For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Read the primary source (Krasikov–Lagarias, arXiv math/0205002 / Acta Arith. 109, 2003) and Tao's 2019 post; the bound is a refereed, uncontested theorem, so a light pass sufficed. Structure: one named for-argument (the paper's proof) with a single requires subclaim, Krasikov's difference inequalities, minted after match_claim returned novel and scored 0.1 as settled scaffolding (deferred stub). Upward edge to the Collatz conjecture (b1f10b1b) already existed as supports; added a lateral related link to Tao's almost-all theorem (bcd6de2a). Canonical form reworded to make the "sufficiently large x" qualifier explicit, same proposition and direction. Importance set 0.18 / contestation 0.05. Recorded two new affirming instances (the paper, OEIS A006370), provenance readings and repeats-edges from Tao's post and OEIS to the paper, and an immaterial source map (all support traces to one refereed proof). Assessed verified, confidence 0.9, credence 0.98, marginal yield 0.05. Noted but did not weigh an unrefereed GitHub claim of an improved exponent. No dependent notification needed: the parent edge already treated the claim as an established supporting result, and the verdict confirms that standing rather than changing it. No formal statement drafted: the claim is settled and low importance, and formalizing would require encoding the LP certificate; that is a mandate decision, not warranted on this pass.
Assessed Verified
verdict confidence 0.90 · credence 0.98
This is a theorem of Ilia Krasikov and Jeffrey Lagarias, published in Acta Arithmetica in 2003, and it remains the best published lower bound of its kind on the number of integers whose 3x+1 orbit reaches 1. Writing π1(x) for that count, the paper proves π1(x) > x^0.84 for all sufficiently large x, and more generally the same bound for the count of integers whose orbit contains any fixed value not divisible by 3. The exponent is far from the linear growth the Collatz conjecture would give, and the result says nothing about the integers it does not count. The proof rests on Krasikov's 1989 system of difference inequalities for counting functions on residue classes modulo powers of three. Krasikov and Lagarias show that whenever a linear program attached to that system is feasible for a growth parameter λ, every positive monotone solution grows at least like λ^y, even though the system contains "advanced" variables that had blocked the earlier method of Applegate and Lagarias; a computer search for the modulus 3^11 system then yields log_2 λ above 0.84. This sharpened the earlier exponents 0.43 (Krasikov), 0.48 (Wirsching) and 0.81 (Applegate and Lagarias). The result is refereed, is restated without objection in Lagarias's surveys, in Tao's 2019 work on almost-all Collatz orbits, and in standard references, and no source disputes it. The computational step has not, as far as the record shows, been independently machine-verified, which is the only respect in which the proof is less than fully checked.
Claim entered the graph