Everything required to settle the Collatz conjecture

Define the Collatz map on the positive integers:1

Conjecture (Collatz). For every integer n ≥ 1, the orbit of n eventually reaches 12. Analysts usually work with the accelerated (Syracuse) map T(n) = (3n+1)/2 for odd n, which folds in the guaranteed halving after every odd step3.

The formal statement is a Π⁰₂ sentenceof arithmetic — “for all n there exists a k with C^k(n) = 1” — and that logical shape matters4. Paul Erdős put a $500 bounty on it and delivered the field's most quoted verdict: “Mathematics may not be ready for such problems.”5

• Open. Neither proved nor disproved, for 89 years and counting.
• Verified by computation for every starting value up to roughly 2⁷¹ ≈ 2.4 × 10²¹6. Every one of those ~10²¹ orbits fell to 1.
• Computation settles nothing here.The conjecture quantifies over all integers; 10²¹ is measure zero of them. Worse, the problem is famous for late-arriving structure — orbits like 27's wander to 9,232 before collapsing, and there is no known bound f such that checking up to f(n) certifies n7.
• The strongest theorem(Tao, 2019) says orbits go “almost bounded” for “almost all” n — a spectacular result that is still, by its own architecture, unable to reach all n (§5, §8).

Vocabulary, then the instrument. For a starting value n: total stopping time σ∞(n) is the number of steps to reach 1; stopping time σ(n) is the number of steps until the orbit first dips below n8; the peakis the orbit's maximum excursion. The chart is log-scale — odd steps (red) kick the value up, even steps (green) grind it down.

Things worth trying above: 27, the canonical wild ride (111 steps, peak 9,232 — a 342× overshoot of its start)9; 837,799, holder of the longest flight below one million (524 steps); and 2⁶⁸ + 1, a value just past the peer-reviewed verification frontier — your browser verifies it in exact arithmetic in milliseconds, which is simultaneously impressive and, per §2, mathematically worthless.

Why do essentially all mathematicians believe the conjecture? Because in aggregate the map is a biased random walk pointed downhill. Follow an odd number through 3n+1 and the subsequent run of halvings: the trailing zeros of 3n+1 behave like fair coin flips, so the expected number of halvings per tripling is 2, giving an expected factor of 3/4 < 1 per odd step10. The same heuristic predicts the average flight length: σ∞(n) ≈ 2 ln n / ln(4/3) ≈ 6.95 ln n — the dashed curve below, threading a cloud it has no right to fit this well.

Annotations on the cloud: the horizontal banding comes from orbit-merging — huge families of n share tail trajectories, so their flight counts differ by small offsets11. The gold record-holders drifting upward roughly logarithmically are exactly what the random-walk model predicts for maxima of ~n independent-ish flights. The heuristic explains everything visible in this chart and proves none of it — the gap between those two clauses is the conjecture.

The honor roll, annotated. Note the shape of every entry: almost all, at least, no cycle shorter than — quantifier concessions that mark exactly where each technique ran out of road.

A settlement must destroy, or exhibit, each of these — and both failure modes really occurin the map's nearest relatives, which is the strongest evidence that no soft argument will ever work:

• Failure A — a nontrivial cycle.Some orbit repeats without ever touching 1. Known constraints: every element > 2⁷¹, length in the hundreds of billions, and no m-cycle shape with m ≤ 91 (§5).
• Failure B — a divergent orbit. Some orbit marches to infinity. Nothing whatsoever excludes this; the density theorems only say such orbits are statistically invisible16.

The moral of the lab: 3n−1(equivalently, Collatz on negative integers) has three known cycles — so “maps of this shape funnel into one cycle” is simply false as a general principle. 5n+1has positive drift and apparently divergent orbits — so “maps of this shape come back down” is also false. Any correct proof must therefore consume the exact constants 3 and +1 and the positivity of the domain17.

The problem has been transplanted into a dozen fields, each yielding structure — and each inheriting the difficulty intact. The major equivalences:

2-adic dynamics: the map is secretly a coin flip

The accelerated map extends to the 2-adic integers ℤ₂, where it is topologically conjugate to the one-sided shift — measure-preserving, ergodic, as random as a dynamical system can be18.

Computability: the neighborhood is undecidable

Conway (1972) proved that iterating generalized Collatz functions (piecewise-linear maps chosen by residue class) can simulate any Turing machine, and built FRACTRAN to demonstrate it; Kurtz–Simon later pinned the generalized problem as Π⁰₂-complete19.

The other transplants

• Diophantine:cycle existence ⇔ solvability of the cycle equation, where linear forms in logarithms of 2 and 3 rule (all of §5's cycle results live here). The relevant open sharpening — how well log₂ 3 can be approximated — connects to deep unsolved problems in transcendence theory.
• Functional equations: Berg–Meinardus recast the conjecture as a statement about solutions of a functional equation for generating functions on the unit disk — analytic, elegant, and so far inert.
• The predecessor tree:Wirsching's program studies the tree of all ancestors of 1; the conjecture says the tree is everything. Yields the x^0.84 density bounds; stuck below exponent 1.
• Rewriting systems / tag systems: Collatz is equivalent to termination of a 3-symbol tag system — connecting it to term-rewriting termination, another land of undecidability20.

