Selection Rules and Channel Structure in a Base Octave Model of Collatz Dynamics

TL;DR

This paper reformulates the Collatz dynamics using a unified parity-controlled transformation, introduces a base octave decomposition, and analyzes structural regularities and constraints leading to eventual confinement of trajectories.

Contribution

It presents a novel unified parity-based model and a base octave decomposition that reveal structural regularities and constraints in Collatz dynamics.

Findings

Identifies growth and decay channels in the Collatz process.

Shows trajectories are confined to a contractive subnetwork.

Establishes a non positive drift in the logarithmic octave coordinate.

Abstract

The Collatz iteration is governed by two distinct update rules, depending on the parity of the current iterate: n(i+1)=3n(i)+1 for odd n(i), and n(i+1)=n(i)/2 for even n(i). We show that these rules can be written equivalently as a single parity controlled transformation, n(i+1)=((2s(i)+1)(2k(i)+s(i))+s(i))/2, where n(i)=2k(i)+s(i) and s(i) is the parity (0 or 1) of n(i), yielding a uniform, step aligned dynamical system in which parity variables are tracked explicitly. This reformulation removes the asymmetry of the traditional presentation and exposes structural regularities that are obscured when odd and even updates are treated separately. Building on this unified rule, we introduce a base octave decomposition, representing every integer uniquely in the form n=B+8(A-1) with B = 1 to 8. The resulting dynamics separate into parity dependent base transitions and affine updates of the octave index, inducing a finite directed transition skeleton lifted across scale levels. Refining the parity description yields a finite 128 state symbolic system that encodes all admissible transitions, including carry effects arising from higher order parity inheritance. Within this framework, we identify growth permitting and decay forcing channels and show that the only persistence mechanism (base 7 transitions in even octaves) is necessarily bounded by the 2 adic valuation of the octave index. An exhaustive enumeration of admissible return paths between persistence episodes establishes a non positive drift in a logarithmic octave coordinate. Because of these finite state constraints, trajectories are eventually confined to a contractive subnetwork associated with the terminal 1,2 cycle. The approach emphasizes structural organization and return map methods, and provides a symbolic framework for analyzing parity driven integer recurrences.