Almost-all and density theorems
The argument
Because almost all orbits drop below their starting value, because almost all orbits, in logarithmic density, fall below any function tending to infinity, and because at least of order x^0.84 integers up to x reach 1, the downward drift predicted by the random-walk heuristic (each Syracuse step multiplies by about 3/4 on average) is rigorously confirmed for typical integers, which supports the conjecture that every integer descends to 1.
Does it work?
Not yet evaluated.
Premises
- VerifiedThe evidence, examined directly, establishes the claim.constitution →For any function f tending to infinity, almost all N in logarithmic density have Collatz orbit minimum below f(N).
- VerifiedThe evidence, examined directly, establishes the claim.constitution →For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.
- VerifiedThe evidence, examined directly, establishes the claim.constitution →For almost all N in natural density, the Collatz orbit of N drops below N.
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 16, 2026
The premises are established theorems, so the argument's weight lies entirely in the inferential step from "almost all" to "all", and that step is inductive rather than deductive: each result, including Tao's logarithmic-density theorem, is compatible with a density-zero set of exceptional orbits, and Tao's own account explains why the method cannot reach every integer. The argument therefore raises confidence in the conjecture without being able to establish it, and it gives no special protection against a single sporadic counterexample of the kind the 5n+1 analogy warns of.