The game of "thermonuclear pennies"

The game of thermonuclear pennies. This is, not surprisingly, related to the game of nuclear pennies; the game of nuclear pennies is in turn related to the Seven trees in one theorem of Blass, which I wrote about back in July. (Very roughly speaking, the "number of trees" can be shown to be "the primitive sixth root of unity", but the seventh power of this number is itself! Objects of Categories as Complex Numbers, by Marcelo Fiore and Tom Leinster, legitimates this.¹ The game of nuclear pennies works as follows; we start with a semi-infinite strip of squares, which have pennies on them. We can "split" a penny on site n into two pennies, one on site n-1 and one on site n+1; we can "fuse" pennies on sites n-1 and n+1 into one on site n. We can analyze positions in this game using arithmetic in the ring of integers with a sixth root of unity, ω, adjoined. Given a position with a_n coins in the nth position, associate the number

Σ_n a_n ωⁿ

with this position; it turns out that two positions can be reached from each other if and only if their associated numbers are the same. (We have the relations ω³ = -1, ω - ω² = 1.)

In thermonuclear pennies the splitting rule is a bit different; it turns out that "sixth root of unity" can be replaced with "fourth root of unity", which is the imaginary unit. Positions in this game represent Gaussian integers. And there's even a way to multiply the positions, which corresponds to multiplication of Gaussian integers.

Challenge (which may or may not be hard, I haven't thought about it all): develop such a penny-fusing game where the positions can be understood as elements of the ring of integers with a fifth root of unity adjoined. (I don't expect the result to be as nice; for example, Z[e^2πi/5] isn't a lattice in the complex plane. I'm not sure why that's relevant, but it might be.)

1. Is "legitimate" a verb, meaning "to make legitimate"? (The stress in the verb would be on the last syllable.) I think it should be, if it's not already. I've heard "precise" used as a verb (I don't remember how it's pronounced), and "legitimate" strikes me as more phonetically akin to verbs than "precise" is, probably because of the -ate suffix.