Geometric Frequency Mixing in Helical Waveguides via a One-Dimensional Covariant Helmholtz Model: Gauge Reduction and Spectral Splitting

This study develops a one-dimensional covariant Helmholtz model for a vector-valued wave field transported along a circular helical centerline and represented in the Frenet–Serret frame. For a helix with constant curvature κ>0 and torsion τ≠0, the geometric coupling is described by a constant skew-symmetric connection matrix Ω∈so(3). The covariant Helmholtz operator is shown to admit an exact gauge reduction to the flat componentwise Helmholtz operator through u(s)=e−Ωsy(s). Thus, within the one-dimensional centerline formulation, the helix preserves the operator spectrum while redistributing the observed Frenet components through parallel transport. The closed-form solutions show that a monochromatic input with wavenumber k is decomposed into a carrier and two geometric sidebands governed by the Darboux rotation rate λ=κ2+τ2. In the sub-geometric regime k<λ, the lower algebraic sideband is represented by the positive observable wavenumber q−=|k−λ|, with associated scale Tbeat=L−=2π/q−. The lossless energy analysis proves conservation of the total averaged energy and its redistribution among the carrier and observable sidebands. A representative helical acoustic-channel design is then examined as a conceptual realization of the centerline model. Monte Carlo perturbations and additive-noise tests show that the predicted sideband locations, lower-sideband scale, and energy partition remain stable under prescribed fabrication tolerances and spectrally identifiable under weak and moderate measurement noise.