  This is the first of two papers devoted to the proof of conformal invariance
of the critical double random current model on the square lattice. More
precisely, we show the convergence of loop ensembles obtained by taking the
cluster boundaries in the sum of two independent currents with free and wired
boundary conditions. The strategy is first to prove convergence of the
associated height function to the continuum Gaussian free field, and then to
characterize the scaling limit of the loop ensembles as certain local sets of
this Gaussian Free Field. In this paper, we identify uniquely the possible
subsequential limits of the loop ensembles. Combined with the second paper,
this completes the proof of conformal invariance.
