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Measurements of beam single-spin asymmetries in semi-inclusive deep-inelastic electron scattering (SIDIS) with positively charged kaons off protons have been performed with 10.6 and 10.2 GeV incident electron beams using the CLAS12 spectrometer at Jefferson Lab. We report an analysis of the electroproduction of positively charged kaons over a large kinematic range of fractional energy, Bjorken $x$, transverse momentum, and photon virtualities $Q^2$ranging from 1 ${\mathrm{GeV}}^2$up to 6 ${\mathrm{GeV}}^2$. This is the first published multidimensionally binned CLAS12 measurement of a kaon SIDIS single-spin asymmetry in the valence quark regime. The data provide constraints on the structure function ratio $F_{LU}^{sin\varphi }/F_{UU}$, where $F_{LU}^{sin\varphi }$is a quantity with a leading twist of twist-3 that can reveal novel aspects of the quark-gluon correlations within the nucleon. The impact of the data on understanding the underlying reaction mechanisms and their kinematic variation is explored using theoretical models for the different contributing twist-3 parton-distribution functions (PDFs) and fragmentation functions (FFs).