For almost all N in natural density, the Collatz orbit of N drops below N.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
This is Terras's theorem (1976), the first rigorous result on the 3x+1 problem: the set of positive integers whose Collatz orbit eventually falls below the starting value has natural density one, or equivalently, the integers with infinite stopping time form a set of density zero. It was proved independently and almost simultaneously by Riho Terras, C. J. Everett (1977), H. Möller (1977), E. Heppner (1978) and Jean-Paul Allouche (1979), and it appears as established background in every survey of the problem, including Lagarias's annotated bibliography and Tao's 2019 paper.
The proof is elementary. The first k parities of an orbit depend only on N modulo 2^k, with each parity pattern realised by exactly one residue class, so the number of odd steps in the first k iterates is binomially distributed; a tail bound for binomial sums shows that all but an exponentially small fraction of residue classes have few enough odd steps that the k-th iterate is smaller than N, and letting k grow with N gives density one. Möller criticised the original write-up, and Terras supplied the missing details in a 1979 note; no objection has stood since. The theorem is also contained in later, stronger results proved in the same natural-density sense, notably Korec's theorem that almost all orbits fall below N^θ for any θ above log 3/log 4. It does not imply the Collatz conjecture, since a density-zero exceptional set may still be infinite, and since the descent it guarantees cannot be iterated directly.
Full reasoning: the evidence and decisions behind this verdict
Standing of the theorem. Terras, "A stopping time problem on the positive integers", Acta Arithmetica 30 (1976), 241–252, proves that the integers with finite stopping time have natural density one; Terras, "On the existence of a density", Acta Arithmetica 35 (1979), 101–102, supplies further details after Möller's 1978 criticism of the original argument. Everett, Advances in Mathematics 25 (1977), obtained the same result independently. Lagarias's annotated bibliography (arxiv.org/pdf/math/0309224, entries 164 and 165) records all of this and calls Terras's paper the first significant research paper on the 3x+1 function. Allouche's 2022 EMS Magazine survey (euromathsoc.org/magazine/articles/64) states the theorem, sketches the proof, and lists the five independent proofs of the late 1970s, remarking that their coincidence suggests the result is not very difficult. Tao's 2019 announcement cites it as the base case of the descent results his theorem improves. All three recorded instances affirm; a search found no source denying the theorem or disputing any of its proofs after 1979.
Direct check of the argument. The written form of the Terras–Everett argument was checked step by step: the parity-vector lemma follows by induction from T(2^(k+1)n + j) ≡ T(j) modulo 2^k; with a odd steps among the first k, T^k(N) = 3^a N / 2^k + β(N) with 0 ≤ β(N) < 3^a, so T^k(N) < N whenever 3^a · (N + 1) < 2^k · N, which holds for a < k·log 2/log 3 minus a bounded correction once N is large relative to k; the fraction of residue classes modulo 2^k with at least that many odd steps is at most η^k for some η < 1 by the Chernoff bound on the binomial distribution, since log 2/log 3 ≈ 0.63 exceeds the mean 1/2. Choosing k of order log N (or any function tending to infinity slowly enough that 2^k stays below N) makes the exceptional set of integers up to x of size O(x·η^k) plus a negligible boundary term, hence density zero. The two material subclaims weigh as follows: the parity-vector bijection is required and is undisputed textbook material; Korec's N^θ theorem is independent corroboration, since it strictly implies the claim for N ≥ 2 and was proved by a different route in a refereed journal (Math. Slovaca 44, 1994), though the claim's verdict does not depend on it.
Formal evidence. A public Lean 4 development (a September 2026 pull request in a small community repository) reports a complete formal proof that the set of positive integers with finite Collatz stopping time has natural density one, with an axiom list confined to Lean's standard classical foundation. It was not built or independently reviewed in this pass and is recorded only as corroboration of the ordinary kind; the verdict rests on the refereed literature and the direct check above. No formal statement is published on this claim.
What would change the verdict: a demonstrated gap common to all the independent proofs, which is implausible for an argument this short and this often re-derived, or a computed counterexample to density one, which is ruled out by the proof itself. Credence is placed just short of certainty only for the generic residual uncertainty of any informally checked proof.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because each of the 2^k parity patterns of length k is realised by exactly one residue class modulo 2^k, the number of odd steps among the first k iterates of the compact Collatz map T is binomially distributed over N, and for the exceptional N with at least k·log 2/log 3 odd steps the Chernoff–Hoeffding tail bound gives a proportion at most η^k for some η < 1. Writing T^k(N) = 3^a N / 2^k + β with β bounded by 3^a shows that any N large relative to k with a < k·log 2/log 3 satisfies T^k(N) < N; letting k grow slowly with N, the exceptional set has natural density zero, so almost all N descend below themselves.
The inference goes through and was checked directly: given the equidistribution of parity patterns, the binomial tail bound and the explicit form of the k-th iterate together force descent below N for all but an exponentially small fraction of residue classes, and letting k grow with N gives density one. The argument rests entirely on the parity-vector bijection modulo 2^k, an elementary induction that nobody disputes; the remaining steps are standard estimates carried in the written form. This is the proof found independently by Terras, Everett, Möller, Heppner and Allouche, and the one gap raised against Terras's original write-up was filled in his 1979 note.
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.
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.
Provenance
Where this claim has been said, linked to its canonical form.
it was shown by Terras that for almost all {N} (in the sense of natural density), one has {\mathrm{Col}_{\min}(N) < N}
Asserted without evidence of the source's own. Tao's announcement states Terras's theorem as established background with attribution and no proof; the evidence for it lies in Terras's 1976 paper and the independent proofs of the same period, not in this post. The stored copy of the post drops its typeset formulas, so the recorded passage's inequality cannot be checked mechanically against it. The quoted passage was not found in the stored copy of this source.
Theorem. The lower density of S is equal to 1, and the same holds for the density of S. The proof is based upon the study of residual classes modulo 2^a.
Survey of results toward the Collatz conjecture preceding Tao's theorem; S is defined as the set of n > 0 whose orbit contains some f^k(n) < n. Allouche sketches the parity-vector proof, notes his own 1979 proof, and credits independent proofs by Everett (1977), Möller (1977), Heppner (1978) and Terras (1976, 1979).
The source's own evidence bears what it asserts. Allouche states the theorem and gives the shape of its proof: the parity of the first a iterates depends only on the residue class modulo 2^a, the number of odd steps is distributed binomially, and a tail estimate for binomial sums finishes the argument. He also records that the result was found independently by Everett, Möller, Heppner and Terras, and remarks that the coincidence suggests it is not very difficult. Worth reading closely: It is the most accessible complete sketch of the proof and the clearest account of who proved the theorem independently and when.
He shows that the set of integers having a finite stopping time has natural density one. Some further details of this proof were supplied later in Terras (1979).
Annotated bibliography entry 164 on Terras (1976), "A stopping time problem on the positive integers", Acta Arithmetica 30, 241–252. Lagarias notes the main result was obtained independently and contemporaneously by Everett (1977), and (entry 165) that Terras (1979) supplied additional details after Möller (1978) criticized the 1976 proof.
Asserted without evidence of the source's own. Lagarias's annotated bibliography records the theorem as the main result of Terras's paper, notes that Everett obtained it independently and contemporaneously, and notes that Terras supplied further details of the proof in 1979 after Möller criticised the original. It is a summary of the primary literature rather than a proof.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.