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1. Where can we find prior work on the subject?

The model of exponential fields 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 by @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.

5. Can $\pi$ be naturally introduced as a term of $\mathbb{H}_n$?

If my understanding of Schanuel’s conjecture is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.

7. Is this interesting?

This is interesting for many different reasons:

First, it provides a recursive set of objects $\mathbb{H_n}$ that have very simple definitions yet unique properties. Among them, the existence of $n$ group structures and $n-1$ field structures. And because of the recursive nature of the construction, many properties exhibited at level $n$ become true at any level. This provides a very powerful model to work with.

Second, the construction centered around $e$ introduces this transcendental number in a very natural manner. This is important, because $e$ is one of the few common numbers that is believed not to be a period. Therefore, the combination of Hypertype Theory and Period Theory provides a solid framework for the classification of transcendental numbers.

Third, the introduction of a commutative power law sheds a new light on the ntion of power laws and exponential operators, which are traditionally non-commutative.

1. Where can we find prior work on the subject?

The model of exponential fields 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. Therefore, exponential fields and hypertypes are quite different.

Many of the ideas outlined above where first proposed by @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 prposed here is to provide a recursive sequence of larger and larger objects that are isomorphic to subsets of $\mathbb{R}$ while providing recursive group and field structures.

5. Can $\pi$ be naturally introduced as a term of $\mathbb{H}_n$?

If my understanding of Schanuel’s conjecture is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.

7. Is this interesting?

This is interesting for several reasons:

First, it provides a recursive set of objects $\mathbb{H_n}$ that have very simple definitions yet unique properties. Among them, the existence of $n$ group structures and $n-1$ field structures. And because of the recursive nature of the construction, many properties exhibited at level $n$ become true at any level. This provides a very powerful model to work with.

Second, the construction centered around $e$ introduces this transcendental number in a very natural manner. This is important, because $e$ is one of the few common numbers that is believed not to be a period. Therefore, the combination of Hypertype Theory and Period Theory provides a solid framework for the classification of transcendental numbers.

Third, the adoption of a commutative power law sheds a new light on the notion of power laws and exponential operators, which are traditionally non-commutative. The introduction of a different viewpoint on a fundamental operation might foster innovation or lead to some interesting discoveries.

The model of exponential fields 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 by @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 is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.

1. Where can we find prior work on the subject?

The model of exponential fields 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 by @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.

5. Can $\pi$ be naturally introduced as a term of $\mathbb{H}_n$?

If my understanding of Schanuel’s conjecture is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.

7. Is this interesting?

This is interesting for many different reasons:

First, it provides a recursive set of objects $\mathbb{H_n}$ that have very simple definitions yet unique properties. Among them, the existence of $n$ group structures and $n-1$ field structures. And because of the recursive nature of the construction, many properties exhibited at level $n$ become true at any level. This provides a very powerful model to work with.

Second, the construction centered around $e$ introduces this transcendental number in a very natural manner. This is important, because $e$ is one of the few common numbers that is believed not to be a period. Therefore, the combination of Hypertype Theory and Period Theory provides a solid framework for the classification of transcendental numbers.

Third, the introduction of a commutative power law sheds a new light on the ntion of power laws and exponential operators, which are traditionally non-commutative.

The model of exponential fields seems to be related, but these are defined with an exponential function instead of a commutative operator, and no inverse function.

Also, many of the ideas outlined above where first proposed by @goblin back in 2013, but only in the context of $\mathbb{R}$. 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 $\mathbb{R}$.

Furthermore, if my understanding of Schanuel’s conjecture is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.

The model of exponential fields 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 by @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 is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.