros, refereed one another, and downloaded 54 arXiv papers to confirm novelty. Anthropic mathematicians Levent Alpöge and Ralph Furman validated the result, external experts Brian Conrey and Dan Goldston reviewed it on short notice, and the Lean formalization passes the comparator checker in the public anthropics/formal-math repository. On September 4, 2026, Anthropic published the first complete computer-checked proof of Fermat’s Last Theorem: 11 days, largely autonomous, 13 million lines of Lean, 30,300 theorems proved along the way, about six billion output tokens from an internal research model described as roughly comparable to Claude Fable 5.1, reviewed by Kevin Buzzard. The FLT effort succeeded only after switching to Prove2Me, a platform maintaining a directed acyclic graph of theorem statements so parallel agents stop losing track of project state. Anthropic’s own zeta post footnote also credits a Claude model with disproving the Jacobian conjecture, which press coverage attributes to Fable 5 via Alpöge’s announcement.

Status
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Very active and widening, as of 2026-10-11: September alone brought the nine-loop amplitude in N=4 super Yang-Mills (September 25, guest-posted by the physicist who issued the challenge) and, beyond mathematics, a new life-sciences laboratory whose agents discovered a novel CRISPR-like enzyme system (September 23). The models are unreleased research versions, so the loop is not reproducible by outsiders today. The artifacts are public and machine-checkable: the formal-math repository (297 stars as of 2026-10-11) is Apache-2.0 with pinned toolchains and CI that builds with no sorry outside trusted statement files, and on October 6 it absorbed a new formalization of a 3SUM/APSP algorithms-paper result (truly subquadratic 3SUM via triangles in sparse lopsided graphs), widening the artifact chain beyond pure mathematics.

Community footprint is strong, with the zeta result drawing a 282-point Hacker News discussion. A three-person experiment formalized Vinogradov’s three primes theorem in three days on consumer Claude Max plans, and the October 1 Claude-shaped science post documents the loop’s most copyable instance yet: 36 manuscripts in 18 fields in three months, driven through Claude Code sessions on cloud VMs and an open-source harness (BootLoops), with human collaborators supplying taste and expert review. The widening continued on October 8 with the first complete UV map of the sky, built by Johns Hopkins astrophysicist and Anthropic researcher Brice Ménard with Claude Science: merged GALEX-era datasets, cross-wavelength inpainting for the never-observed third of the sky, and over 100 million Gaia-inferred stars. The loop has also escaped the vendor: Nobel laureate Giorgio Parisi and Francesco Zamponi proved the decade-old jamming identity a + b = 1 with Claude and published the full transcript alongside their paper (October 10), and Tom Verhoeff’s proof of Lehmer’s permutation conjecture (arXiv 2610.01240, submitted October 1) reports Claude Opus 5.5 autonomously finding two core concepts that cut a 150-page human-AI proof under 30 pages. Independent human follow-up has started too: Youness Lamzouri re-derived the 67.2 percent zeta bound with a simpler argument drawn from the same prior work, added a corollary (at most 11 percent of zeros neither simple nor on the line), and James Maynard held up the episode as the positive path for AI-assisted mathematics.

Strengths
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Every headline result ships with a machine-checkable artifact, not just a blog post.
The methodology notes read like engineering postmortems: failed attempts, state loss, and the DAG fix are documented.
Novelty checking is built into the loop, including downloading and reading prior work before claiming a result.
Failures still contributed: about 7 percent of the final FLT lines came from abandoned attempts.
The nine-loop solve stayed inside an academic budget, about $100 of compute on 96 CPUs for a week on the bootstrap path, and was validated by Lance Dixon, the researcher whose team had queued the problem.

Cautions
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The results are vendor-validated first and community-validated slowly, and the zeta bound’s external review was two experts on short notice.
A counterexample-style disproof can be a weak result: mathematician Andrew Blumberg called the Jacobian counterexample unilluminating even while granting the achievement.
The encouraging-prompt detail (a non-mathematician telling Claude to keep going) is charming but also a reminder that the model’s own calibration failed here.
The program’s own results still come from the vendor’s write-ups, though independent follow-up has begun: Lamzouri re-derived and simplified the zeta bound, and Parisi and Zamponi published their full transcript.
The independent reads cut against the breakthrough framing: von Hippel’s own takeaway is that Claude solved his challenge with known methods on more compute, and Song He’s group matched most of the nine-loop result days later with GPT-6 assistance, so the low-hanging-fruit reading beats the new-capability reading.
The Claude-shaped science post is also a failure-mode catalog: the author documents Claude declaring victory with an unproven lemma, having no sense of time, and needing every result looked at by a human.

Pricing
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The published artifacts are free under Apache-2.0. The compute is internal and not priced publicly, and Anthropic sells the ingredients (Claude subscriptions plus science grant programs) rather than the loop itself.

Compared to
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Math Inc. Gauss: a purpose-built autoformalization agent with an open harness and audited benchmarks; choose Gauss for formalization throughput, the Claude loop for genuinely new mathematics.
OpenAI for Science: the rival lab program, currently defined by a disputed Navier-Stokes claim; Anthropic’s publish-the-artifacts-and-invite-review pattern is the sharper contrast.
Claude Code: the harness underneath, which you can run today with your own subagent fleets.

Bottom line
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Recommended for anyone orchestrating long-horizon multi-agent work: this is the best-documented public example of subagent fleets, DAG coordination, and verification gates applied to real research. Not for anyone expecting to rerun the exact loops, since the models are unreleased as of 2026-09-18.

Changes
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2026-09-13 - Created as the Anthropic member of the new Automated research category.
2026-10-05 - Recorded the September-to-October widening: the nine-loop N=4 super Yang-Mills amplitude (September 25, Fable 5.1 in Claude Science, Dixon-validated), the life-sciences laboratory and its ART enzyme discovery (September 23), and the 36-manuscript BootLoops campaign (October 1); formal-math stars refreshed to 253.
2026-10-07 - Added the anthropics/formal-math star history chart to the Status section.
2026-10-11 - Recorded the formal-math repository’s October 6 addition of a 3SUM/APSP algorithms-paper formalization and refreshed the star count to 297.
2026-10-12 - Recorded the October 8 UV sky map, the first third-party Claude proofs (Parisi-Zamponi’s jamming identity with full transcript, Verhoeff’s Lehmer conjecture), and the independent zeta follow-up (Lamzouri’s simpler re-derivation).