The model of [exponential fields](https://en.m.wikipedia.org/wiki/Exponential_field) seems to be related, but these are defined with an exponential function instead of a commutative operator, and no inverse function. Furthermore, an exponential field defines a single field structure, while $\mathbb{H}_n$ defines $n-1$ field structures on the same object, alongside $n$ group structures.

Also, many of the ideas outlined above where first [proposed](https://math.stackexchange.com/a/383157/55205) by [@goblin](https://math.stackexchange.com/users/42339/goblin) back in 2013, but only in the context of $\mathbb{R}$, while distributivity was questioned but not established. One of the main benefits of the Hypertype Theory outlined above is to provide a recursive sequence of larger and larger objects that are isomorphic to subsets of $\mathbb{R}$ while providing full field structures.

Furthermore, if my understanding of [Schanuel’s conjecture](https://en.m.wikipedia.org/wiki/Schanuel%27s_conjecture) is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.