 Grid-separated additive coordinates

For five pairs of continuous functions on $[0,1]$, one can arrange that at every arbitrarily fine scale there are closed interval families whose product rectangles cover each point at least three times, and within each family the images under $a_q(x)+b_q(y)$ are mutually disjoint. To see this, put gaps near five shifted fine grids; gap sets for different shifts are disjoint. Each coordinate misses at most one family, so a pair misses at most two. Approximate any prescribed pair of functions by constants on the complementary intervals, interpolating across gaps. Perturb the finitely many constants to avoid all equalities between distinct pair sums. The rectangle images are then separated by a positive distance, a property preserved under sufficiently small uniform perturbations. For each required mesh size this property defines a dense open subset of $C([0,1]{)}^{10}$. The Baire category theorem gives a tuple satisfying every scale simultaneously.