{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-tenth-problem-rationals-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-hilbert-tenth-problem-rationals-claim/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":"2026-10-06-openai-hilbert-tenth-problem-rationals-claim","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Hilbert's tenth problem over ℚ is undecidable","deck":null,"takeaway":"Family 004 of OpenAI's 6 Oct 2026 release claims that no algorithm decides whether an integer polynomial has a rational zero, the rational-number version of Hilbert's tenth problem, open since Matiyasevich settled the integer case in 1970. The 62-page proof avoids defining the integers inside the rationals; it reduces integer solvability to infinitely many finite rational tests. It is not formalized and leans on three companion manuscripts. Frank Calegari said it 'threw me for a loop', and Hector Pasten called the strategy 'original'.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Hilbert's tenth problem over the rational numbers (6 Oct version)","url":"https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI: A pointwise 2-converse for elliptic curves with rational two-torsion","url":"https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI: Fontaine–Mazur modularity at the prime 2 (input to the proof)","url":"https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"arXiv 2610.20691: Zhi-Wei Sun, 86 unknowns for Hilbert's tenth problem over the rationals","url":"https://arxiv.org/abs/2610.20691","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Scientific American: The most exciting claims from OpenAI's heap of new proofs (Pasten on Hilbert's tenth)","url":"https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","type":"press","group":"press","domain":"scientificamerican.com"},{"n":6,"title":"Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct)","url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","type":"discussion","group":"reactions","domain":"galoisrepresentations.org"},{"n":7,"title":"Bill Gasarch: Conjecture that I thought would not be solved in my lifetime....were (11 Oct)","url":"https://blog.computationalcomplexity.org/2026/10/conjecture-that-i-thought-would-not-be.html","type":"discussion","group":"reactions","domain":"blog.computationalcomplexity.org"},{"n":8,"title":"oai-math-revised: community notes on the Hilbert's tenth manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims a negative answer to Hilbert's tenth problem over the rationals: no algorithm can decide whether a polynomial equation has a rational solution","summary":"\"Hilbert's tenth problem over the rational numbers\" (OpenAI, version of 6 Oct 2026, first dated 24 Sep, family 004) states: \"There is no algorithm which, given a polynomial f ∈ ℤ[X1, …, Xn], with n part of the input, decides whether f has a zero in ℚⁿ.\" The integer case was settled negatively in 1970, but over ℚ the problem stayed open because the expected route, an existential definition of ℤ in ℚ, is believed to be impossible (Mazur's conjecture). The manuscript instead builds, for each polynomial, an effective sequence of finite rational-solvability tests that all succeed exactly when the polynomial has an integer zero. It relies on companion papers on 2-converses and on Fontaine–Mazur modularity at 2, has no Lean formalization and has not been refereed.","key_facts":["Theorem 1.1 (verbatim): 'There is no algorithm which, given a polynomial f ∈ Z[X1, . . . , Xn], with n part of the input, decides whether f has a zero in Qn.' Paper: 62 pages, version of 6 Oct (citation update of the 24 Sep original); companion 'A pointwise 2-converse for elliptic curves with rational two-torsion' (91 pages, 7 Oct version)","Strategy (paper, §1.2): for each nonconstant f, 'an effectively generated sequence of finite tests' answerable by rational-solvability queries, with 'f has an integer zero ⇐⇒ every test succeeds'. Ingredients: compactness, a rank-one elliptic group whose point indices encode integers, local comparisons of elliptic logarithms, and 'a height bound from odd contact with five points' proved through a family of abelian surfaces and modularity","Why the obvious route was closed (paper, §1.1): Mazur conjectured that the real closure of the rational points of a variety has finitely many components, which 'would rule out a Diophantine definition of Z in Q'; Koenigsmann's universal definition of ℤ and Poonen's results on subrings ℤ[S⁻¹] did not give the existential definition. The number-field case was finished by Koymans–Pagano and by Alpöge–Bhargava–Ho–Shnidman, but ℚ is not a finitely generated ℤ-algebra","Dependencies inside the release: the paper cites the pointwise 2-converse, 'Fontaine–Mazur modularity at the prime 2' (family 010) and the Goldfeld/2-converse manuscript (family 006); 'Its modularity step uses the odd regular Fontaine–Mazur theorem at 2'. None of these is formalized","Frank Calegari (Persiflage, 8 Oct, verbatim): 'This one threw me for a loop.' 'As for Hilbert’s tenth problem itself, it seemed plausible to me that it was actually solvable over the rational numbers. I don’t think I would have put money on that, however.' He still hopes the problem is decidable for curves, which 'should be a consequence of conjectures that we believe'","Hector Pasten (Scientific American, 8 Oct): the proof uses two of his ideas from different areas, and 'even he hadn’t realized they could be put together in this way'. Verbatim: 'It’s extremely surprising to me that they connect … Their general strategy is original.'","Bill Gasarch (Computational Complexity blog, 11 Oct): 'OpenAI showed that this problem is also undecidable. It has not been Lean-formalized. Judging from the table of contents, the proof looks like it uses deep math.'","Follow-up: Zhi-Wei Sun, '86 unknowns for Hilbert's tenth problem over the rationals' (arXiv 2610.20691, 14 Oct): 'Recently, Hilbert's Tenth Problem over the field Q of rational numbers was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction'; he