  Subject to hypotheses based on the matroid structure theory of Geelen,
Gerards, and Whittle, we completely characterize the highly connected members
of the class of golden-mean matroids and several other closely related classes
of quaternary matroids. This leads to a determination of the eventual extremal
functions for these classes. One of the main tools for obtaining these results
is the notion of a frame template. Consequently, we also study frame templates
in significant depth.
