Interpreting map-based $E$/$B$ spectral properties of CMB foregrounds

Map-space $E$/$B$ decompositions of linear polarization are attractive for foreground and CMB analyses because they isolate the $B$-family patterns that contaminate primordial tensor searches from $E$-family patterns that trace coherent Galactic structures. However, the $E$/$B$ transform is non-fully-local and induces apparent spectral complexity in projected fields even when the underlying sky is spectrally simple in $\underset{‾}{P}=Q+iU$. We quantify this effect for synchrotron emission. We introduce a complex-parameter description of the frequency dependence of $\underset{‾}{P}$, its spin-preserving projections ${\underset{‾}{P}}_E$ and ${\underset{‾}{P}}_B$, and the scalar $\underset{‾}{S}=E+iB$, using complex log–Taylor and moment expansions (with simple transformation rules under $E$/$B$ projection) and linking their coefficients to spectral-index variations, line-of-sight mixing, synchrotron ageing, and Faraday effects. Using a toy model and a PySM template, we find that scalar combinations, especially $\mathrm{\mid }E\mathrm{\mid }$ and $\mathrm{\mid }B\mathrm{\mid }$, acquire the largest induced complexity, while $\underset{‾}{S}$ is less affected but lacks a directly interpretable amplitude and angle. By contrast, ${\underset{‾}{P}}_E$ and ${\underset{‾}{P}}_B$ retain a clear geometric meaning and exhibit only moderate spectral distortions, while satisfying the closure relation $\underset{‾}{P}={\underset{‾}{P}}_E+{\underset{‾}{P}}_B$ (which extends to all spectral orders in the moment formalism). Finally, with three frequency channels, we compare low-order spectral truncations and propose diagnostics to test whether the data favour a single power law in $P$ or independent power laws in $(P_E,P_B)$. This work is intended to be of practical relevance for both Galactic science and CMB $B$-mode analyses and lays the conceptual foundation for a series of papers applying the framework to observational data.