AI Proof WatchKeep up with AI contributions

Explicit positive lower density for the Collatz Conjecture

Kenny Lau

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 for this record

What was already known

Mazur had already reported positive-density convergence to 1 in logarithmic time. Earlier density and almost-bounded-orbit results address distinct conclusions. This work quantifies and formalizes the predecessor construction.

AI involvement AI-autonomous

Lau set the goal of formalizing and improving Mazur’s estimates. The provenance statement says AxiomProver autonomously produced the new mathematics and Lean formalization, while Lau takes responsibility for the introduction and summarized argument.

The provenance explicitly says AxiomProver autonomously produced the new explicit positive-density mathematics from Lau’s stated goal; Lau supplied exposition and responsibility.

Importance Notable

Provides explicit constants and thresholds, plus a scoped reported Lean checking record, for a previously qualitative positive-density statement.

Limitations

The constants and threshold are enormous and the guaranteed proportion extremely small. This does not prove convergence for every integer or density-one convergence. Ordinary and accelerated Collatz steps are distinct; for even prescribed targets the accelerated statement concerns the odd part. The manuscript discloses human understanding of some parts. AIProofWatch has not rebuilt the proof, audited every dependency, or established outside acceptance.

Technical verification Formal (Lean)

A completed rigorous check is documented for the scoped statement. Palomar: Lean, NanoDa and con-ron (reported): The compared CollatzPosDens statements at 3a38bfbb70379e051b51155def1423846fab3a90, including Theorem 1.3, Remark 1.4 and Corollary 1.2; logarithmic-time positive-density bounds, not convergence for all starts; standard Lean axioms only.

Formal checking

Reported checking of the stated formal scope; not reproduced by AIProofWatch. Separate kernels and author-produced checkers do not establish outside researcher review.

Checker
Palomar: Lean, NanoDa and con-ron (reported)
Scope
The compared CollatzPosDens statements at 3a38bfbb70379e051b51155def1423846fab3a90, including Theorem 1.3, Remark 1.4 and Corollary 1.2; logarithmic-time positive-density bounds, not convergence for all starts; standard Lean axioms only.
Basis
Reported by the linked checker
Check date
2026-10-15
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

Named human researchers explain or reconstruct the scoped central mechanism in the cited paper/account; credit for that explanation is separate from technical checking. Lau set the goal of formalizing and improving Mazur’s estimates. The provenance statement says AxiomProver autonomously produced the new mathematics and Lean formalization, while Lau takes responsibility for the introduction and summarized argument.

Correction and publication history

2026-10-16 · publication

Primary statement, hypotheses, prior bounds, AI role and evidence scope reviewed on 16 October 2026 against every current catalog/update page and existing immutable revisions. New recent scoped contribution; ordinary notifying editor publication.

Read this published revision