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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.

Tier 3 MajorMethodology 0.1
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Current score

+1.48

10 base · Major (tier 3 of 5, 10 pts)
× 0.5500 attribution · Critical contribution
× 0.5500 evidence · Peer review or substantial independent validation
× 0.7000 realization · Manuscript plus machine-verifiable formalization
× 0.7000 durability · Useful for many years
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). Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source.

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. Update (2026-09-14): Watch AI materials-science and bioscience abilities closely. Article URL: https://www.lesswrong.com/posts/SCtkSz4nQ9icLZ4uq/watch-ai-materials-science-and-bioscience-abilities-closely Comments URL: https://news.ycombinator.com/item?id=49694871 Points: 6 # Comments: 1 Update (2026-09-14): The AI job market in 2026. Article URL: https://www.ilinmaks.com/blog/en/ai-jobs-market-2026 Comments URL: https://news.ycombinator.com/item?id=49696867 Points: 7 # Comments: 0 Update (2026-09-14): Largest known Roman mosaic, beneath Baths of Trajan, opens to the public. Article URL: https://www.theartnewspaper.com/2026/09/14/largest-roman-mosaic-opens-to-the-public Comments URL: https://news.ycombinator.com/item?id=49696132 Points: 16 # Comments: 3 Update (2026-09-14): iOS 27, iPadOS 27, and macOS 27. Article URL: https://www.apple.com/newsroom/2026/09/major-updates-for-apples-software-platforms-are-now-available/ Comments URL: https://news.ycombinator.com/item?id=49701004 Points: 137 # Comments: 117 Update (2026-09-14): Dropping eBPF CPU Cost by About 90% with Memoization (Not AI Gen). Article URL: https://nathannaveen.dev/posts/dropping-ebpf-cpu-cost-by-90/ Comments URL: https://news.ycombinator.com/item?id=49697477 Points: 11 # Comments: 2 Update (2026-09-15): Ex-FTC boss Khan: break out the handcuffs for AI CEOs, citing 1934 precedent. Article URL: https://www.theregister.com/ai-and-ml/2026/09/14/ex-ftc-boss-khan-urges-uncle-sam-to-break-out-the-handcuffs-for-ai-ceos-citing-1934-precedent/5296325 Comments URL: https://news.ycombinator.com/item?id=49706223 Points: 38 # Comments: 10 Update (2026-09-15): OpenAI buys smartphone camera maker Glass Imaging for $300M. Article URL: https://techcrunch.com/2026/09/14/openai-buys-smartphone-camera-maker-glass-imaging-for-300-million-report-says/ Comments URL: https://news.ycombinator.com/item?id=49711240 Points: 20 # Comments: 3 Update (2026-09-17): The Painful Truth: The RAM Crisis Is Only Just the Beginning. Article URL: https://www.madshrimps.be/news/the-painful-truth-the-ram-crisis-is-only-just-the-beginning/ Comments URL: https://news.ycombinator.com/item?id=49736220 Points: 44 # Comments: 34 Update (2026-09-17): LLM Classification Is Feature Engineering. Article URL: https://minimallysufficient.com/posts/llm-classification-is-feature-extraction/ Comments URL: https://news.ycombinator.com/item?id=49742437 Points: 5 # Comments: 0 Update (2026-09-17): Towards Self-Driving Codebases. Article URL: https://blog.detail.dev/posts/towards-self-driving-codebases/ Comments URL: https://news.ycombinator.com/item?id=49743527 Points: 5 # Comments: 1 Update (2026-09-17): I Hate You Microsoft. Article URL: https://henriquenunez.eu/posts/you_did_it_again_ms/ Comments URL: https://news.ycombinator.com/item?id=49746986 Points: 4 # Comments: 0 Update (2026-09-17): How to Write with an LLM. Article URL: https://sockpuppet.org/blog/2026/09/17/how-to-write-with-an-llm/ Comments URL: https://news.ycombinator.com/item?id=49747070 Points: 12 # Comments: 2 Update (2026-09-18): Warren Buffett Steps Down as Berkshire Chairman, Names Son to Replace Him. Article URL: https://www.nytimes.com/2026/09/18/business/warren-buffett-berkshire-chairman.html Comments URL: https://news.ycombinator.com/item?id=49752614 Points: 23 # Comments: 7 Update (2026-09-19): How come AI-related posts get so many points on HN?. Have we all decided to rebrand Hacker News as a platform for discussing AI? I think that the too many people love to discuss harshly about AI Comments URL: https://news.ycombinator.com/item?id=49764057 Points: 5 # Comments: 4 Update (2026-09-20): The senior engineer death spiral. Article URL: https://sunilpai.dev/posts/the-senior-engineer-death-spiral/ Comments URL: https://news.ycombinator.com/item?id=49776130 Points: 10 # Comments: 1 Update (2026-09-22): Can gzip be a language model?. Article URL: https://nathan.rs/posts/gzip-lm/ Comments URL: https://news.ycombinator.com/item?id=49797323 Points: 36 # Comments: 2 Update (2026-09-23): Jev in 25 Lines of Python. Article URL: https://www.nobodywho.ai/posts/jev-in-25-lines/ Comments URL: https://news.ycombinator.com/item?id=49812769 Points: 57 # Comments: 22 Update (2026-09-23): Two Git ignore files nobody told me about. Article URL: https://mihai.dinculescu.dev/posts/two-git-ignore-files-nobody-told-me-about/ Comments URL: https://news.ycombinator.com/item?id=49814432 Points: 19 # Comments: 7

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.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 17 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 18 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.

  • 19 Sep 2026 · 1.48 → 1.48

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  • 20 Sep 2026 · 1.48 → 1.48

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  • 22 Sep 2026 · 1.48 → 1.48

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  • 23 Sep 2026 · 1.48 → 1.48

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  • 23 Sep 2026 · 1.48 → 1.48

    Ingest update: added a follow-up source.