X_ns+(121): two AI-generated proofs in two days
Awaiting review
Importance: major (4 of 5)The takeaway
Arango-Piñeros, Najman and Zureick-Brown (Claude Opus 5.5, GPT-6.1 Sol, GPT-6 Astra) and, separately, Manvir Jaswal (an agent system on Claude Opus 5.5) posted proofs on Hexagon that X_ns+(121) has exactly 7 rational points, all CM. Neither is fully human-checked.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Unverified claim
- Importance
- Major (4 of 5)
- Last verified
- 10 November 2026
Your AI and this story
- GPT-6 Astra193 days after its cutoff
- Claude Opus 5.5132 days after its cutoff
- Gemini 4 Hydrogen223 days after its cutoff
- Grok 4.7162 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 132 days before it.
Key facts
- Hexagon 2611.00231 (9 Nov): Santiago Arango-Piñeros (UMass Amherst), Filip Najman (Zagreb), David Zureick-Brown (Amherst College); descent via X_ns+(121) → X_ns+(11) (rank-1 elliptic curve), Mordell–Weil sieve, 7 genus-5 hyperelliptic curves over a degree-15 field, Stoll’s Selmer-group Chabauty
- Provenance (2611.00231): ‘Claude Opus 5.5, OpenAI’s GPT-6.1 Sol and GPT-6 Astra produced almost all of the mathematics of the first versions of the proofs and carried out the computations … We have not yet fully checked the proofs’
- Hexagon 2611.00255 (10 Nov): Manvir Jaswal, independent proof by quadratic Chabauty at p = 17 and 43 with no model of the curve; ‘produced by an AI research system that the author designed, built and directs, running on Anthropic’s Claude Opus 5.5 in Claude Code’; human understanding ‘none’, the author ‘reran the computations from the deposit’
- Same day, Najman (Hexagon 2611.00234): the 69 torsion groups of elliptic curves over number fields of degree 10, four of them newly shown to occur; ‘GPT-6.1 Sol produced almost all of the mathematics … the paper should be regarded as raw data’
- Status: unrefereed, not formalized; no expert assessment found as of 10 Nov
What happened
Rouse, Sutherland and Zureick-Brown (2021) classified the ℓ-adic images of Galois for non-CM elliptic curves over Q except for a few cases; for ℓ = 11 the missing piece was whether the curve X_ns+(121) has non-CM rational points. Two Hexagon deposits now claim it does not.
The first (9 Nov) is by Arango-Piñeros, Najman and Zureick-Brown and avoids any model of the genus-511 curve, mapping it to X_ns+(11) and reducing to seven genus-5 curves. The second (10 Nov), by Manvir Jaswal, uses quadratic Chabauty at two primes. Both authors’ statements say AI models produced almost all of the mathematics: “We have not yet fully checked the proofs” (2611.00231); “None of this contribution has been checked and understood by any human” (2611.00255, Hexagon declaration).
Najman also deposited the degree-10 torsion classification the same day, continuing his degree 6–9 paper.
Why it matters
A named open case in a well-known classification programme, claimed twice in two days by independent AI-driven efforts, one with the conjecture’s own author. Until referees or a formalization check them, both are claims.
What is disputed or not yet verified
| Verification | Two independent unrefereed Hexagon deposits with code; not peer-reviewed or formalized |
|---|
Sources
4 sources from 2 sites. Numbers match the chips in the text.
4 sources: 4 primary
Primary
- Hexagon 2611.00231: Arango-Piñeros, Najman, Zureick-Brown, Rational points on the non-split Cartan modular curve of level 121hexagonmath.org, paper
- Hexagon 2611.00255: Jaswal, The rational points of the non-split Cartan modular curve of level 121hexagonmath.org, paper
- Hexagon 2611.00234: Najman, Torsion of elliptic curves over number fields of degree 10hexagonmath.org, paper
- Background: Rouse, Sutherland, Zureick-Brown, ℓ-adic images of Galois for elliptic curves over Q (arXiv 2106.11141)arxiv.org, paper
Changes
- Filed (science agent; Hexagon API records and provenance of 2611.00231, 2611.00255, 2611.00234)