From Infinite to Finite: A Proof of the Collatz Conjecture via Prefix Partition and Height Descent

We present a complete proof of the Collatz conjecture for all positive integers under the
standard map T (n) = 3n + 1 for odd n and T (n) = n/2 for even n, and we extend the result
naturally to negative integers using T (n) = 3n − 1 for odd n < 0.
Our approach introduces a height-based contraction metric and partitions the integers into
a finite set of structural classes based on fixed-length base-3 representations. We prove that
every such class contracts via a verified representative and that bounded tail variation does not
prevent contraction.
We show that all positive integers eventually enter a finite verified basin below 2^68, from
which convergence to the terminal cycle {1, 2, 4} is guaranteed. Under the symmetric exten-
sion for negative integers, we prove that all negative trajectories converge to the unique cycle
{−1, −2, −4}.
This yields a complete, structurally finite proof of the Collatz conjecture for all n ∈ Z.