[analytic_kernel]

I think your analysis of dominion spread has some merit, but it seems to be based on a fixed number of sources that do not scale with time (temple checks occurring at each turn) or space (more temples built as number of provinces increases). If you are building new temples, then the n in f(n) is not constant, but rather n(t). Also, irregular tile (province) geometry and variable numbers of neighbors per province make it more of a discrete or statistical problem that might have some difficulty being modeled with a simple polynomial function.