how do we handle the case where each symbol are located periodicity with certain coordinates as mentioned?:)

the problem why i need to cross post is that no practical solution is achieved in MO for a long time and here I believe better answer could be achieved...

I edited the format and added the explicit question, could i remove my hold?

@vzn Thanks for your comment. Indeed, this problem has been tackled partially in physics and some results has been obtained. However, the previous approach is done on a heavy manual manner (non-automizable) and I converted this problem into a tiling problem of Wang tiles. Once this tiling on a finite lattice problem is solved, all previous work can be reproduced and larger problem could be tackled automatically. That's why I resort to mathematics and computer science....

@vzn Thanks a lot for your reply, the actual problem is to understand the ground state of a physical crystal structure under pair interactions up to some ranges. It is likely that a physical crystal pattern is simply an order tiling of atomic ("cluster tiles") represented by Wang Tile... But the nature is usually well order (usually periodic), my goal is to review this pattern by computational force rather than ("imagination or manual construction of tiles")....