Phase factors connect the lane spectra
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Each local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes,
\[
\widehat x(k)=\sum_{r=0}^{2}e^{-2\pi ikr/12}
\widehat{E_rx}(k\bmod4).
\]
The identity follows by writing \(n=r+3m\) in the full DFT sum. Its phase factors retain the distinction between the lanes. The diagnostic replay compares this assembly with an independent direct twelve-point DFT at a declared tolerance of \(10^{-12}\).