{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-11-09-non-split-cartan-121-two-ai-proofs/","md":"https://postcutoff.com/e/2026-11-09-non-split-cartan-121-two-ai-proofs/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-11-09-non-split-cartan-121-two-ai-proofs","date":"2026-11-09","date_precision":"day","short_title":"X_ns+(121): two AI-generated proofs in two days","deck":null,"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2611.00231: Arango-Piñeros, Najman, Zureick-Brown, Rational points on the non-split Cartan modular curve of level 121","url":"https://hexagonmath.org/abs/2611.00231","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Hexagon 2611.00255: Jaswal, The rational points of the non-split Cartan modular curve of level 121","url":"https://hexagonmath.org/abs/2611.00255","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":3,"title":"Hexagon 2611.00234: Najman, Torsion of elliptic curves over number fields of degree 10","url":"https://hexagonmath.org/abs/2611.00234","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":4,"title":"Background: Rouse, Sutherland, Zureick-Brown, ℓ-adic images of Galois for elliptic curves over Q (arXiv 2106.11141)","url":"https://arxiv.org/abs/2106.11141","type":"paper","group":"primary","domain":"arxiv.org"}],"official":4,"filed":"2026-11-10","updated":"2026-11-10","orgs":["Anthropic","OpenAI","University of Zagreb","UMass Amherst","Amherst College"],"title":"Two independent, mostly AI-generated proofs that the non-split Cartan modular curve X_ns+(121) has only CM rational points, completing the 11-adic images of Galois","summary":"On 9 and 10 Nov 2026 two Hexagon deposits claimed, by different methods, that the genus-511 modular curve X_ns+(121) has exactly seven rational points, all CM. With Rouse, Sutherland and Zureick-Brown's earlier work this would finish the classification of 11-adic Galois images of non-CM elliptic curves over Q. Both papers state the mathematics was produced mainly by AI models and neither has been fully checked by a human; one is co-authored by Zureick-Brown, whose conjecture it settles.","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"],"key_numbers":[],"tags":["math","number-theory","modular-curves","galois-representations","claude-opus-5-5","gpt-6-1-sol","gpt-6-astra","hexagon"],"science":{"field":"mathematics","subfield":"number theory (arithmetic geometry)","problem":"Rational points on X_ns+(121); the level-121 case of the Rouse–Sutherland–Zureick-Brown conjecture on ℓ-adic images","result":"Claimed: X_ns+(121)(Q) consists of 7 CM points, so the 11-adic images of non-CM elliptic curves over Q are determined","open_since":"","ai_system":["Claude Opus 5.5","GPT-6.1 Sol","GPT-6 Astra","Claude Code"],"human_role":"Humans chose the problem and directed; AI produced almost all the mathematics, computations and text; partial (2611.00231) or no (2611.00255) human checking","verification":{"key":"none","label":"Unverified claim","text":"Two independent unrefereed Hexagon deposits with code; not peer-reviewed or formalized"},"status":"pending","shock":"Two groups reached the same open case by different methods within a day, both mainly through AI."},"body_md":"## What happened\n\nRouse, Sutherland and Zureick-Brown (2021) classified the ℓ-adic images of Galois for non-CM elliptic curves over Q except for a\nfew cases; for ℓ = 11 the missing piece was whether the curve X_ns+(121) has non-CM rational points. Two Hexagon deposits now\nclaim it does not.\n\nThe first (9 Nov) is by Arango-Piñeros, Najman and Zureick-Brown and avoids any model of the genus-511 curve, mapping it to\nX_ns+(11) and reducing to seven genus-5 curves. The second (10 Nov), by Manvir Jaswal, uses quadratic Chabauty at two primes.\nBoth authors' statements say AI models produced almost all of the mathematics: \"We have not yet fully checked the proofs\"\n(2611.00231); \"None of this contribution has been checked and understood by any human\" (2611.00255, Hexagon declaration).\n\nNajman also deposited the degree-10 torsion classification the same day, continuing his degree 6–9 paper.\n\n## Why it matters\n\nA named open case in a well-known classification programme, claimed twice in two days by independent AI-driven efforts, one with\nthe conjecture's own author. Until referees or a formalization check them, both are claims.","disputed":[{"label":"Verification","text":"Two independent unrefereed Hexagon deposits with code; not peer-reviewed or formalized","sources":[]}],"related":[{"id":"2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman","url":"https://postcutoff.com/e/2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman/","date":"2026-11-04","date_precision":"day","short_title":"Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol","deck":null,"takeaway":"Filip Najman deposited on Hexagon a paper that determines every group occurring as the torsion of an elliptic curve over a number field of degree 6, 7, 8 or 9, including three groups not previously known to occur.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-05","updated":"2026-11-06","orgs":["OpenAI","Anthropic","University of Zagreb"]},{"id":"2026-10-06-hexagon-repository-llm-assisted-math","url":"https://postcutoff.com/e/2026-10-06-hexagon-repository-llm-assisted-math/","date":"2026-10-06","date_precision":"day","short_title":"Hexagon launches: a new arXiv-style repository for LLM-assisted mathematics papers","deck":null,"takeaway":"It is the first dedicated infrastructure for AI-generated mathematics backed by leading mathematicians, and it accepts work that no human claims to understand.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-29","orgs":["Hexagon Mathematics Foundation"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-10","type":"filed","text":"Created (science agent; Hexagon API records and provenance of 2611.00231, 2611.00255, 2611.00234)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-10","run":null,"sources_read":null,"updated":"2026-11-10","human_review":null,"version":{"date":"2026-11-10"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":193,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":132,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":223,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":162,"in_training_data":false}],"short_url":"https://postcutoff.com/s/x-ns-121-two"}