{"schema":"postcutoff/event@1","as_of":"2026-10-18T22:10:00+02:00","url":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/","md":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/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":"2026-09-06-collatz-positive-density-log-time-mazur-codex","date":"2026-09-06","date_precision":"day","short_title":"Collatz positive-density convergence, AI-developed, Lean-checked","deck":null,"takeaway":"The first linear lower bound for Collatz-convergent integers (earlier bounds were X^0.84, then X^0.90): an explicit positive proportion of n < X reach 1 within 10.46 ln n steps. Developed mainly by AI agents, Lean-checked twice, not yet peer reviewed. The Collatz conjecture itself stays open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Paper: Explicit Positive-Density Collatz Convergence in Logarithmic Time (v2.1, PDF)","url":"https://www.proofatlas.ai/papers/positive-density-log-time-collatz/Mazur_Explicit_Positive_Density_Collatz_Convergence_in_Logarithmic_Time_v2.pdf","type":"paper","group":"primary","domain":"proofatlas.ai"},{"n":2,"title":"ProofAtlas formalization page","url":"https://www.proofatlas.ai/formalizations/positive-density-log-time-collatz/","type":"code","group":"primary","domain":"proofatlas.ai"},{"n":3,"title":"AxiomMath/CollatzPositiveDensity (independent Lean formalization)","url":"https://github.com/AxiomMath/CollatzPositiveDensity","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Lech Mazur on X: announcement (Sept 6, 2026)","url":"https://x.com/LechMazur/status/2096464472973975782","type":"discussion","group":"reactions","domain":"x.com"}],"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["ProofAtlas","OpenAI","Axiom Math"],"title":"Collatz: a positive fraction of integers provably reach 1 in 10.46 ln n steps; AI-agent proof (ProofAtlas + Codex) by Lech Mazur, Lean-formalized and re-formalized by Axiom","summary":"On Sept 6, 2026 Lech Mazur posted (X, ~204k views) \"Explicit Positive-Density Collatz Convergence in Logarithmic Time\": for explicit c > 0 and X₀, every X ≥ X₀ has at least cX positive integers n < X that reach 1 within (523/50) ln n ordinary Collatz steps. Previous counting bounds for convergent integers were sublinear (Krasikov–Lagarias X^0.84; Mazur's own July 2026 X^0.90). The paper's disclosure says generative AI \"contributed substantially to mathematical exploration, proof development, computational testing, formalization, and exposition\", using the ProofAtlas.ai harness with OpenAI GPT models through Codex. The theorem has a Lean 4 formalization (~42k added lines), and on Oct 9 Axiom Math published an independent Lean formalization (AxiomMath/CollatzPositiveDensity) checked with the Lean comparator. The constants are astronomically small/large and the conjecture remains open. No peer review yet.","key_facts":["Theorem 1.1: explicit rational c > 0 and integer X₀ with #{1 ≤ n < X : τ(n) ≤ (523/50) log n} ≥ cX for every X ≥ X₀; so convergence to 1 within 10.46 log n steps holds on a set of positive lower natural density","Prior bounds: Krasikov & Lagarias (Acta Arith. 2003) X^0.84 convergent integers ≤ X; Mazur, 'Certified x^0.90 Lower Bounds for Collatz Predecessor Sets' (ProofAtlas, July 17, 2026); Tao (Forum Math. Pi 2022): almost all orbits (log density) attain almost bounded values, which does not give convergence to 1","Paper's own limits: 'The density bound is extremely small and the cutoff extremely large; no practical numerical range is claimed'; 'The result does not settle the Collatz conjecture'. Axiom's README states the thresholds as N ≥ 2^2^2^2^87640 and fraction 1/2^2^2^87640","AI disclosure (quoted): 'Generative AI contributed substantially to mathematical exploration, proof development, computational testing, formalization, and exposition, not only to language editing. Lech Mazur directed the AI-assisted research and is the author responsible for the manuscript and its claims. The ProofAtlas.ai harness was used for the AI-assisted research and verification workflow, with OpenAI GPT models accessed through Codex. Anthropic Claude was also used for manuscript review.'","Mazur on X: 'Developed primarily by AI agents using the ProofAtlas.ai harness and Codex, with some high-level guidance from me (which may or may not have helped)'","Lean (paper §9): main theorem formalized with no admitted proof or project axiom, only propext, Classical.choice, Quot.sound; builds on earlier formalizations of Tao's almost-boundedness and Mazur's natural-density extension (July 21, 2026). ProofAtlas page: 'Lean checked', 'Unfinished proof steps: None', 'Publication: Review pending'","Axiom Math repo AxiomMath/CollatzPositiveDensity (created 2026-10-09) formalizes the theorem plus variants for other targets s coprime to 6 and t not divisible by 3, with a challenge file verified by the Lean comparator"],"key_numbers":[],"tags":["math","number-theory","collatz","dynamical-systems","lean","codex","proofatlas","axiomprover","ai-for-math"],"science":{"field":"mathematics","subfield":"number theory / arithmetic dynamics (Collatz 3x+1 problem)","problem":"Collatz conjecture: lower bounds on the number of integers below X whose orbit reaches 1","result":"A positive lower natural density of integers reach 1 within 10.46 ln n ordinary Collatz steps (linear count cX, with explicit but astronomically small c), improving earlier sublinear bounds X^0.84 and X^0.90.","open_since":"1937","ai_system":["OpenAI GPT models via Codex","ProofAtlas.ai harness","Claude (manuscript review)","Axiom agent tooling (second formalization)"],"human_role":"AI-led: 'developed primarily by AI agents' with high-level guidance from Lech Mazur, who directed the work and is responsible for the claims","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (ProofAtlas build, then an independent Axiom Math formalization with comparator check); not peer reviewed; no published expert assessment found"},"status":"pending","shock":"A linear lower bound on Collatz-convergent integers, a step past bounds that had stood since 2003, came from an amateur directing AI agents."