The most metal-deficient stars hold important clues about the early build-up and chemical evolution of the Milky Way, and carbon-enhanced metal-poor (CEMP) stars are of special interest. However, little is known about CEMP stars in the Galactic bulge. In this paper, we use the large spectroscopic sample of metal-poor stars from the Pristine Inner Galaxy Survey (PIGS) to identify CEMP stars ([C/Fe] > +0.7) in the bulge region and to derive a CEMP fraction. We identify 96 new CEMP stars in the inner Galaxy, of which 62 are very metal-poor ([Fe/H] < -2.0); this is more than a ten-fold increase compared to the seven previously known bulge CEMP stars. The cumulative fraction of CEMP stars in PIGS is $42^{,+14,}_{,-13} %$ for stars with [Fe/H] < -3.0, and decreases to $16^{,+3,}_{,-3} %$ for [Fe/H] < -2.5 and $5.7^{,+0.6,}_{,-0.5} %$ for [Fe/H] < -2.0. The PIGS inner Galaxy CEMP fraction for [Fe/H] < -3.0 is consistent with the halo fraction found in the literature, but at higher metallicities the PIGS fraction is substantially lower. While this can partly be attributed to a photometric selection bias, such bias is unlikely to fully explain the low CEMP fraction at higher metallicities. Considering the typical carbon excesses and metallicity ranges for halo CEMP-s and CEMP-no stars, our results point to a possible deficiency of both CEMP-s and CEMP-no stars (especially the more metal-rich) in the inner Galaxy. The former is potentially related to a difference in the binary fraction, whereas the latter may be the result of a fast chemical enrichment in the early building blocks of the inner Galaxy.
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