> 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
This is a good point; I had been thinking of the problem differently.
In this model, the diamond is viewed as just being a window onto the infinite grid, and we're not interested in whatever properties it might have considered as a finite graph. Thinking about it this way, the idea that the coloring in your article disproves is that "there is a 14-packing-coloring of the infinite grid which uses a chessboard pattern for color 1".
But then, since (by your later result) there is no 14-packing-coloring of the infinite grid, it's not too surprising that there isn't such a coloring that also features a chessboard.
What bothers me in this discussion of the chessboard conjecture is the switch: you say "at this point, we were conjecturing that using color 1 in the chessboard pattern that yields density 1/2 was optimal, meaning that you could packing-color a graph D_r with k colors (...) if, and only if, you could do it enforcing the chessboard pattern of 1's". And this seems plausible, and the illustration provided as a counterexample isn't actually a counterexample to this statement. You can packing-color the graph shown there with 14 colors, and you can do so while enforcing the chessboard pattern of 1s.
In your explanation here, you're viewing D_r as a subgraph of the successfully-colored infinite grid, not as a finite graph in its own right. That handily explains why the chessboard pattern goes around the center of D_r instead of through it. The illustration in the post does then show that there's a problem. But to my eye, the problem is in the premise of a successfully 14-packing-colored infinite grid, not in the chessboard conjecture. In between stating the chessboard conjecture and providing the counterexample, you changed what the chessboard conjecture said -- there is no reference to the infinite grid in your statement of the chessboard conjecture, but the illustration of a "counterexample" is a counterexample to a statement about the infinite grid. It feels wrong. :-/
This is a good point; I had been thinking of the problem differently.
In this model, the diamond is viewed as just being a window onto the infinite grid, and we're not interested in whatever properties it might have considered as a finite graph. Thinking about it this way, the idea that the coloring in your article disproves is that "there is a 14-packing-coloring of the infinite grid which uses a chessboard pattern for color 1".
But then, since (by your later result) there is no 14-packing-coloring of the infinite grid, it's not too surprising that there isn't such a coloring that also features a chessboard.
What bothers me in this discussion of the chessboard conjecture is the switch: you say "at this point, we were conjecturing that using color 1 in the chessboard pattern that yields density 1/2 was optimal, meaning that you could packing-color a graph D_r with k colors (...) if, and only if, you could do it enforcing the chessboard pattern of 1's". And this seems plausible, and the illustration provided as a counterexample isn't actually a counterexample to this statement. You can packing-color the graph shown there with 14 colors, and you can do so while enforcing the chessboard pattern of 1s.
In your explanation here, you're viewing D_r as a subgraph of the successfully-colored infinite grid, not as a finite graph in its own right. That handily explains why the chessboard pattern goes around the center of D_r instead of through it. The illustration in the post does then show that there's a problem. But to my eye, the problem is in the premise of a successfully 14-packing-colored infinite grid, not in the chessboard conjecture. In between stating the chessboard conjecture and providing the counterexample, you changed what the chessboard conjecture said -- there is no reference to the infinite grid in your statement of the chessboard conjecture, but the illustration of a "counterexample" is a counterexample to a statement about the infinite grid. It feels wrong. :-/