{"schema":"postcutoff/event@1","as_of":"2026-10-16T12:39:00+02:00","url":"https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/","md":"https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/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":"2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","date":"2026-10-15","date_precision":"day","short_title":"All 16,453,493 imaginary quadratic fields with class number ≤ 2000 determined unconditionally, using OpenAI's Siegel-zero proof made explicit in Lean with Claude Opus 5.5","deck":null,"takeaway":"The first downstream theorem built on OpenAI's claimed exclusion of Landau–Siegel zeros: the unconditional class-number table jumps from h ≤ 100 (Watkins, 2004) to h ≤ 2000, with the key bounds checked in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.21511: Imaginary quadratic fields of class number at most 2000 (Kimura)","url":"https://arxiv.org/abs/2610.21511","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.21099: A simpler proof that there are no Landau–Siegel zeros (Kerr)","url":"https://arxiv.org/abs/2610.21099","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv 2610.15483: On the OpenAI proof that Landau-Siegel zeros do not exist (Banks)","url":"https://arxiv.org/abs/2610.15483","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"OpenAI: Uniform exclusion of Landau–Siegel zeros (1 Oct manuscript)","url":"https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"}],"official":4,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic"],"title":"All 16,453,493 imaginary quadratic fields with class number ≤ 2000 determined unconditionally, using OpenAI's Siegel-zero proof made explicit in Lean with Claude Opus 5.5","summary":"arXiv 2610.21511 (15 Oct 2026) by Iwao Kimura makes the argument of OpenAI's October manuscript on Landau–Siegel zeros explicit, (1 − β) log q ≥ 5811 × 10⁻⁷ for every real zero of every primitive real character, proves it and the resulting class-number bound in Lean, and with a sieve computation determines all imaginary quadratic fields with class number at most 2000: exactly 16,453,493, the largest |d| being 1,062,350,083. Unconditional lists previously stopped at h ≤ 100. Claude Opus 5.5 (via Claude Code) did the explicit constants, Lean files, sieve program and most of the text under the author's direction. The same day Bryce Kerr posted a simpler ChatGPT-assisted proof of the Siegel-zero theorem (2610.21099).","key_facts":["Paper: 'Imaginary quadratic fields of class number at most 2000', Iwao Kimura, arXiv 2610.21511 [math.NT], 15 Oct 2026","Theorem 1 (Lean): for χ primitive real of conductor q ≥ 3 and L(β, χ) = 0, (1 − β) log q ≥ 5811 · 10⁻⁷, and ≥ 1193/200000 once log q ≥ 35.0131","Theorem 2: exactly 16,453,493 imaginary quadratic fields have class number ≤ 2000; largest |d| = 1,062,350,083 (h = 1978). For h ≤ 1000: 4,115,897 fields, largest |d| = 227,932,027 (h = 996). Matches the GRH-conditional data of Cremona–Sutherland and Holmin et al.","Prior state: Watkins (2004) determined the fields with h ≤ 100; 'To our knowledge, no unconditional list beyond h ≤ 100 was known'","Status of the parts: the class-number bound (Theorem 10) is proved in Lean and checked by the Lean kernel, a replay and an independent kernel (nanoda), reusing OpenAI's Lean formalization of the Siegel-zero manuscript, so it 'does not depend on the correctness of the text of [SZ]'; the sieve computation for 6·2²⁸ ≤ |d| ≤ 1.65·10¹⁵ is argued in the paper but its program is not formally verified; small discriminants come from Mosunov–Jacobson tables","AI statement (Section 8): 'The following was done with Claude Opus 5.5 (Anthropic), used through Claude Code, under the author's direction: (a) the explicit form of the argument of [SZ] … (b) the Lean files … (c) the sieve program … (d) the text of Sections 2 to 7 and of this section, written from the author's outline'. 'The author posed the problem and made the decisions that shaped the work' and 'worked through the proofs in the paper'","Companion item the same day: Bryce Kerr, 'A simpler proof that there are no Landau–Siegel zeros', arXiv 2610.21099 (4 pages): replaces OpenAI's interpolation-determinant analysis with the pigeonhole principle and a result of Masser; 'ChatGPT assisted with the mathematical development and preparation of this paper'; Kerr thanks William Banks for a draft of his exposition"],"key_numbers":[],"tags":["math","number-theory","class-number-problem","siegel-zeros","lean","claude","claude-opus-5-5","openai"],"science":{"field":"mathematics","subfield":"analytic and computational number theory","problem":"Gauss class number problem: list all imaginary quadratic fields of a given class number (effectively, unconditionally)","result":"Unconditional determination of all imaginary quadratic fields with class number ≤ 2000 (16,453,493 fields), via an explicit, Lean-verified version of OpenAI's Landau–Siegel zero exclusion.","open_since":"1801","ai_system":["OpenAI internal model (Siegel-zero proof and its Lean formalization)","Claude Opus 5.5 (Claude Code)"],"human_role":"Human-directed: the author chose the problem, checks and trusted base; Claude did constants, Lean, code and most text; the underlying theorem is OpenAI's","verification":{"key":"lean","label":"Lean-verified","text":"Key theorems Lean-verified (two kernels); sieve computation not formally verified; unrefereed preprint; OpenAI's manuscript itself unrefereed"},"status":"pending","shock":"Two weeks after an AI lab posted a claimed proof that Siegel zeros do not exist, a single author used another lab's model to turn it into a 20-fold extension of a classical table, resting on machine-checked proofs."