{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra/","md":"https://postcutoff.com/e/2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra/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":"2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra","date":"2026-10-28","date_precision":"day","short_title":"Kisin proves Serre's 1960 Artin-character conjecture with GPT Astra","deck":null,"takeaway":"In 1960 Serre conjectured that a function built from fixed-point multiplicities of a finite group acting on a regular local ring is the character of an ℓ-adic representation. It is the higher-dimensional analogue of the Artin character of local fields. Tomoyuki Abe proved the positive-characteristic case (Inventiones, 2025). On 28 Oct Harvard's Mark Kisin posted the mixed-characteristic case on Hexagon, which finishes the conjecture. He writes that 'extensive use of GPT Astra was made both to arrive at the proofs … and to write a first draft', and that 'significant human effort' went into checking them. Part of the work was done at 'The Overhang', a San Francisco meeting on 9–10 Sept 2026.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2610.00597: Serre's conjecture on the Artin character for a regular local ring (Kisin)","url":"https://hexagonmath.org/abs/2610.00597v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Background: T. Abe, On the Serre conjecture for Artin characters in the geometric case (arXiv 2405.19601)","url":"https://arxiv.org/abs/2405.19601","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Background: Beraldo, Pippi, Proof of the Deligne–Milnor conjecture (arXiv 2410.02327)","url":"https://arxiv.org/abs/2410.02327","type":"paper","group":"primary","domain":"arxiv.org"}],"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Harvard University","OpenAI"],"title":"Mark Kisin completes Serre's 1960 conjecture on the Artin character of a regular local ring, with 'extensive use of GPT Astra' for the proofs and the first draft (Hexagon)","summary":"\"Serre's conjecture on the Artin character for a regular local ring\" (Mark Kisin, Harvard; Hexagon 2610.00597, 28 Oct 2026) proves that for a finite group G acting on a regular local ring A with isolated fixed points, and every prime ℓ invertible in A, the Artin function a_{A,G} is the character of a finite-dimensional Q_ℓ-representation of G, as Serre conjectured in 1960. The equal-characteristic-p case is due to Abe, and residue characteristic zero is easy. Kisin treats mixed characteristic with an equivariant deformation that reduces to isolated singularities, where Beraldo and Pippi's twisted conductor formula applies. Hexagon labels: \"A mix of human-written and AI-generated text\", \"Human understanding: all parts\".","key_facts":["Statement (Theorem 1.1, verbatim): 'For every prime ℓ invertible in A, a_{A,G} is the character of a finite-dimensional Q_ℓ-representation of G.' The paper: 'The following assertion was conjectured by Serre in 1960 [18, §6].' For a discrete valuation ring a_{A,G} is the classical Artin character","Prior cases, per the paper: rings containing a field of positive characteristic by Tomoyuki Abe (Invent. Math. 241, 2025); residue characteristic zero by 'an easy calculation'; arithmetic surfaces by Abbes (2000) via the vanishing-cycles approach of Kato, Shuji Saito and Takeshi Saito","Method: after completion and approximation, an equivariant deformation replaces Spec A by a regular flat scheme whose non-smooth locus is a single point, so that Beraldo–Pippi's twisted conductor formula (from their proof of the Deligne–Milnor conjecture) gives the character","Provenance statement (verbatim): 'Extensive use of GPT Astra was made both to arrive at the proofs of the results presented here, and to write a first draft of the manuscript. Significant human effort was needed to check and refine the arguments, and make the paper more readable. Part of this work was done at The Overhang, held in San Francisco 9-10, September 2026.'","Status: Hexagon preprint (no peer review, no formalization), posted 28 Oct; no expert comment found as of 29 Oct"],"key_numbers":[],"tags":["math","number-theory","algebraic-geometry","commutative-algebra","serre-conjecture","artin-character","hexagon","gpt-6-astra"],"science":{"field":"mathematics","subfield":"arithmetic geometry, commutative algebra","problem":"Serre's 1960 conjecture: the Artin function of a finite group acting on a regular local ring with isolated fixed points is the character of an ℓ-adic representation","result":"Proof in mixed characteristic, completing the conjecture (positive characteristic: Abe 2025; residue characteristic zero: classical)","open_since":"1960","ai_system":["GPT-6 Astra"],"human_role":"Kisin directed the work; GPT Astra was used extensively to find the proofs and write the first draft; Kisin checked, refined and rewrote","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed Hexagon preprint by a leading number theorist"},"status":"pending","shock":"A 66-year-old conjecture of Serre is finished by one of the field's leading figures, who says a model did much of the proving and the first draft."},"body_md":"## What happened\n\nFor a local field, the Artin character measures how a Galois group acts on the ring of integers, and its being a true character is a\nclassical theorem. In 1960 Serre conjectured the same for any finite group acting on a regular local ring with an isolated fixed point:\nthe function built from the lengths of the fixed-point loci should be the character of an ℓ-adic representation. Abe settled the case of\nrings over a field of characteristic p in 2025, leaving mixed characteristic open.\n\nMark Kisin's 18-page Hexagon paper does the mixed-characteristic case. It deforms the action to one with an isolated singularity and then\napplies Beraldo and Pippi's twisted conductor formula. In the provenance statement he says GPT Astra was used extensively to find the\nproofs and draft the paper, and that checking and refining the arguments took significant human effort. Part of the work was done at\n\"The Overhang\", a San Francisco meeting on 9–10 Sept.\n\n## Why it matters\n\nIt closes a conjecture of Serre's, and the paper is signed by a leading arithmetic geometer who puts it on Hexagon with a plain account\nof the model's role. Kisin says he understands all of it, and it is one of the most senior uses of a model on a classical problem that we\nhave recorded. It is not yet refereed.","disputed":[{"label":"Verification","text":"Unrefereed Hexagon preprint by a leading number theorist","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing-2","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-2/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct 2026 (part 2)","deck":"Claude 'largely carried out' Lipshitz's Khovanov research, ChatGPT found Soundararajan's proof and wrote Etingof's and Mathew's papers, George Elliott's free-product classification 'carried out using ChatGPT', and a first read of Hexagon's backlog","takeaway":"A second pass over the 28 Oct arXiv announcement and a first full read of Hexagon (hexagonmath.org), which the sweep had never covered, found results with large AI roles signed by senior mathematicians. Robert Lipshitz writes that his Khovanov paper's research 'was largely carried out by various Claude models'. Kannan Soundararajan's proof of a problem from Montgomery's CBMS book 'was found by ChatGPT 6'. ChatGPT 6 Pro wrote Pavel Etingof's answer to a 1999 question and produced Akhil Mathew's answer to a question of Drinfeld. George Elliott's classification of reduced free products of nuclear C*-algebras 'was carried out using ChatGPT'. Eight items have their own entries: the unforced 2D Euler non-uniqueness claims, the Erdős–Lovász covering-number bound, UPP^cc under intersection, Kisin's Serre conjecture, Linnik's constant ≤ 3.99, 3D Yang–Mills on a torus, random k-SAT and Chenglong Ma's Hovey–Strickland provenance claim.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":30,"official":29,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"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":"2026-10-29","type":"filed","text":"Created (Hexagon record, provenance statement and PDF; found in a first full read of the Hexagon API)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Hexagon record, provenance statement and PDF; found in a first full read of the Hexagon API","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":181,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":120,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":211,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":150,"in_training_data":false}],"short_url":"https://postcutoff.com/s/kisin-proves-serre-1960"}