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Benefit Ledger · provisional · Mathematics

Claude, with Alpöge and Howell, posts rank ≥30 and ≥31 elliptic curves over Q

On 20 and 23 August 2026 the ICARM elliptic-curve rank leaderboard recorded new records of rank at least 30 and 31, with public witness points, credited to Claude working with Levent Alpöge and Ava Howell. Exact rank 30/31 is reported only under GRH and BSD.

20 Aug 2026Tier 3 MajorMethodology 0.1

Current score

+1.48

10 base · Major (tier 3 of 5, 10 pts)
× 0.5500 attribution · Critical contribution
× 0.5500 evidence · Peer review or independent validation
× 0.7000 realization · Independently validated or deployed
× 0.7000 durability
Event-level product before credit split: 1.48

Breaking the 18-year rank record with checkable points is major arithmetic geometry (tier 3). Humans are co-credited (0.55). Evidence is an NSF-institute leaderboard with witnesses (0.55). Realization is certified lower bounds (0.70).

What happened

Elkies–Klagsbrun had held rank ≥29 since 2024. ICARM verifies submissions. Curve commentary credits Claude, Alpöge, and Howell. Unconditional claims are lower bounds via independent rational points; exact ranks assume GRH+BSD. The compiled sheet calls the Claude an internal Anthropic model; the leaderboard says Claude. Identity is therefore version-unspecified Claude in the Claude family, not Fable 5 unless a later source names it. This is a record construction, not a proof that ranks are unbounded.

Model attribution

Claude
Version unspecified
+1.48

Claude

Claude system credited on the ICARM leaderboard with Alpöge and Howell for the rank-30 and rank-31 curves.

Public leaderboard comments name Claude with two human mathematicians; no Fable/Opus SKU is given, so identity stays version-unspecified in the Claude family.

Attribution 0.5500 · Credit share 100% · Anthropic

Claims

  • ICARM recorded elliptic curves over Q of rank at least 30 (20 August) and 31 (23 August), with witness points, credited to Claude, Alpöge, and Howell.

    outcome · supported

  • These curves prove that ranks of elliptic curves over Q are unbounded.

    significance · disputed

Sources

primary sources

independent sources

Revision history

  • 13 Sep 2026 · 0.00 → 1.48

    Imported events under methodology 0.1.