{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-02-linnik-constant-3-99-naslund-claude-gpt-astra/","md":"https://postcutoff.com/e/2026-10-02-linnik-constant-3-99-naslund-claude-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":"2026-10-02-linnik-constant-3-99-naslund-claude-gpt-astra","date":"2026-10-02","date_precision":"day","short_title":"Linnik's constant ≤ 3.99, written by Claude Opus 5.5 and GPT Astra","deck":null,"takeaway":"Linnik proved in 1944 that the least prime congruent to a mod q is at most q^L for an absolute constant L. Heath-Brown brought L down to 5.5 in 1992 and Xylouris to 5. On 2 Oct Eric Naslund posted on Hexagon a computer-assisted proof that L ≤ 3.99. It rests on a weighted form of a Heath-Brown lemma that Heath-Brown said no neat way of achieving 'seems available'. The paper says it was written, and the research carried out, by Claude Opus 5.5 and GPT Astra 6 under Naslund's prompting and supervision, with some formalization by Grok. Two components are Lean-verified on Palomar; the final bound itself is not. Hexagon lists human understanding as 'some parts', and we found no expert assessment as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2610.00018: Linnik's constant is at most 3.99 (Naslund; Claude Opus 5.5, GPT Astra 6, Grok)","url":"https://hexagonmath.org/abs/2610.00018v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Palomar PALOMAR-2026-10-01-000020: Lean components of the Linnik 3.99 proof","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-10-01-000020&version=1","type":"code","group":"primary","domain":"palomar-registry.org"},{"n":3,"title":"Hexagon 2610.00151: Square-difference-free sets of exponent 0.7580758 (Naslund)","url":"https://hexagonmath.org/abs/2610.00151v1","type":"paper","group":"primary","domain":"hexagonmath.org"}],"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","xAI"],"title":"Linnik's constant is at most 3.99: an AI-written, computer-assisted proof posted by Eric Naslund lowers Xylouris's exponent 5 for the least prime in an arithmetic progression","summary":"\"Linnik's constant is at most 3.99\" (contributed by Eric Naslund; Hexagon 2610.00018, 2 Oct 2026) claims P(a, q) < q^3.99 for every sufficiently large q, uniformly in a, where P(a, q) is the least prime ≡ a (mod q). The previous record exponent was 5 (Xylouris). The main new ingredient is a graded near-density estimate for zeros of Dirichlet L-functions, a weighted form of Heath-Brown's Lemma 12.1 of 1992, combined with the zero-location and far-density method of Heath-Brown and Xylouris and a much larger machine-checked case analysis. Provenance: \"This paper was written by Claude Opus 5.5 and GPT Astra 6 under my prompting and supervision.\" Lean (Palomar PALOMAR-2026-10-01-000020) covers the graded near lemma and the soundness of the certificate checker, not the 3.99 bound.","key_facts":["Claim (abstract, verbatim): 'We give a computer-assisted proof that P(a,q) ≪ q^3.99, improving the exponent 5 of Xylouris. More precisely, P(a,q) < q^3.99 for every sufficiently large q, uniformly in a.'","New ingredient (abstract): 'a graded near-density estimate for zeros of Dirichlet L-functions, a weighted form of Heath-Brown's Lemma 12.1 of the kind he asked for in 1992.' The Palomar record adds that Heath-Brown 'remarks (section 16, item 9) no neat way of achieving' it 'seems available'","Scale: Heath-Brown and Xylouris 'closed their final case analyses with 14 and 21 main cases, each by a chain of inequalities evaluated in floating point'; the new proof uses a far larger certified case analysis (leaf linear programs with exact certificates)","Provenance statement (verbatim): 'This paper was written by Claude Opus 5.5 and GPT Astra 6 under my prompting and supervision. The research was carried out by the same models under my prompting and supervision, with some of the formalization work also done by xAI's Grok. My recommendation is for the reader to explore this paper with AI as well, and use their favorite model to ask the paper questions.' Hexagon labels: 'Primarily AI-generated text', 'Human understanding: some parts'","Formal status (Palomar abstract): two components are Lean-verified, the graded near lemma and a sound exact checker for the leaf-LP certificates; 'That bound itself