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Benefit Ledger · provisional · Mathematics

An unreleased Claude raises the critical-line zero bound for zeta from 41.6% to 67.2%

On 10 August 2026 Anthropic reported that an unreleased research Claude, prompted to attempt the Riemann hypothesis, instead improved the proven lower bound on zeros on the critical line from 41.6% to 67.2%. Anthropic mathematicians and two outside experts examined the argument. The Riemann hypothesis remains open.

Tier 3 MajorMethodology 0.2
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Current score

+5.06

10 base · Major (tier 3 of 5, 10 pts)
× 0.8500 attribution · Primary causal contribution
× 0.7000 evidence · Peer review or substantial independent validation
× 0.8500 realization · Manuscript plus machine-verifiable formalization
× 1.0000 durability · Effectively permanent
Event-level product before credit split: 5.06

A large improvement to a classical zeta-zero proportion is major analytic number theory (tier 3), not a Riemann Hypothesis solution. Attribution is to the unreleased Claude (0.85). Evidence is a lab paper, named expert readers, and a reported Lean formalization (0.70). Realization is a public argument plus machine-checkable artifact (0.85). A correct bound is permanent (1.00).

What happened

A staff member asked Claude to take a real stab at the Riemann hypothesis. The unreleased research model did not prove RH. Over two Claude Code sessions (~31 million output tokens, ~60 subagents) it combined work of Aryan, Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh, Montgomery, and Bombieri to raise the classical proportion of zeros known to lie on the critical line. Levent Alpöge and Ralph Furman wrote an informal expert note; Brian Conrey and Dan Goldston examined the paper on short notice; a Lean formalization is reported to pass Comparator. Anthropic says it does not expect the method to prove RH. This is an unreleased Claude, not the version-unspecified Claude row and not a Millennium solution.

Model attribution

+5.06

Claude

Unreleased research Claude that produced the 67.2% bound while attempting RH; humans validated and wrote an informal note.

Anthropic attributes the proof search to an unreleased research version of Claude, not to a named Opus/Sonnet/Fable release.

Attribution 0.8500 · Credit share 100% · Anthropic

Claims

  • An unreleased Claude improved the proven lower bound on zeta zeros on the critical line from 41.6% to 67.2%.

    outcome · supported

  • This proves the Riemann hypothesis.

    significance · disputed

Sources

primary sources

Revision history

  • 13 Sep 2026 · 0.00 → 1.49

    Imported events under methodology 0.1.

  • 14 Sep 2026 · 1.49 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 17 Sep 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 18 Sep 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 19 Sep 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 21 Sep 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 6 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 7 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 11 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 11 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 21 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 24 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 27 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

  • 28 Oct 2026 · 5.06 → 5.06

    Methodology 0.2: keep Major / not RH; raise evidence, realization, and durability for expert notes and Lean.

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