  We present a novel technique to obtain base independent expressions for the
matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric
ambient spaces. These can be used to systematically construct elliptically and
genus one fibered Calabi-Yau $d$-folds that lead to desired gauge groups and
spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is
based on the calculation of fiber Gopakumar-Vafa invariants using the
Batyrev-Borisov construction of mirror pairs and application of the so-called
Frobenius method to the data of a parametrized auxiliary polytope. In
particular for fibers that generically lead to multiple sections, only
multi-sections or that are complete intersections in higher codimension, our
technique is vastly more efficient than classical approaches. As an application
we study two Higgs chains of six-dimensional supergravities that are engineered
by fibrations of codimension two complete intersection fibers. Both chains end
on a vacuum with $G=\mathbb{Z}_4$ that is engineered by fibrations of
bi-quadrics in $\mathbb{P}^3$. We use the detailed knowledge of the structure
of the reducible fibers that we obtain from GV-spectroscopy to comment on the
corresponding Tate-Shafarevich group. We also show that for all fibers the
six-dimensional supergravity anomalies including the discrete anomalies
generically cancel.