},"body_md":"## What happened\n\nSiegel's 1935 theorem says class numbers of imaginary quadratic fields grow, but its constant is ineffective because a real\n\"Siegel zero\" of a Dirichlet L-function could not be ruled out. That is why complete lists of fields with a given class number\nstopped at h ≤ 100 (Watkins, 2004), apart from GRH-conditional tables.\n\nOpenAI's internal model claimed on 1 Oct 2026 that Siegel zeros do not exist, with a Lean formalization but no explicit constant.\nWilliam Banks wrote out an explicit version on 11 Oct. On 15 Oct Iwao Kimura posted [arXiv 2610.21511](https://arxiv.org/abs/2610.21511),\nwhich proves its own explicit constant, (1 − β) log q ≥ 0.0005811, in Lean (reusing OpenAI's formalization, not Banks's text), derives\nan explicit class-number lower bound, and runs Watkins's sieve up to |d| ≈ 1.65 · 10¹⁵. Result: exactly 16,453,493 imaginary\nquadratic fields have class number at most 2000, the largest discriminant in absolute value being 1,062,350,083.\n\nThe AI roles are listed in a dedicated section. OpenAI's model produced the Siegel-zero manuscript and its Lean proof (\"We took no\npart in them\"). Claude Opus 5.5 in Claude Code worked out the explicit constants, wrote the Lean files and certificates, the sieve\nprogram and comparison scripts, and drafted Sections 2–7 \"from the author's outline\". The author made the design decisions,\nincluding basing the result on a formal proof rather than on OpenAI's text, and checked the proofs and sources.\n\nThe same listing has Bryce Kerr's four-page [arXiv 2610.21099](https://arxiv.org/abs/2610.21099), \"A simpler proof that there are no\nLandau–Siegel zeros\", which replaces OpenAI's interpolation determinant with a pigeonhole argument and Masser's theorem. \"ChatGPT\nassisted with the mathematical development and preparation of this paper.\"\n\nNo Hacker News thread or press coverage was found on 16 Oct (Algolia, web search); X was not searched.\n\n## Why it matters\n\nThis is the first concrete consequence drawn from OpenAI's Siegel-zero claim, and it is built so that it does not trust the claim's\nprose: the needed bounds are machine-checked. If the claim holds, effective results that number theory waited 90 years for become\nroutine computations. Two independent rewrites (Banks, Kerr) and now a Lean-based application within two weeks also show how fast\nspecialists are absorbing and checking the October release.\n\nCaveats: unrefereed; the large sieve computation is not formally verified (it agrees with GRH-conditional tables where they overlap).","disputed":[{"label":"Verification","text":"Key theorems Lean-verified (two kernels); sieve computation not formally verified; unrefereed preprint; OpenAI's manuscript itself unrefereed","sources":[]}],"related":[{"id":"2026-10-11-landau-siegel-zeros-banks-explicit-account","url":"https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/","date":"2026-10-11","date_precision":"day","short_title":"William Banks writes out OpenAI's Landau–Siegel proof with explicit constants","deck":"(1 − β) log q ≥ 0.003 for every real character, and says he verified it in full","takeaway":"The first detailed human account of OpenAI's claim that Landau–Siegel zeros do not exist gives an explicit, effective constant. Banks used Claude to draft it but says he worked through every line.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-16","orgs":["OpenAI","Anthropic","University of Missouri"]},{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":9,"filed":"2026-10-07","updated":"2026-10-15","orgs":["OpenAI"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-15","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-16","type":"filed","text":"Created from the PDFs of arXiv 2610.21511 and 2610.21099"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-16","run":null,"sources_read":"The PDFs of arXiv 2610.21511 and 2610.21099","updated":"2026-10-16","human_review":null,"version":{"date":"2026-10-16"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":168,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":107,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":198,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":137,"in_training_data":false}],"short_url":"https://postcutoff.com/s/all-16453493-imaginary-quadratic"}