A Rigorous Vanishing-Density Theorem for Modular Collatz Sieves: Explicit Construction and Density Estimates

The main result of this work is an explicit construction of a composite modulus M=∏p≤XpM = \prod_{p \le X} pM=∏p≤Xp such that the fraction of residue classes modulo MMM whose Collatz trajectories never reach 1 is arbitrarily small. Consequently, the set of integers that are excluded by all such modular sieves has natural density zero.