A Two-Operator Calculus for Arithmetic-Progression Paths in the Collatz Graph

Abstract

A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single step, (ii) gives a closed formula for every set of seeds that realises a prescribed parity word, (iii) yields a one line affine invariant that forbids trajectories consisting of infinitely many odd moves, and (iv) reduces the non-trivial-cycle problem to a pair of linear congruences.