Assume you know that 13 colors aren't enough and you want to prove that 14 aren't enough either. Any 14 packing-coloring of the grid must use color 6 somewhere, as otherwise it would be only using 13 colors that are no better than colors {1, ..., 13}, and thus not enough. Now, you can imagine the diamond you're considering is centered around one of these 6s. This is breaking a different kind of symmetry; a translational one, as it's breaking the original symmetry of centering your diamond on any possible color, by choosing an arbitrary one. Granted, this is different form of symmetry-breaking as it's not about auto-morphisms on the set of solution. Also, it's totally possible that "symmetry breaking" was a bad choice of words; as long as you agree that it's a helpful idea as it significantly reduces the search space, we are on the same page.
formalizing this might be tricky, I mean something like a bijection from (the set of mappings from an infinite packing-coloring to a fixed colored sub-graph) to itself.
formalizing this might be tricky, I mean something like a bijection from (the set of mappings from an infinite packing-coloring to a fixed colored sub-graph) to itself.