The obstructions, stated as sharply as the literature allows:

• No conserved or monotone quantity.The standard way to prove termination is a Lyapunov/ranking function that provably decreases. None is known for Collatz, and the map's violent excursions (27 → 9,232) show any candidate must tolerate enormous transient growth — it would have to encode, in effect, the answer itself21.
• The 2-adic/3-adic incompatibility. The even step is perfectly natural in base 2; the odd step in base 3. The map shuttles information between two incommensurable number systems, and mathematics has famously weak tools for statements coupling multiplicative structure in 2 and 3 simultaneously22.
• Statistical methods cannot finish.Terras-to-Tao arguments control density; the conjecture demands totality. An exceptional set of density zero can still be infinite — and one exception is fatal. Closing “almost all” to “all” requires per-orbit, non-statistical control that no one has23.
• The verification asymmetry (§1, note 4).No finite computation can even in principle certify a divergent orbit — so the “experimental” route to disproof is half-closed, and the proof route gets no inductive foothold from the 10²¹ verified cases.

The deliverable, itemized. There are exactly three exits, and each has a non-negotiable bill of materials:

Path A — prove it true

1. [ ]Exclude all nontrivial cycles, for every period.Current transcendence bounds cap out at m-cycles, m ≤ 91. A full exclusion needs either radically stronger lower bounds on |2^b − 3^a| (beyond what Baker-type theory delivers) or a structurally new cycle argument.
2. [ ]Exclude divergent orbits — every single one.The hard core. Requires per-orbit control: a decreasing ranking function, a covering-system induction that closes, or an entirely new mechanism. Nothing in the current literature — including Tao's theorem — even claims a path to this.
3. [ ]Pass the no-go filter of §6.The argument must visibly fail for 3n−1 and for 5n+1, and must fail on the negative integers. If you cannot point to the exact step where it does, the proof is wrong before refereeing begins.

Path B — prove it false

1. [ ]Exhibit a counterexample orbit.A cycle: every element > 2⁷¹, hundreds of billions of terms, verifiable but unfindable by any current search strategy. Or divergence: which needs a proof of escape for a specific orbit — a type of theorem never yet achieved for any qn+r map (§6, note 16). Disproof is, if anything, harder than proof.

Path C — prove it unprovable

1. [ ]Show independence from PA (or ZFC).Conway's universality makes this less absurd than it sounds, but no independence technique (forcing, incompleteness coding) has ever been made to grip a specific, natural Π⁰₂ arithmetic statement like this one. Note the fine print: independence would notmean “true” — that shortcut works for Π⁰₁ statements only, and Collatz is not one.

Anything presented as a settlement that does not check one of these boxes — in full, with the quantifier “all” and not “almost all” — is a contribution to §5 at best and to the crackpot file at worst24.

The canonical shelf, in reading order:

• J. C. Lagarias, The 3x+1 Problem and Its GeneralizationsAmerican Mathematical Monthly 92 (1985). The classic survey; still the right first read. His two annotated bibliographies (arXiv math/0309224 and math/0608208) index essentially the entire literature.
• J. C. Lagarias (ed.), The Ultimate Challenge: The 3x+1 ProblemAMS, 2010. The book of record on the problem.
• T. Tao, Almost all orbits of the Collatz map attain almost bounded valuesForum of Mathematics, Pi (2022); arXiv:1909.03562. The state of the art, with an unusually candid discussion of why the method cannot finish the job.
• R. Terras, A stopping time problem on the positive integersActa Arithmetica 30 (1976). Where density-1 began.
• I. Krasikov & J. C. Lagarias, Bounds for the 3x+1 problem using difference inequalitiesActa Arithmetica 109 (2003). The x^0.84 bound.
• J. L. Simons & B. M. M. de Weger, Theoretical and computational bounds for m-cycles of the 3n+1 problemActa Arithmetica 117 (2005); extended by C. Hercher to m ≤ 91 (2023).
• J. H. Conway, Unpredictable IterationsProc. 1972 Number Theory Conf., Boulder. Undecidability enters; see also his FRACTRAN paper (1987) and Kurtz–Simon (2007) for Π⁰₂-completeness.
• G. J. Wirsching, The Dynamical System Generated by the 3n+1 FunctionSpringer Lecture Notes in Mathematics 1681 (1998).
• D. Barina, Convergence verification of the Collatz problemJournal of Supercomputing (2020), plus the live project records.

This dossier grew a research wing: a live, iterated attack campaign on the conjecture, run under strict ground rules — no fabricated steps, dead ends logged as results, corrections kept on the record. Twenty-five paths and counting, all executing in the browser at the attack log, with the full iteration log kept in the standalone Collatz Lab app. Headlines so far:

• Certificate program exhausted: modular Lyapunov tables (any modulus), Syracuse valuation tables, all finite-state machines over (residue, parity history), and both one-counter classes — each decided empty with machine-verified witnesses, all killed by the same obstruction: the negative-domain cycles at −1, −5, −17.
• Certified density bounds: the Applegate–Lagarias tree-search method pushed far past its published-era depth — γ ≥ 0.7748, certifying 1.6×10¹⁸ integers below 2^78.1 reach 1, territory beyond the 2⁷¹ verification frontier.
• A literature-grade cycle bound assembled: Hercher 2023 + Barina 2025 combine to establish K ≥ 1.375×10¹¹ odd members in any nontrivial cycle — and our exact-convergent engine independently derived the same integer, 137,528,045,312.
• Unifications and corrections: the covering tail and the divergence barrier shown to be one large-deviation object (entropy rate 0.05004 bits/step, confirmed against exact combinatorics); an evolutionary adversary that converges onto the proven obstruction map; four of our own overclaims caught by our own instruments and downgraded in public.

The status line has not moved: the conjecture is open. What the campaign adds is an unusually precise, machine-checked map of why — and a standing triage kit for every future claimed proof.