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In the largest, currently known, class of one Quadrillion globally consistentF-theory Standard Models with gauge coupling unification and no chiral exotics,the vector-like spectra are counted by cohomologies of root bundles. In thiswork, we apply a previously proposed method to identify toric base 3-folds,which are promising to establish F-theory Standard Models with exactly threequark-doublets and no vector-like exotics in this representation. The basespaces in question are obtained from triangulations of 708 polytopes. Bystudying root bundles on the quark doublet curve$C_{(\mathbf{3},\mathbf{2})_{1/6}}$ and employing well-known results aboutdesingularizations of toric K3-surfaces, we derive a \emph{triangulationindependent lower bound} $\check{N}_P^{(3)}$ for the number $N_P^{(3)}$ of rootbundles on $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ with exactly three sections. Theratio $\check{N}_P^{(3)} / N_P$, where $N_P$ is the total number of roots on$C_{(\mathbf{3},\mathbf{2})_{1/6}}$, is largest for base spaces associated withtriangulations of the 8-th 3-dimensional polytope $\Delta^\circ_8$ in theKreuzer-Skarke list. For each of these $\mathcal{O}( 10^{15} )$ 3-folds, weexpect that many root bundles on $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ areinduced from F-theory gauge potentials and that at least every 3000th root on$C_{(\mathbf{3},\mathbf{2})_{1/6}}$ has exactly three global sections and thusno exotic vector-like quark-doublet modes.

Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs