21,864 Unresolved, Low-mass Binaries Identified via their Overluminosity in $\mathit{\text{Gaia}}$ DR3 and a Catalog of 347,440 Systems within 100 pc of the Sun

The fundamental parameters of a low-mass star can potentially be determined from its photometry and astrometry. This is complicated by the fact that 10-20 percent of low-mass stars are predicted to be equal-mass binaries. These unresolved systems appear more luminous compared to single stars with the same fundamental parameters. We present a method to differentiate binary stars from single-star main sequence K and M dwarfs using their $\mathit{\text{Gaia}}$ DR3 XP spectra. We assemble a training set of stars which have pristine astrometry and photometry, are located within 100pc of the Sun, and exclude stars with $\mathit{\text{Gaia}}$ DR3 flags suggesting they may be unequal mass systems, thereby leaving stars that are predominantly either single- or equal-mass binaries. We then iteratively train Random Forest Regression (RFR) models to predict absolute magnitude and color given the RP spectral coefficients of a star. After each model, we remove the stars that have absolute magnitudes significantly brighter than their predicted values. This method converges on a model trained only on single stars. We then use this model to identify the ``overluminous’’ K and M stars in $\mathit{\text{Gaia}}$ DR3 within 100 parsecs, with some quality cuts. We find that $\sim 13\mathrm{\%}$ of the sample is significantly overluminous and assume these to be unresolved binaries. We aggregate several multiplicity surveys across different projected separations and incorporate our overluminous binaries to create a general $\mathit{\text{Catalog}}\mathit{\text{of}}\mathit{\text{Systems}}$ within 100 pc. We use this $\mathit{\text{Catalog}}$ to provide lower limits on the multiplicity fraction for stars between $0.1$ and $0.7M_{\odot }$.