A Coordinate System for Collatz Dynamics

Abstract

It is well-established that every odd positive integer $n$ can be written uniquely as $n = \lambda \cdot 2^a \cdot 3^b - 1$ where $\gcd(\lambda, 6) = 1$ and $a \geq 1$. Building from this 3-smooth factorization, we introduce a partition of the nonnegative integers into countably many infinite triangles where each row $k$ forms a Collatz chain of alternating parity. The partition admits a coordinate system as a skeleton $\mathcal{L}_\lambda$ using the pair $(a, b)$ for odd positive integers within a geometric structure where row $k$ corresponds to $k = a + b$. Each position $(a, b)$ maps to $(a-1, b+1)$, a deterministic diagonal flow requiring no number-theoretic input. At the boundary $a = 1$, the trajectory exits to another skeleton depending on the factorization of $\lambda \cdot 3^{b+1} - 1$. The coordinate system is new. As a concrete application, we prove that rows $k \equiv 2 \pmod 4$ with $k \geq 6$ in the principal skeleton $\mathcal{L}_1$ contain no primes, and show this is the unique residue class admitting complete algebraic obstruction. Our contribution is the framework that makes visible which nonnegative integers these arguments apply to, with all results independent of the Collatz conjecture.