  Root bundles appear prominently in studies of vector-like spectra of 4d
F-theory compactifications. Of particular importance to phenomenology are the
Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze
a superset of the physical root bundles whose cohomologies encode the
vector-like spectra for the matter representations $(\mathbf{3},
\mathbf{2})_{1/6}$, $(\mathbf{\overline{3}}, \mathbf{1})_{-2/3}$ and
$(\mathbf{1}, \mathbf{1})_{1}$. For the family $B_3( \Delta_4^\circ )$
consisting of $\mathcal{O}(10^{11})$ F-theory QSM geometries, we argue that
more than $99.995\%$ of the roots in this superset have no vector-like exotics.
This indicates that absence of vector-like exotics in those representations is
a very likely scenario.
  The QSM geometries come in families of toric 3-folds $B_3( \Delta^\circ )$
obtained from triangulations of certain 3-dimensional polytopes $\Delta^\circ$.
The matter curves in $X_\Sigma \in B_3( \Delta^\circ )$ can be deformed to
nodal curves which are the same for all spaces in $B_3( \Delta^\circ )$.
Therefore, one can probe the vector-like spectra on the entire family $B_3(
\Delta^\circ )$ from studies of a few nodal curves. We compute the cohomologies
of all limit roots on these nodal curves.
  In our applications, for the majority of limit roots the cohomologies are
determined by line bundle cohomology on rational tree-like curves. For this, we
present a computer algorithm. The remaining limit roots, corresponding to
circuit-like graphs, are handled by hand. The cohomologies are independent of
the relative position of the nodes, except for a few circuits. On these
\emph{jumping circuits}, line bundle cohomologies can jump if nodes are
specially aligned. This mirrors classical Brill-Noether jumps. $B_3(
\Delta_4^\circ )$ admits a jumping circuit, but the root bundle constraints
pick the canonical bundle and no jump happens.

High Energy Physics - Theory;Algebraic Geometry