Briefly, at a given [$T_{\rm eff}$]{}, [$\log~g$]{}, and [\[Fe/H\]]{} (this metallicity estimate is determined from the above $\chi^{2}$ minimization), we generate by interpolation a series of synthetic spectra by varying \[C/Fe\] by 0.01 dex, from –1.0 to $+$1.0 dex from the above determined value over the spectral range 4290–4318 Å (in which the prominent CH $G$-band feature exists), and check how the $\chi^{2}$ values from the differences between the synthetic spectra and the observed spectrum change. We fit a spline function to the distribution of the $\chi^{2}$ values against [\[C/Fe\]]{}, in order to establish the local minimum point where \[C/Fe\] is determined. We demand that the extrema of the $\chi^2$ values are at least 10% above the local minimum. In this case, we raise a ‘D’ flag, indicative of a clear detection of CH $G$-band.

In cases where no minimum is found, the $\chi^{2}$ behavior typically reveals a continuously declining trend of $\chi^{2}$ toward the edge of the grid. We fit to the portion of the declining function of $\chi^{2}$ with a Gaussian function having a mean value of [\[C/Fe\]]{} at the minimum of declining $\chi^{2}$, in order to estimate the full width half maximum (FWHM) of this variation. After 