  The study of vector-like spectra in 4-dimensional F-theory compactifications
involves root bundles, which are important for understanding the Quadrillion
F-theory Standard Models (F-theory QSMs) and their potential implications in
physics. Recent studies focused on a superset of physical root bundles whose
cohomologies encode the vector-like spectra for certain matter representations.
It was found that more than 99.995\% of the roots in this superset for the
family $B_3( \Delta_4^\circ )$ of $\mathcal{O}(10^{11})$ different F-theory QSM
geometries had no vector-like exotics, indicating that this scenario is highly
likely.
  To study the vector-like spectra, the matter curves in the F-theory QSMs were
analyzed. It was found that each of them can be deformed to nodal curve that is
identical across all spaces in $B_3( \Delta^\circ )$. Therefore, from studying
a few nodal curves, one can probe the vector-like spectra of a large fraction
of F-theory QSMs. To this end, the cohomologies of all limit roots were
determined, with line bundle cohomology on rational nodal curves playing a
major role. A computer algorithm was used to enumerate all limit roots and
analyze the global sections of all tree-like limit roots. For the remaining
circuit-like limit roots, the global sections were manually determined. These
results were organized into tables, which represent -- to the best knowledge of
the author -- the first arithmetic steps towards Brill-Noether theory of limit
roots.
