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CJM Problem Sketch No. 002

The Collatz Conjecture as a Structural Reachability Problem

Conceptual approach only. No proof or solution is claimed.

Keunsoo Yoon
Independent Research Group (Seoul, Republic of Korea)
austiny@gatech.edu, austiny@snu.ac.kr

Aug 6, 2026

โ… . The Problem

The Collatz Conjecture begins with any positive integer n. If n is even, divide it by two. Otherwise, multiply it by three and add one.

T(n) =
{
n / 2 if n is even
3n + 1 otherwise

Repeated application of this rule is conjectured to eventually reach the value 1 for every positive integer. The central question is therefore not merely how to compute the next step, but how global reachability is structurally organized.

โ…ก. Structural Reinterpretation

The traditional formulation follows one arithmetic trajectory:

n โ†’ T(n) โ†’ T2(n) โ†’ ยทยทยท โ†’ 1

The CJM approach instead represents the first N iterations as one finite layered reachability structure. Each layer corresponds to one iteration, and each reachable integer becomes a candidate state. The problem is therefore viewed as a global structural network rather than as a single temporal computation.

โ…ข. SAT โ†’ 3SAT Encoding

Let Rk,v denote that value v is reachable at layer k. The encoding imposes four basic conditions.

  1. The initial value is placed in Layer 0.
  2. Every transition follows the Collatz rule.
  3. Every transition is uniquely determined.
  4. Reachability propagates across successive layers.

These Boolean constraints are first expressed as SAT and then normalized through the common CJM pipeline.

Collatz Structure โ†’ SAT โ†’ 3SAT โ†’ CJM

โ…ฃ. The CJM Question

For a finite horizon N, the classical question

Does the sequence eventually reach 1?

is reformulated by CJM as:

Is reaching 1 structurally admissible within the first N layers?

โ…ค. Expected Changbal Region

By varying the initial value or the structural depth, one may observe transitions in reachability, branch interaction, and structural admissibility. A single Collatz trajectory is not itself a Changbal Jump. A stronger candidate would be a repeatable critical region where many trajectories simultaneously exhibit a structural transition.

Discussion Note.

A natural objection is that the Collatz map is deterministic, so the forward SAT encoding may seem like a mere restatement of a fixed computation rather than a genuine constraint problem. However, the stopping time for arbitrary n is not known in closed form, so treating N as a free parameter keeps the admissibility question nontrivial.

A more clearly combinatorial alternative would reverse the direction and build the inverse Collatz tree, where each value v admits predecessor 2v, and additionally (v โˆ’ 1)/3 when v โ‰ก 4 (mod 6). This reverse formulation is left for future work.

โ…ฅ. Limitation

This sketch does not prove the Collatz Conjecture, nor does it establish that every positive integer reaches 1. Its purpose is only to illustrate how an iterative arithmetic process may be reformulated as a finite 3SAT structure and examined through CJM as a problem of global structural reachability.