Step inside the Collatz research map.

Your first move

Here is a puzzle you can start solving in ten seconds. Choose six. If the number is even, divide it by two. If it is odd, multiply by three and add one. Six becomes three. What comes next?

You found a path

Ten! Then five, sixteen, eight, four, two, one. You have checked one starting number. The Collatz conjecture says that every positive whole number eventually reaches one. Try seven yourself after the video. The rules stay simple. The journey can get surprisingly complicated.

The difficult word: every

Imagine testing an enormous crowd of starting numbers. Every one comes home. That still leaves infinitely many untested starts. A proof must exclude both an endless escape and a different cycle. One stubborn exception would be enough to defeat the conjecture.

Step inside the question

Now step inside our research map. These twisting ribbons borrow the visual language of protein diagrams. Their shape helps us explore ideas; it predicts no solution. Turn the structure, select an argument, and follow its connections. Color tells you whose work you are reading. Empty nodes mark open steps.

Tao: a powerful almost-all result

The green branch follows Terence Tao. His theorem says that almost all starts reach values below any bound that grows to infinity, however slowly. Here, almost all has a precise meaning: logarithmic density. This is a major result. But exceptional starts remain possible, and the theorem does not say that every path reaches one.

Earlier foundations

The red branch credits other researchers. Riho Terras and C. J. Everett established almost-everywhere descent below the starting number. Bernstein and Lagarias developed a precise two-adic description of the dynamics. These are different tools for different parts of the problem. This selection is only a small part of the literature.

Our blue working archive

Blue opens our working archive: exact identities, restricted lemmas, audits, and failed routes. Khamit Kadyrbekov and Daniyal Kadirbekov organize this project with AI assistance. The new weighted-fiber drafts isolate a residual part that still needs control. Blue means a project record, not a claim that every ingredient is new. Internal checks are not external peer review.

The missing universal step

Here is one way a proof could finish. Show that every odd starting number greater than one eventually falls below itself. Strong induction would then carry every start to one. We do not yet have that universal step. A large archive does not tell us what percentage is solved, or when the missing idea will arrive.

Your next experiment

This is the Ingenium approach: begin with a small example, make a prediction, test your reasoning, then find the real research question. Open the map from the description. Rotate it. Enter a branch. Read an original source. Then ask: exactly which missing statement would connect this argument to a proof for every number?