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

TL;DR

This paper introduces a novel calculus based on two elementary operators that simplifies the analysis of the Collatz map by splitting progressions, providing formulas, invariants, and reducing cycle problems to linear congruences.

Contribution

It presents a new two-operator calculus for analyzing Collatz trajectories, enabling simplified splitting, formulas, invariants, and cycle problem reduction.

Findings

Splits progressions into even and odd subsequences in one step

Provides a closed formula for seeds with a given parity pattern

Reduces cycle detection to solving linear congruences

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.