= Frequency-separated Fourier block series
{title2=$f=\sum_j q_j,\qquad \sum_j\|q_j\|_\infty<\infty$}
If <trigonometric polynomials> have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its <Fourier partial sum> contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.
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