{"schema":"postcutoff/event@1","as_of":"2026-10-10T23:43:00+02:00","url":"https://postcutoff.com/e/2026-09-07-erdos-1122-monotone-additive-gpt-6-astra/","md":"https://postcutoff.com/e/2026-09-07-erdos-1122-monotone-additive-gpt-6-astra/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-07-erdos-1122-monotone-additive-gpt-6-astra","date":"2026-09-07","date_precision":"day","short_title":"Erdős's 1946 conjecture on almost-monotone additive functions (Erdős problem #1122) proved in an anonymous Zenodo preprint whose proofs and Lean code were generated by GPT-6 Astra; Mangerel builds on it to settle Sárközy's 2001 conjectures","deck":null,"takeaway":"On Sept 7, 2026 an anonymous author posted \"Almost everywhere monotone additive functions\" on Zenodo.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Zenodo: Almost everywhere monotone additive functions (Anonymous, Erdős Problem 1122)","url":"https://doi.org/10.5281/zenodo.22651918","type":"paper","group":"primary","domain":"doi.org"},{"n":2,"title":"arXiv 2610.08424: A. P. Mangerel, On Sárközy's Local Extrema Conjectures","url":"https://arxiv.org/abs/2610.08424","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv 2108.12351: A. P. Mangerel, Additive functions in short intervals, gaps and a conjecture of Erdős","url":"https://arxiv.org/abs/2108.12351","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"Erdős Problem #1122 (T. F. Bloom, erdosproblems.com)","url":"https://www.erdosproblems.com/1122","type":"discussion","group":"reactions","domain":"erdosproblems.com"}],"official":3,"filed":"2026-10-10","updated":"2026-10-10","orgs":["OpenAI"],"title":"Erdős's 1946 conjecture on almost-monotone additive functions (Erdős problem #1122) proved in an anonymous Zenodo preprint whose proofs and Lean code were generated by GPT-6 Astra; Mangerel builds on it to settle Sárközy's 2001 conjectures","summary":"On Sept 7, 2026 an anonymous author posted \"Almost everywhere monotone additive functions\" on Zenodo. It claims a proof of Erdős's 1946 conjecture (Erdős problem #1122) that an additive function which decreases only on a set of density zero must be c·log n. The paper says GPT-6 Astra proposed the proofs and generated a Lean 4 formalization (cited results taken as hypotheses). On Oct 6 Alexander Mangerel, who had made the earlier partial progress, called it \"an AI-assisted proof\" and used its method to resolve two 2001 conjectures of Sárközy (arXiv 2610.08424).","key_facts":["Problem (Erdős 1946, Ann. of Math. 47; erdosproblems.com #1122): if f is additive and #{n ≤ X : f(n+1) < f(n)} = o(X), must f(n) = c log n? Erdős proved it when f never decreases or when f(n+1) − f(n) → 0","Earlier partial progress: Mangerel (arXiv 2108.12351, published 2022) proved it when the decreases number ≪ X/(log X)^(2+c) and f(p) is not too large","Preprint: Anonymous, 'Almost everywhere monotone additive functions', Zenodo 10.5281/zenodo.22651918 (v1 Sept 7, 2026; v2 Sept 22 adds erdos-1122-lean.zip). About 64 views and 32 downloads by Oct 10","Method (abstract): deduce finite concentration without a growth assumption on higher prime powers, via a truncated normalization and bounded clipping that reduce to Mangerel's first-moment short-interval theorem, Ruzsa's second-moment estimate and Elliott's fourth-moment inequality","AI disclosure: 'GPT-6 Astra was used to propose the mathematical proofs and generate the Lean formalization. GPT-5.6 Sol and Claude Opus 5 were used for editorial review of the exposition. The author finished the final manuscript and takes full responsibility for its content.'","Lean 4: the formalization treats the cited results of Mangerel, Ruzsa, Erdős and Hildebrand as hypotheses, so it is not a full formal proof from Mathlib. We have not compiled it","Oct 6, 2026: Alexander P. Mangerel, 'On Sárközy's Local Extrema Conjectures' (arXiv 2610.08424): 'Recently, an AI-assisted proof of problem (ii') was obtained [1] that relied crucially on this work'. He says his paper is 'inspired by the method in [1]' and that the AI paper does not adequately describe its context. He resolves, in a strong form, Sárközy's two 2001 conjectures on multiplicative functions with finitely many local maxima or minima","erdosproblems.com/1122 (checked Oct 10): one proof claim listed, but the page was last edited April 1, 2026 and the problem is not yet marked solved. New proof claims there were suspended in early October"],"key_numbers":[],"tags":["math","number-theory","additive-functions","erdos-problems","lean","formal-verification","gpt-6-astra","ai-for-math"],"science":{"field":"mathematics","subfield":"analytic / probabilistic number theory (additive functions)","problem":"Erdős problem #1122 (Erdős 1946): additive functions that decrease only on a density-zero set must be c·log n","result":"Claimed proof that any real additive f with #{n ≤ X : f(n+1) < f(n)} = o(X) equals c·log n with c ≥ 0. Lean 4 formalization conditional on cited theorems. An expert in the area builds on it to settle Sárközy's 2001 local-extrema conjectures.","open_since":"1946","ai_system":["GPT-6 Astra","GPT-5.6 Sol","Claude Opus 5"],"human_role":"AI-led: by the disclosure, GPT-6 Astra proposed the proofs and wrote the Lean code; an anonymous human finished the manuscript. GPT-5.6 Sol and Claude Opus 5 did editorial review.","verification":{"key":"lean","label":"Lean-verified","text":"Preprint on Zenodo with a Lean 4 formalization that assumes cited results as hypotheses; used by a specialist (Mangerel) in follow-up work; not peer-reviewed; not marked solved on erdosproblems.com"},"status":"pending","shock":"An 80-year-old Erdős conjecture was settled anonymously on Zenodo with almost no attention, and it surfaced only when the field's leading expert cited it as the inspiration for his own new theorem."