Diagonal and branchwise criteria for universal Fourier series

We characterize diagonal profiles admitting universal real trigonometric Fourier series with factors in ${\mathcal{E}}_s={\bigcap }_{p>s}{\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^{s,q}$, $2\le s<\mathrm{\infty }$, $1\le q\le \mathrm{\infty }$. For nonincreasing profiles we obtain exact dyadic criteria and logarithmic endpoints. For independently regular odd/even sine/cosine branches we characterize simultaneous subseries, sign, and rearrangement universality for finite almost-everywhere targets. One function in every $L^p$, $1\le p<2$, supplies the coefficients. A separation example admits this realization although every compatible nonincreasing majorant with bounded relative cost belongs to ${\mathrm{\ell }}^1$. Residue-class decompositions extend the criterion; a weakest-model condition permits extended-real rearrangement targets. A Walsh–Paley analogue gives the same diagonal tests for ternary modifications. Trigonometric realization uses Pogosyan’s majorant theorem; the multiplier, factorization, projection-transfer, separation, and Walsh Fourier-origin arguments are proved here.