proves undecidability already for 86 rational unknowns","Questions raised: on Calegari's blog Maarten Derickx (10 Oct) pointed to a Facebook post by Jennifer Park saying the paper cites her work 'in a completely wrong way'; an anonymous reply said the point 'doesn’t seem at all a cogent point' and that the ideas need study. We did not see Park's post. The paper cites Eisenträger–Miller–Park–Shlapentokh on subrings of ℚ","Felipe Voloch (comment, 11 Oct) listed conjectural routes to decidability for curves of genus > 1 (ABC/Vojta with explicit constants, the section conjecture, Tate + Hodge + Fontaine–Mazur) and finiteness of Sha for genus 1"],"key_numbers":[],"tags":["math","number-theory","logic","computability","hilbert-problems","hilbert-tenth-problem","elliptic-curves","openai","openai-math-release","unverified","claim"],"science":{"field":"mathematics","subfield":"number theory / mathematical logic (Diophantine decidability)","problem":"Hilbert's tenth problem over ℚ: is there an algorithm deciding whether a polynomial with integer coefficients has a rational zero?","result":"Claimed negative answer: rational solvability is undecidable (number of variables part of the input), via a Turing reduction from integer solvability to finitely many rational tests","open_since":"1970","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed manuscript; no Lean formalization; depends on unverified companion manuscripts (families 006, 010)"},"status":"pending","shock":"Calegari had found it plausible that the rational case was decidable, and Pasten had not seen that his ideas could be combined this way; the proof works around the expected impossibility of defining ℤ existentially in ℚ."},"body_md":"## What happened\n\nHilbert's tenth problem (1900) asked for a procedure deciding whether a polynomial equation with integer coefficients has an\ninteger solution. Davis, Putnam, Robinson and Matiyasevich showed in 1970 that none exists. The same question with rational\nsolutions has been open ever since. The natural route, defining the integers inside the rationals by a polynomial condition, is\nexpected to fail: Mazur's conjecture on the topology of rational points would rule it out.\n\nOpenAI's manuscript takes a different route. For each polynomial it generates an infinite list of finite tests, each answerable\nwith finitely many questions about rational solvability, such that the polynomial has an integer zero exactly when every test\npasses. A rational-solvability algorithm would then decide integer solvability, which is impossible. The arithmetic work is to show\nthat some test fails whenever there is no integer zero. For this the paper encodes integers as multiples of a point on a rank-one\nelliptic curve and controls heights through a five-point covering construction, abelian surfaces and modularity. That modularity\nstep uses the release's own Fontaine–Mazur theorem at 2, and the argument also cites the release's 2-converse papers.\n\n## Why it matters\n\nA negative answer over ℚ would close one of the best-known questions left over from Hilbert's list, and it would do so without the\nexistential definition of ℤ that most experts thought unavailable. Because the proof is long, unformalized and depends on other\nunrefereed manuscripts in the same release, it should be treated as a serious claim awaiting expert verification.\n\n## Reactions\n\nSpecialists were surprised. Frank Calegari wrote that the result \"threw me for a loop\" and that he had thought the problem might be\ndecidable over ℚ. Hector Pasten told Scientific American that the proof combines two of his own ideas in a way he had not\nanticipated. Zhi-Wei Sun, who had reduced the integer problem to few unknowns, took the result as given and brought the rational\nversion down to 86 unknowns. On Calegari's blog a commenter relayed a complaint that the paper miscites earlier work on subrings of\nℚ; another commenter doubted that this mattered. No line-by-line check has been published.","disputed":[{"label":"Verification","text":"Unrefereed manuscript; no Lean formalization; depends on unverified companion manuscripts (families 006, 010)","sources":[]}],"related":[{"id":"2026-10-16-ai-assisted-results-october-15-listing","url":"https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/","date":"2026-10-16","date_precision":"day","short_title":"arXiv results with disclosed AI help, 15 Oct 2026 listing","deck":"Monodromy conjecture, Fried counterexamples, Godefroy–Lancien–Zizler question and more","takeaway":"One day's maths listing brought five results big enough for their own entries (Smale's mean value constant, Hopf with T³, class numbers ≤ 2000, the Delsarte rate, LLM-found quantum circuits) and about two dozen more with explicit AI statements.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":12,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim","url":"https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims BSD in Selmer corank ≤ 1 and Goldfeld's conjecture","deck":null,"takeaway":"Families 002 and 006 of OpenAI's 6 Oct 2026 maths release claim the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose p-power Selmer group has corank 0 or 1 at some prime, and Goldfeld's conjecture that half of all quadratic twists have analytic rank 0 and half rank 1. Together they would give full BSD for 100% of the quadratic twists of every curve. None of it is formalized, and the experts who have commented read it as a strong technical result in the rank ≤ 1 range that leaves the general conjecture untouched.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 001–125; manuscript introduction and references, Persiflage digest and comments, SciAm, Gasarch and Sun's arXiv abstract read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 001–125; manuscript introduction and references, Persiflage digest and comments, SciAm, Gasarch and Sun's arXiv abstract read","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":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-hilbert-tenth-problem"}