  Triply-periodic networks are among the most complex and functionally valuable
self-assembled morphologies, yet they form in nearly every class of biological
and synthetic soft matter building blocks. In contrast to simpler assembly
motifs -- spheres, cylinders, layers -- TPN assemblies require molecules to
occupy variable local domain shapes, confounding attempts to understand their
formation. Here, we examine the double-gyroid (DG) network phase of block
copolymer (BCP) melts, a prototypical soft self-assembly system, by using a
geometric formulation of the strong stretching theory (SST) of BCP melts. The
theory establishes the direct link between molecular BCP packing,
thermodynamics of melt assembly and the {\it medial map}, a generic geometric
measure of the center of complex shapes. We show that "medial packing" is
essential for thermodynamic stability of DG in strongly-segregated melts,
reconciling a long-standing contradiction between infinite- and
finite-segregation theories, corroborating our SST predictions at
finite-segregation via self-consistent field calculations. Additionally, we
find a previously unrecognized non-monotonic dependence of DG stability on the
elastic asymmetry, the comparative entropic stiffness of matrix-forming to
tubular-network forming blocks. The composition window of stable DG --
intermediate to competitor lamellar and columnar phases -- widens both for
large and small elastic asymmetry, seemingly overturning the heuristic view
that packing frustration is localized to the tubular domains. This study
demonstrates utility of geometric optimization of {\it medial tesselations} for
understanding in soft-molecular assembly. As such, the particular medial-based
approach deployed here is readily generalizable to study packing frustration
far beyond the case of DG morphologies in neat BCP assemblies.
