Why Collatz WorksExploring the 3n+1 Problem

Interactive explorations of the Collatz conjecture — uncovering structural patterns, deep connections, and a possible path to proof.

Start the Journey

Connections

Research Roadmap

1. The Puzzle

Pick any number, apply the rules, watch it reach 1. Try to find one that doesn't.

2. The Binary Engine

Watch bits get destroyed step by step. Every drop eats information.

3. No Loops

The irrationality of log₂3 prevents any orbit from cycling. Every candidate fails.

4. The Hidden Rotation

In log₆ coordinates, Collatz is an irrational rotation on a circle. Chaos becomes order.

5. The Countdown

The +1 carry propagation is a deterministic timer. Not random. Not statistical. Algebraic.

6. Finite Fuel

Natural numbers have finite bits. The carry reads faster than the orbit generates. The fuel runs out.

Status

This is an exploration, not a completed proof. The results here include proved theorems, verified computations, and structural conjectures. Some arguments have gaps — notably the finite propagation bound needs full algebraic verification, and the asymptotic cycle elimination needs a rigorous uniformity bound. We describe both what we've proved and what remains open. Peer review and collaboration are welcome.

What This Site Is

An amateur mathematician's multi-year exploration of the Collatz conjecture, presented as interactive visualizations you can play with. The goal is not to claim a proof, but to share genuinely interesting structural discoveries:

The Collatz map is a thermodynamic system — with a conservation law, a dissipation rate, and a critical threshold. Among all $nx+c$ , $x/y$ systems, $3x+1$ is the only nontrivial convergent one, because 3 is the only odd prime less than $2^2 = 4$ . Read more →
The transfer operator has exactly 4 non-zero eigenvalues — the cube roots of $4/3$ , spaced at $120°$ intervals. This connects to the Hilbert-Polya conjecture and Eisenstein integers. Read more →
Orbits trace walks on the Eisenstein lattice — and convergence becomes a geometric question: does a biased random walk on $\mathbb{Z}[\omega]$ always end above a geodesic? Read more →
Carry propagation is a countdown timer — the $+1$ in $3n+1$ reads bits of $n$ at a rate that exceeds the orbit's ability to generate new ones. This is verified computationally but not yet fully proved for all integers. Read more →

Two Paths Through This Site

The Proof Journey — 7 interactive chapters. For anyone who knows basic math and binary. Explore WHY the conjecture should be true by playing with the dynamics yourself. Start here →

The Research — Proved results, structural connections, and the roadmap of what's done and what remains. For mathematicians. Proved results → | Connections → | Roadmap →

Prior Work

This exploration grew out of several years of self-published work by an amateur mathematician working in industry. The earlier writings developed the dropping set framework, the geometric correspondence, and the base-6 rotation discovery. Read more →