Ab initio density-functional theory calculations, photoemission spectroscopy (PES), scanning tunneling microscopy, and spectroscopy (STM, STS) have been used to solve the 2sqrt[3]×2sqrt[3]R30 surface reconstruction observed previously by LEED on 0.5 ML K/Si:B. A large K-induced vertical lattice relaxation occurring only for 3/4 of Si adatoms is shown to quantitatively explain both the chemical shift of 1.14 eV and the ratio 1/3 measured on the two distinct B 1s core levels. A gap is observed between valence and conduction surface bands by ARPES and STS which is shown to have mainly a Si-B character. Finally, the calculated STM images agree with our experimental results. This work solves the controversy about the origin of the insulating ground state of alkali-metal/Si(111):B semiconducting interfaces which were believed previously to be related to many-body effects.