  We provide a quantitative three-dimensional vortex approximation construction
for the Ginzburg-Landau functional. This construction gives an approximation of
vortex lines coupled to a lower bound for the energy, optimal to leading order,
analogous to the 2D ones, and valid for the first time at the
$\varepsilon$-level. These tools allow for a new approach to analyze the
behavior of global minimizers for the Ginzburg-Landau functional below and near
the first critical field in 3D, followed in two forthcoming papers. In
addition, they allow to obtain an $\varepsilon$-quantitative product estimate
for the study of Ginzburg-Landau dynamics.
