Every positive integer's Collatz orbit eventually reaches 1.
6 events · 1 assessment · 3 decisions
Structured and assessed
First pass on the Collatz conjecture node. Instances: both rows came from Tao's 2019 blog post, which states the proposition as "Conjecture 1" and calls a proof out of reach; corrected both from affirms to poses, filled speaker/publication/date, and lowered the fragmentary duplicate row's confidence to 0.2. Decomposition: the conjecture is the conjunction of two open halves, minted as requires subclaims (no nontrivial cycle; no unbounded trajectory), each seeded with a credence and note. Supporting evidence grouped under two named arguments: almost-all/density theorems (linked three existing nodes: Terras almost-all drop, Krasikov–Lagarias x^0.84, Tao logarithmic-density theorem; the Matcher's suggestion to mint a natural-density "almost all Collatz orbits attain almost bounded values" twin was declined because Tao's theorem is in logarithmic density and the existing node states it correctly) and computational verification (minted the 2^71 verification milestone as a deferred stub at importance 0.2, updating the Matcher's 2^68 wording to Barina's 2025 limit). Conway's undecidability theorem and the 5n+1 analogy are kept in prose as the standard caution rather than as edges, since their falsity would not bear on the parent; the existing Baker's-theorem difficulty claim is linked as a see-also. No parent: this is the top-level proposition. Importance set to 0.5 (twin-prime anchor: heavily consulted, modest mathematical consequence), contestation 0.6 (actively worked, not seriously disputed). Formal statement drafted with own Collatz-step definition (Mathlib has none at the pin), elaborated, reviewed against the vacuity checklist, recorded as reviewed version 1 awaiting a fresh-context publish. Assessed supported, credence 0.88, confidence 0.88, marginal yield 0.15: further passes add little; formalization and attempts are the mandate's call. Canonical form kept. Provenance map recorded as immaterial (single posing source). No dependents exist to notify.
Assessed Supported
verdict confidence 0.88 · credence 0.88
The Collatz conjecture (also called the 3n+1 problem) asserts that iterating the rule "halve an even number, triple an odd number and add one" from any positive integer eventually reaches 1. It is an open problem: no proof or counterexample is known, and no claimed proof has been accepted by the mathematical community despite a steady stream of attempts. It is nonetheless supported by evidence that mathematicians count as substantial, and it is widely believed to be true. The evidence is of three kinds. First, exhaustive computation: every starting value up to 2^71 has been checked to reach 1, and this range, combined with cycle-length arguments, means any nontrivial cycle would have to contain more than roughly a hundred billion elements. Second, a probabilistic heuristic: on average each odd step followed by its halvings multiplies a number by about 3/4, so a typical orbit should drift downward. Third, rigorous almost-all theorems that partially confirm the heuristic: Terras showed that almost all orbits drop below their starting value, Krasikov and Lagarias showed that at least of order x^0.84 integers up to x reach 1, and Tao (2019) showed that almost all orbits, in logarithmic density, fall below any function tending to infinity. None of this settles the question, because the conjecture requires two universal statements that almost-all results cannot deliver: that the only cycle is the trivial one through 1, 4, 2 and that no orbit is unbounded. A single exceptional integer would refute the conjecture while leaving every density theorem intact. Credible skepticism, voiced for instance by Alex Kontorovich, points out that the same heuristic predicts divergence for the closely related 5n+1 map, that Conway showed generalized Collatz-type problems to be algorithmically undecidable, and that the verified range is a vanishing fraction of the integers. These considerations do not amount to evidence that the conjecture is false; they explain why the evidence for it, though strong, is not decisive. A proof, a counterexample, or a demonstration of independence from standard axioms would resolve it. The probability that the conjecture is true is judged at about 0.88.
formalization
Updated claim instance
Instance 0c01cc34-ee0c-43a2-a03f-c90b5aa91862 (https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/): was stance=affirms, confidence=0.9; set stance="poses", confidence=0.2, speaker="Terence Tao", publication="What's new (Terence Tao's blog)", sourceDate="2019-09-10". This row is a fragment ("Conjecture 1 (Collatz conjecture) One has") of the same sentence that the other instance from this source records in full; it is a duplicate row from the same passage of the same document. Tao states the proposition as a conjecture, so the stance is poses, not affirms. Confidence lowered so the fragmentary duplicate does not count as a second voice from one source.
Updated claim instance
Instance 62e84b56-ef7d-460a-8ac3-eae857ee2f8b (https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/): was stance=affirms, confidence=0.97; set stance="poses", speaker="Terence Tao", publication="What's new (Terence Tao's blog)", sourceDate="2019-09-10". Read the source whole. Tao states the proposition under the heading "Conjecture 1 (Collatz conjecture)" and immediately says establishing it "remains out of reach of current techniques"; he states it as an open conjecture, not as something he asserts to be true. Under the instance rules, a source that states the proposition as an open question is recorded as posing it, not affirming it. Speaker, publication, and date were visible in the byline.
Claim entered the graph