The Wardany Framework for the Collatz Conjecture: A Density-1 Approach

We present a novel framework for attacking the Collatz conjecture,

based on iterative expansion of verified sets. Starting from a compu-

tationally verified base set S0 = {1, 2, . . . , Q}, we define a sequence

of sets Si such that Si+1 contains numbers that reach Si within a

bounded number of Collatz steps. We prove that each Si contains

all even numbers up to twice the bound of Si, and we identify the

inclusion of odd numbers as the key remaining challenge. By incorpo-

rating geometric families of known convergent numbers and leveraging

Tao’s recent result on almost bounded orbits, we argue that the union

S∞= i Si has natural density 1. This would imply that the Collatz

conjecture holds for almost all positive integers. While the full con-

jecture remains open, our framework provides a clear path toward a

density-1 result, which would itself be a historic breakthrough.