Discrete dynamical systems and elementary number theory

Collatz conjecture

Collaboration beta

Starting from any positive integer, repeatedly halve even values and send odd values to (3n+1)/2. The question is whether every orbit reaches 1; the source's two-gate route has not proved this.

\forall n \in ℤ_{> 0}, \exists k \geq 0 : T^{k} (n) = 1.

Known results and sources

The Collatz problem asks whether every positive-integer orbit eventually reaches 1.

Research problem

Exact mathematical statement

Define the one-bit Collatz map on positive integers by

T(n)=\begin{cases}n/2,&n\text{ even},\\(3n+1)/2,&n\text{ odd}.\end{cases}

The Collatz conjecture asserts that for every positive integer $n$ , some iterate $T^k(n)$ equals $1$ . Equivalently in the source's two-gate formulation, every input above one has eventual homogeneous-coefficient descent below one, and that event forces a later orbit value below the starting value. Both gates remain open.

Problem infographic

Problem at a glance

A landscape explainer defines the one-bit Collatz map, traces sample orbits, and separates the open universal question from the source's two unproved gates. — The one-bit Collatz map is simple to state; the source decomposes universal descent into two still-open gates.

Current mathematical picture

Where work on Collatz conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureExact-level excess-six coverage

A high action can cross a threshold with positive overshoot, so the clock need not attain the exact level. A first-passage argument that controls both exact exits and overshoot jumps remains viable.

Route status · Narrowed route

Main reductionCurrent reduction

The source separates universal Collatz descent into Gate II coefficient descent and Gate I-star conversion of coefficient descent into ordinary descent.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Collatz conjecture in numbers

4.4kretained lines of mathematical investigation4,359 in the current working snapshot

Argument development: 3,757 · 86%
Explored or eliminated routes: 72 · 2%
Computational analysis: 51 · 1%
Open obligations: 283 · 6%
Definitions and setup: 196 · 4%

7selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.

Suggested move: Classify exact bridge exits, high-action overshoots, subcritical loops, and bridge-free addresses under one authenticated state convention.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeExact-level excess-six coverage

A high action can cross a threshold with positive overshoot, so the clock need not attain the exact level. A first-passage argument that controls both exact exits and overshoot jumps remains viable.

Route status · Narrowed route

Narrowed routeLow-credit direct canonical-source splice

The source's first-high coefficient is already below one in those cases. A different lower source, earlier reverse credit, or a direct LE1/LE2 arithmetic contradiction may handle the low-credit branch.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.Suggested move: Classify exact bridge exits, high-action overshoots, subcritical loops, and bridge-free addresses under one authenticated state convention.

Ready to work on

Resolve the low-credit cases where j is at most 2 or the first high action exceeds the available canonical-source credit.Suggested move: Construct an alternate lower source with adequate coefficient credit or derive an exact LE1/LE2 arithmetic incompatibility.

Ready to work on

Close the other Gate II exits and the independent Gate I-star branches after any equality-branch elimination.Suggested move: Return to the complete residual ledger and prove each remaining branch without transferring conclusions from the reset equality analysis.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The universal assertion that every positive-integer Collatz orbit reaches 1 remains open. Peer-reviewed almost-all-orbit descent results do not prove the universal statement.

[1][2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2022
Peer reviewedTao proved that, in logarithmic density, almost every starting value eventually falls below any prescribed function tending to infinity; this is not universal convergence to 1.[2]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction1 of 71
lemma2 of 72
equivalence1 of 71
negative result2 of 72

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included

retained route statementDoes every positive Collatz orbit reach 1?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementTwo-gate equivalenceintermediate
retained route statementAll target statuses remain openintermediate
retained route statementFirst-high credit barrierintermediate
retained route statementExcess may skip levelsintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureExact-level excess-six coveragereported failure
Useful failureLow-credit direct canonical-source splicereported failure
Research targetProve the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.open
Research targetResolve the low-credit cases where j is at most 2 or the first high action exceeds the available canonical-source credit.open
Research targetClose the other Gate II exits and the independent Gate I-star branches after any equality-branch elimination.open
Research targetReset first-passage diagonal theoremsuperseded
Research targetIndependent branches remainsuperseded
Narrowed routeExact-level excess-six coverageA high action can cross a threshold with positive overshoot, so the clock need not attain the exact level. A first-passage argument that controls both exact exits and overshoot jumps remains viable.
Narrowed routeLow-credit direct canonical-source spliceThe source's first-high coefficient is already below one in those cases. A different lower source, earlier reverse credit, or a direct LE1/LE2 arithmetic contradiction may handle the low-credit branch.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.

Collatz conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Starting from any positive integer, repeatedly halve even values and send odd values to (3n+1)/2. The question is whether every orbit reaches 1; the source's two-gate route has not proved this.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
The Ultimate Challenge: The 3x+1 Problemsurvey or monograph · Jeffrey C. Lagarias (editor) · American Mathematical Society · 2010 · accessed Aug 14, 2026
2
Almost all orbits of the Collatz map attain almost bounded valuespeer reviewed result · Terence Tao · Cambridge University Press · 2022 · ARXIV 1909.03562 · DOI 10.1017/fmp.2022.8 · accessed Aug 14, 2026

Important qualifications

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No priority, authorship, proof, formalization, or rights conclusion is inferred from external metadata.

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Collatz conjecture

Exact mathematical statement

Problem at a glance

Where work on Collatz conjecture stands

Collatz conjecture in numbers

Prove the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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