},"body_md":"## What happened\n\nIn 1946 Erdős proved that an additive function (f(ab) = f(a) + f(b) for coprime a, b) that never decreases must be a constant\nmultiple of log n. He conjectured that the same holds if it decreases only on a set of natural density zero. The question is\nlisted as problem #1122 on Thomas Bloom's erdosproblems.com. Alexander Mangerel made the best partial progress (2021–22).\n\nOn Sept 7, 2026 an anonymous author posted an 8-page proof on Zenodo, tagged \"Erdős Problem 1122\". A second version on Sept 22\nadded a Lean 4 project. The paper's \"Declaration of generative AI use\" says OpenAI's GPT-6 Astra proposed the proofs and\ngenerated the Lean formalization. The record drew little attention (about 60 views by Oct 10).\n\nIt surfaced on Oct 6, when Mangerel posted \"On Sárközy's Local Extrema Conjectures\" on arXiv. He cites the Zenodo paper as \"an\nAI-assisted proof\" of Erdős's conjecture that \"relied crucially\" on his earlier work. He describes its two-step strategy (finite\nconcentration, then rigidity), says his own paper is inspired by it, and uses the method to classify all positive multiplicative\nfunctions with rare local maxima or minima. This settles two 2001 conjectures of András Sárközy in strong form. Mangerel's\npaper itself has no AI-use statement that we found.\n\nWe found this through the Oct 10 arXiv sweep. It flagged Mangerel's paper only as \"claiming to settle conjectures\", because the\nAI mention sits in his introduction, not in a disclosure section. We found no press or social-media discussion.\n\n## Why it matters\n\nIt fits the autumn 2026 pattern of Erdős problems and old conjectures settled by frontier models and posted outside arXiv,\nanonymously or by unknown authors. It is also an early case of a specialist openly building new human research on an AI-generated\nproof. The proof has not been peer-reviewed, and the Lean code assumes the cited analytic theorems as hypotheses.","disputed":[{"label":"Verification","text":"Preprint on Zenodo with a Lean 4 formalization that assumes cited results as hypotheses; used by a specialist (Mangerel) in follow-up work; not peer-reviewed; not marked solved on erdosproblems.com","sources":[]}],"related":[{"id":"2026-10-06-erdos-problems-site-freezes-proof-claims","url":"https://postcutoff.com/e/2026-10-06-erdos-problems-site-freezes-proof-claims/","date":"2026-10-06","date_precision":"day","short_title":"erdosproblems.com freezes proof claims and drops 'open/solved' labels and solver credits after a wave of unexplained AI proofs","deck":null,"takeaway":"erdosproblems.com was where AI-for-math claims were counted and disputed, from the GPT-5 controversy of October 2025 to the 2026 waves of GPT-6 Astra and Claude results.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":2,"official":2,"filed":"2026-10-08","updated":"2026-10-08","orgs":["erdosproblems.com"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-09-30","updated":"2026-10-10","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-17-zeta-5-irrational-fauzan-lean-verified","url":"https://postcutoff.com/e/2026-09-17-zeta-5-irrational-fauzan-lean-verified/","date":"2026-09-17","date_precision":"day","short_title":"ζ(5) proved irrational","deck":"Aabir Fauzan's Zenodo preprint, the first such result since Apéry's ζ(3) in 1978, is formally verified in Lean within a week, one formalization written by Claude","takeaway":"It is the most famous number-theory result of the AI-assisted 2026 wave: a problem experts had worked on for about 48 years.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":5,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Aalto University","Google DeepMind","Anthropic"]}],"people":[{"id":"thomas-bloom","name":"Thomas Bloom","url":"https://postcutoff.com/person/thomas-bloom/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-10","type":"filed","text":"Created from the Oct 10 arXiv sweep (Mangerel 2610.08424 → Zenodo record)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-10","run":null,"sources_read":"The Oct 10 arXiv sweep (Mangerel 2610.08424 → Zenodo record)","updated":"2026-10-10","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":130,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":69,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":160,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":99,"in_training_data":false}],"short_url":null}