Cepheid Metallicity in the Leavitt Law (C--MetaLL) survey. IX. Metallicity dependence of period-Wesenheit relations based on a homogeneous spectroscopic sample

The C-MetaLL project has provided homogeneous spectroscopic abundances of 290 Classical Cepheids (DCEPs) for which we have the intensity-averaged magnitudes in multiple optical and NIR bands, periods, pulsation modes, and Gaia parallaxes corrected for individual zero-point (ZP) biases. Our goal is to derive updated period--Wesenheit--metallicity (PWZ) relations using the largest and most homogeneous metallicity sample ever used for such analyses, covering a range of -1. 3< Fe/H <+0. 3 dex, and to assess the metallicity dependence of these relations. We computed several optical and NIR Wesenheit magnitudes adopting both Cardelli et al. and Fitzpatrick reddening laws, and transformed Johnson-Cousins Wesenheit magnitudes into their HST equivalents using empirical relations. Using 275 DCEPs with reliable parallaxes, we applied a robust photometric parallax technique, which simultaneously fits all parameters -- including the global ZP counter-correction to parallaxes -- and handles outliers via a Cauchy likelihood to account for the sample's excess variance. Gaia We find a stronger metallicity dependence (γ ≈ -0. 5 mag/dex in optical, -0. 4 mag/dex in NIR) than recent literature reports. Gaia parallax ZP counter-correction (ε) varies moderately across bands, with an average value of ∼10 μas, aligning with previous determinations. Applying our PWZ relations to ∼4500 LMC Cepheids yields distances generally consistent within 1σ with geometric estimates. The choice of reddening law has a small impact, while using only fundamental-mode pulsators significantly increases the uncertainties. Including α element corrections increases |γ| and reduces ε. However, we find 1σ consistency γ values with the literature, particularly for the Wesenheit magnitude in the HST bands, by restricting the sample to brighter (i. e. closer) objects, or by including only pulsators with -0. 7< Fe/H <0. 2 dex. Our results hint at a large γ or a non-linear dependence on metallicity of DCEP luminosities at the metal-poor end, which is difficult to quantify with the precision of parallaxes of the present dataset.