},"body_md":"## What happened\n\nLech Mazur (who posted the GPT-5.6 Pro Sendov proof in August) announced on Sept 6, 2026 that a fixed positive fraction of all\nintegers below every large enough cutoff reach 1 under the Collatz map within 10.46 ln n ordinary steps. Until then, unconditional\ncounting results only gave sublinear numbers of convergent integers: X^0.84 (Krasikov–Lagarias 2003) and X^0.90 (Mazur's own\nAI-assisted ProofAtlas paper of July 2026). Tao's 2019 theorem is a different kind of statement: almost every orbit gets below any\nslowly growing function, which does not give convergence to 1. The paper combines finite ternary mixing, a weighted inverse-orbit\nconstruction from one fixed convergent seed, and a counting estimate. Both constants are given by finite expressions, but they are\nastronomically extreme.\n\nThe disclosure credits generative AI with the exploration, proof development, testing, formalization and exposition, through the\nProofAtlas.ai harness with OpenAI GPT models in Codex, and Claude for manuscript review. ProofAtlas lists the Lean build as passed\nwith no unfinished steps and the publication as \"review pending\". On Oct 9, Axiom Math published a separate Lean formalization\nthat also covers other targets, with a comparator-checked challenge file.\n\nWe missed this for six weeks. It was not on arXiv and had no Techmeme or press coverage. The sweep found it through Axiom's GitHub\nrepo (which cites the X post) while checking the AxiomMath account on Oct 18.\n\n## Why it matters\n\nCollatz is one of the best-known open problems. This does not prove it, but it is the first result showing that convergent\nintegers make up a positive proportion, with a logarithmic time bound. It was produced mainly by AI agents directed by a\nnon-academic, and checked in Lean twice. No expert review or published reaction was found by Oct 18, so its standing in the\nfield is unknown.","disputed":[{"label":"Verification","text":"Formal proof in Lean (ProofAtlas build, then an independent Axiom Math formalization with comparator check); not peer reviewed; no published expert assessment found","sources":[]}],"related":[{"id":"2026-10-16-openai-2d-area-law-lean-formalized-axiom","url":"https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/","date":"2026-10-16","date_precision":"day","short_title":"OpenAI's 2D area law proof Lean-verified by Axiom","deck":null,"takeaway":"One of the physics claims in OpenAI's October maths release, an entropy area law for gapped 2D lattice ground states (open since Hastings' 1D proof in 2007), now has a complete Lean 4 proof with a frozen statement layer and comparator check, produced by AI agents under a human auditor.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math","OpenAI"]},{"id":"2026-08-05-sendov-conjecture-proved","url":"https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/","date":"2026-08-05","date_precision":"day","short_title":"Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro","deck":"Tao simplifies and formalises it","takeaway":"It is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},{"id":"2026-07-28-lean-kernel-soundness-bug-collatz","url":"https://postcutoff.com/e/2026-07-28-lean-kernel-soundness-bug-collatz/","date":"2026-07-28","date_precision":"day","short_title":"Lean kernel soundness bug #14576","deck":"An AI-assisted 'disproof' of the Collatz conjecture passes Lean's kernel and nanoda; the 'Summer of Soundness Bugs'","takeaway":"\"Verified in Lean\" has become the main evidence behind AI labs' math claims: OpenAI's Navier–Stokes blow-up, 300 of 719 results in its October release, and Anthropic's formal-math repository.","category":"research","category_label":"Research","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":9,"official":5,"filed":"2026-10-09","updated":"2026-10-09","orgs":["Lean FRO","OpenAI"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[{"id":"2026-09-06-lechmazur-collatz-positive-density","title":"Lech Mazur: AI agents prove a positive fraction of integers reach 1 under Collatz in 10.46 ln n steps (Lean-formalized)","url":"https://postcutoff.com/p/2026-09-06-lechmazur-collatz-positive-density/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-18","type":"filed","text":"Created (found via AxiomMath/CollatzPositiveDensity during the evening science check; paper PDF, ProofAtlas page and X post via fxtwitter read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-18","run":null,"sources_read":"Found via AxiomMath/CollatzPositiveDensity during the evening science check; paper PDF, ProofAtlas page and X post via fxtwitter read","updated":"2026-10-18","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":129,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":68,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":159,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":98,"in_training_data":false}],"short_url":"https://postcutoff.com/s/collatz-positive-density"}