02/16/2021, 08:40 AM
So I posted a proof of \( C^\infty \) before I started working today--I noticed a small expansion error, which I hastily fixed--so I deleted the post. I figured I'd do a couple more run throughs of the result before I posted. It was a dumb mistake towards the end where I got ahead of myself, but everything still works; there was one download on the file, so whoever downloaded it I apologize in advance. (Unless the one download was my own download, not sure how it works here.)
I am in the process of rewriting my entire paper to focus on \( C^\infty \) hyper-operations; whereby it's a long proof by induction. But the initial step is to prove that this tetration is \( C^\infty \). Now I can most definitely show this tetration is \( C^\infty \); the trouble I'm having is making the proof as general as possible; so that we can create a proof by induction showing \( e \uparrow^k t \) is \( C^\infty \). The proof I'm posting now is intended to be abstract because I intend to use it as a template for the inductive process.
Now, the proof is a little rough around the edges. I haven't fleshed out everything, but honestly it'd only take more words, not more work. Everything definitely works. This isn't much more than what my initial post in this thread was; but it's far better explained. I'm posting this theorem here, without the full paper; essentially to gauge how well explained it is. If anyone has any questions, or any comments or hangups, I'm beyond happy to answer them (and they'll help because they'll teach me how to better write the paper).
Despite my errors at assuming this construction would be analytic; I promise no mistakes were made in this circumstance. Somethings may be unclear though, and if they are, please tell me so I can correct myself.
I am in the process of rewriting my entire paper to focus on \( C^\infty \) hyper-operations; whereby it's a long proof by induction. But the initial step is to prove that this tetration is \( C^\infty \). Now I can most definitely show this tetration is \( C^\infty \); the trouble I'm having is making the proof as general as possible; so that we can create a proof by induction showing \( e \uparrow^k t \) is \( C^\infty \). The proof I'm posting now is intended to be abstract because I intend to use it as a template for the inductive process.
Now, the proof is a little rough around the edges. I haven't fleshed out everything, but honestly it'd only take more words, not more work. Everything definitely works. This isn't much more than what my initial post in this thread was; but it's far better explained. I'm posting this theorem here, without the full paper; essentially to gauge how well explained it is. If anyone has any questions, or any comments or hangups, I'm beyond happy to answer them (and they'll help because they'll teach me how to better write the paper).
Despite my errors at assuming this construction would be analytic; I promise no mistakes were made in this circumstance. Somethings may be unclear though, and if they are, please tell me so I can correct myself.
