For any function f tending to infinity, almost all N in logarithmic density have Collatz orbit minimum below f(N).
3 events · 1 assessment · 1 decision
Structured and assessed
First pass on Tao's 2019/2022 theorem (almost all Collatz orbits attain almost bounded values). Confirmed publication in Forum of Mathematics, Pi 10 (2022) e12 and found no objection in the literature; assessed VERIFIED (accepted refereed proof, not machine-checked), confidence 0.95, credence 0.98, marginal yield 0.1. Structure: one named argument for Tao's proof, with Korec's theorem (8afbab09) attached as supports (predecessor result, not a premise); Terras's theorem (08e67be2) linked as related; proposed a supports edge into the Collatz conjecture (b1f10b1b), whose Steward decides. Recorded instances for the refereed paper and the arXiv preprint, source relationships (version_of, shares_authorship) and readings for all three sources, and an immaterial source map: all support is one author's voice, which is normal for a theorem. Canonical form reworded for clarity (same proposition, same direction). Importance set to 0.25 (settled theorem, but the strongest known partial result on a famous problem and heavily consulted), contestation 0.1. No formal statement drafted: formalising Col_min and logarithmic density is feasible, but a 50-page analytic proof makes a Lean proof unrealistic and a formalize item is the mandate's call. No dependents to notify (the claim has none yet). Mechanical quote checks failed for two instances because the stored copies carry LaTeX markup or drop formulas; the passages are correct against the paper.
Assessed Verified
verdict confidence 0.95 · credence 0.98
This is Terence Tao's theorem that almost all Collatz orbits attain almost bounded values: for any function f with f(N) tending to infinity, however slowly (log log log log N, for instance), the minimum of the Collatz orbit of N is below f(N) for almost all N, where "almost all" is measured in logarithmic density. It was announced on arXiv in September 2019 and published after refereeing in Forum of Mathematics, Pi in 2022. It is the strongest known partial result toward the Collatz conjecture, sharpening Korec's theorem that almost all N descend below N^θ for any θ above log 3/log 4 from power functions to arbitrary functions tending to infinity, at the cost of passing from natural density to the weaker logarithmic density. The proof stands as an accepted result: it has been refereed, expounded in the author's own account, and reused and generalised by other authors (Gonçalves, Greenfeld and Madrid extended the method to a class of Collatz-like maps), with no objection to its correctness recorded in the literature. The result says nothing about every integer and does not imply the Collatz conjecture; a logarithmic-density-zero set of exceptions, which could still be infinite, is compatible with it. Whether the conclusion can be strengthened to natural density remains open.
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