Alright, I get the difference between Rank and City Score now.
The only thing I don't understand yet, is this:
As the program walks the hard-coded spiral, once a week seems about right, the system would consider the 3x3 grid that's centered on YOUR city. Those 9 cities would be sorted by the UNDERLYING CITY SCORE, with the highest scoring city ending up in the highest ranked sector in the 3x3 grid, ... , and with the lowest scoring city ending up in the lowest ranked sector in the 3x3 grid.
Could this really be possible? If I understand you correctly, Rank #15 would share his grid with Ranks #3-5, 14, 16, 33-34, while Rank #3 would share his grid with Ranks #1-2, 4, 11-15, as he has another grid centered on his city.
I'll try to explain my concern with an example. Let's assume that Rank #3 turns out to have the highest City Score when the movement takes place. This would mean that he gets to be at the centre of the mountain, as Rank #1, as he has beaten the score of the previous Rank #1. However, in the grid of Rank #15, Rank #3 also played a role. As he obviously has the highest score in that grid, shouldn't he stay at the same place as he was?
Another example. Let's assume that Rank #3 removes a lot of buildings and turns out to have the lowest City Score when the movement takes place. He would then take the place of Rank #15, the lowest Rank in his grid. But what if the previous Rank #16 would also have a higher score? Wouldn't this affect the placing of the lowered Rank #3? Rank #3 and Rank #16 didn't share a grid before the movement, so in theory they wouldn't affect each other, but this would mean that the placement of the other Ranks in your grid don't have to make sense in terms of City Scores. A player with a low City Score could be placed more at the center of the World Map than a player with a higher City Score, just because he did have a high score earlier (and it takes a while before your city would be placed at the edge of the World Map).
Somehow I think it could work to let each Rank have his own grid, but I still need some more convincing. I know it would be difficult, but could you explain how the map would change after one movement? Perhaps you could start with the easy *unrealistic) scenario that the Ranks are the same as the rankings of City Scores and then change the City Scores a bit to show how the cities would move.