Explicit positive-density bounds for Collatz convergence
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A Collatz step sends an even n to n/2 and an odd n to 3n+1. What positive fraction of starting integers can be proved to reach 1 in logarithmically many ordinary steps, with explicit constants and an explicit starting threshold?
Why it matters and what counts as progress
Why it matters: Explicit density, time and threshold constants strengthen a qualitative positive-density Collatz result without proving all-orbit convergence.
What counts as progress: Increase the proved density, sharpen step or threshold constants, or extend target-reaching results. Positive density does not settle convergence for every starting value.
How the problem developed
Lech Mazur, with AI agents using ProofAtlas and Codex Announced that at least cX starting values n < X reach 1 within 10.46 log n ordinary Collatz steps for a fixed c > 0, with a Lean formalization.
- 14 Oct 2026 · Explicit refinement [2]
Kenny Lau; AxiomProver autonomously generated and formalized the mathematics Gave an explicit tower-sized lower bound for the number of odd n < N reaching 1 within 10.46 log n ordinary steps, and a 3.4881 log n bound in accelerated steps.
AI contribution
Explicit positive lower density for the Collatz Conjecture
Lau reports an explicit refinement of Mazur’s positive-density Collatz result: writing L=2^(2^(2^87640)), for every X≥2^L at least X/L positive integers n<X reach 1 within floor(10.46 log n) ordinary Collatz steps. The formal development also gives an odd-starting-value version with accelerated-step bounds and target-reaching variants for prescribed positive targets not divisible by three.
Sources
- Lech Mazur, Explicit positive-density Collatz convergence in logarithmic time
- Original statement and human–AI provenance
- Lech Mazur, Explicit positive-density Collatz convergence in logarithmic time
- Pinned convergence, step-count and target-specific statements
- Explicit thresholds and relation to Mazur’s theorem
Website timeline
Publication dates and editorial changes.
Lau–AxiomProver make positive-density Collatz convergence bounds explicit