This paper establishes an equivalent geometric wave model for the Collatz sequence. The sole foundation of the construction is Euler's formula e^i = -1, from which one derives (n) = (-1) ⁿ for every integer n — an exact parity switch that makes it possible to distribute the three Collatz operations (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 C₃ = (1 - z) /4 — a pure wave of frequency 1/2 containing no z — completely decoupled from the multiplicative 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 v₂ (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 (p) = (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.
Lixin Wang (Sun,) studied this question.
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