Implied by Korec's stronger descent bound
The argument
Because almost all N in natural density have an orbit minimum below N^θ for any fixed θ above log 3/log 4, and since N^θ < N for every N ≥ 2 when θ < 1, every such N has an iterate strictly below N; the claim is therefore the special case of Korec's theorem obtained by weakening the bound N^θ to N.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →For almost all N, Col_min(N) is below N^θ for any fixed θ greater than log3/log4 ≈ 0.7924.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 17, 2026
The inference is immediate: a bound N^θ with θ < 1 is strictly smaller than N for every N ≥ 2, and both theorems are stated in natural density, so Korec's result contains this one. The argument stands or falls with Korec's theorem that almost all orbits fall below N^θ for θ above log 3/log 4, a refereed 1994 result that has not been questioned; it serves here as independent corroboration, since the claim already has its own earlier and simpler proof.