An Equivalent Helix Wave Construction for the Collatz Sequence: From Euler's Formula to Fourier Three-Component Isolation (V2)

This paper establishes an equivalent geometric wave model for the Collatz sequence. The sole foundation of the construction is Euler's formula e^{i\pi} = -1, from which one derives \cos(\pi n) = (-1)^n for every integer n — an exact parity switch that makes it possible to distribute the three Collatz operations (\div 2, 3n, +1) uniquely into three mutually orthogonal Fourier components. Each positive integer n is modeled as the frequency of a positive helix; the helix's continued extension legitimizes the use of Fourier analysis, a legitimacy itself rooted in Euler's formula. The three-component decomposition shows that the entire contribution of +1 falls precisely in C_3 = (1 - \cos\pi z)/4 — a pure wave of frequency 1/2 containing no z — completely decoupled from the multiplicative \div 2 and 3n channels. An exact operational bijection is established between the wave frequency evolution and the Collatz integer sequence, with precision guaranteed at every step by Euler's formula. The worst-case run-length theorem proves that consecutive k=1 chains have exact length v_2(n+1) - 1, and a 2-adic argument shows that no permanent worst-case chain exists in the positive integers. Complete coverage of all operation ratios then gives the main result: for any positive integer frequency, continued Collatz operations necessarily converge to base-wave frequency 1. The same framework is applied to the 5n+1 sequence, proving divergence via the unified criterion \bar\delta(p) = \log(p/4) for the (2j+1)n+1 family, and establishing that p = 3 is the unique odd integer for which all growth modes are excludable. This paper is part of the RHR (Resonant Helix Reductionism) framework; the companion paper on the Riemann Hypothesis is available at DOI 10.5281/zenodo.20398329.