Where We Stand on the
Collatz Conjecture

One rule. Every positive integer.

Choose any positive whole number. If it is even, divide by two. If it is odd, multiply by three and add one. Repeat. Starting at six, the path reaches one. The Collatz conjecture says this happens for every positive starting number. A simple rule. A universal claim.

The word that makes it hard: every.

A number can rise before it falls. Checking a huge collection of starts still leaves infinitely many unchecked. A proof must exclude every alternative: an endless trajectory that escapes to infinity, and a cycle other than four, two, one. A convincing pattern is not yet a universal argument.

Why pursue it?

Why spend effort on this? Collatz tests how much we understand about deterministic arithmetic dynamics: exact rules with complicated long term behavior. Progress connects number theory, probability, and the study of when algorithms terminate. Its value is in understanding and new methods. A practical payoff cannot be promised in advance.

Progress has many authors.

The problem is traditionally associated with Lothar Collatz. In the nineteen seventies, Riho Terras and, independently, C. J. Everett proved that almost every start eventually falls below itself, in natural density. Daniel Bernstein and Jeffrey Lagarias developed a precise two adic coding of the dynamics. These are different advances toward understanding the same problem.

Tao: a major almost-all theorem.

Terence Tao made a major advance in twenty nineteen, published in twenty twenty two. Choose any bound that tends to infinity, however slowly. Almost every orbit eventually goes below that bound, in logarithmic density. This is much stronger control of orbit minima. But it does not say that every orbit reaches one. The exceptional starts remain the central difficulty.

Our map records what is checked.

Our contribution is an evolving research map and a collection of internally reviewed arguments. The current snapshot has three hundred fifty six nodes and five hundred thirty two relationships. Green means checked within the stated assumptions. Unfilled nodes mark open obligations. The count of green nodes measures neither the percentage solved nor the likelihood of success. External review and novelty remain separate questions.

Where the current route stops.

In one restricted family, recent work separates a difficult denominator estimate into two sources: deep divisibility and large multiplicative orders. A further reduction compresses a first coefficient certificate to degree below four d, independent of the full matrix length, under stated assumptions. The needed bounds for the actual coefficients are still missing. Even closing this branch would not settle arbitrary Collatz trajectories.

People, tools, responsibility.

This map and explanation are by Khamit Kadyrbekov and Daniyal Kadirbekov, with artificial intelligence assistance for exploration, drafting, and internal checking. Our workflow separates arithmetic derivation, structural reformulation, critical review, and connection to the full goal. Earlier mathematicians are credited for their published work. They are not collaborators on this project. Responsibility for the claims remains with the authors.

The next proof obligation.

One route to a full proof is to show that every odd start greater than one eventually falls below itself. Strong induction would then finish the argument. We have not established that universal descent. There is no defensible completion date or percentage remaining. Explore the map, inspect the assumptions, challenge a step, or help prove a clearly stated missing lemma. That is how this project can move forward.