is NOT formally verified here: the paper's zero-location arguments, case analysis and exterior regimes have conventional proofs and computer checks only.' The certificates 'are checked by three separate programs, one of them formally verified'","Same contributor: 'Square-difference-free sets of exponent 0.7580758' (Hexagon 2610.00151, 8 Oct), D(N) ≥ N^0.7580758…, above Krachun's 0.752796…; 'written by GPT-6-Astra and revised by Claude Opus 5.5 under my prompting and supervision'; Naslund understands 'the core argument' but has 'not carefully read some of the optimizations'","Not checked: no referee, no expert comment found (web, Hexagon, HN) as of 29 Oct"],"key_numbers":[],"tags":["math","number-theory","analytic-number-theory","linnik-constant","primes-in-progressions","hexagon","lean","palomar","claude-opus-5-5","gpt-6-astra","grok"],"science":{"field":"mathematics","subfield":"analytic number theory","problem":"Linnik's constant: the least L with P(a, q) ≪ q^L for the least prime in every reduced residue class mod q","result":"Claimed L ≤ 3.99 (P(a, q) < q^3.99 for all large q), improving Xylouris's L ≤ 5","open_since":"1944","ai_system":["Claude Opus 5.5","GPT-6 Astra","Grok 4.7"],"human_role":"Naslund prompted and supervised; the models did the research and wrote the paper; Grok helped with formalization","verification":{"key":"lean","label":"Lean-verified","text":"Hexagon preprint; computer-assisted certificates checked by three programs; two components Lean-verified (Palomar), the final bound not formalized"},"status":"pending","shock":"A record that moved from 5.5 to 5 over two decades of human work is claimed down to 3.99 by models, with the human contributor recommending that readers query the paper with AI."},"body_md":"## What happened\n\nEvery reduced residue class mod q contains primes, and Linnik showed in 1944 that the first one is below q^L for a fixed L. Pushing L down\nis a benchmark of analytic number theory, because it depends on how well one controls zeros of Dirichlet L-functions near 1. Heath-Brown\nreached 5.5 in 1992 with an elaborate case analysis, and Xylouris lowered it to 5.\n\nOn 2 Oct Eric Naslund deposited on Hexagon a paper claiming 3.99. Its key lemma is a weighted density estimate that Heath-Brown had wished\nfor, and its case analysis is closed by exact certificates checked by three separate programs. Naslund writes that Claude Opus 5.5 and GPT\nAstra 6 did the research and the writing under his prompting and supervision. A Palomar record holds Lean proofs of the new lemma and of the\ncertificate checker's soundness. It states that the bound itself is not formally verified.\n\n## Why it matters\n\nIf correct, it is the largest improvement to Linnik's constant since Heath-Brown's 1992 paper, and it comes from model-driven research\nwith only partial human understanding claimed. The partial formalization shows where to look: the zero-location arguments and the case\nanalysis outside the certified leaves still need a human or machine check.","disputed":[{"label":"Verification","text":"Hexagon preprint; computer-assisted certificates checked by three programs; two components Lean-verified (Palomar), the final bound not formalized","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"]},{"id":"2026-08-18-palomar-lean-registry","url":"https://postcutoff.com/e/2026-08-18-palomar-lean-registry/","date":"2026-08-18","date_precision":"day","short_title":"Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims","deck":null,"takeaway":"Formal verification became the main way to trust AI mathematics in 2026.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":6,"official":6,"filed":"2026-09-29","updated":"2026-10-29","orgs":["Lean FRO","ICARM"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (Hexagon records of 2610.00018 and 2610.00151, Palomar API record; 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 records of 2610.00018 and 2610.00151, Palomar API record; 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":155,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":94,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":185,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":124,"in_training_data":false}],"short_url":"https://postcutoff.com/s/linnik-constant-3-99-written"}