So, I think I have it complete. I've added about 15 pages to this paper trying to make it air-tight, and drawing out every string. The proofs have increased in length, and all the things I took for granted have been better explained. I'm going to sit on it for maybe 5 more days, and go into proof-reading mode; making sure I haven't made any glaring mistakes. But I thought I'd share the shift in my thinking which allows for a more elegant approach.
Now, as we've done it, we've written our solution:
\(
\lim_{n\to\infty}\log \log \cdots(n\,\text{times})\cdots \log \phi(s+n+\omega) = e \uparrow \uparrow s = \phi(s+\omega) + \tau(s + \omega)\\
\)
Now, as \( \Re(s) \to -\infty \) we can write \( \phi(s) \to 0 \) and \( \tau(s) \to L \)--both geometrically--and \( e^L = L \). This can be done, given that \( \tau(s) \) is holomorphic on \( 0 < \Im(s) < 2\pi \) upto a nowhere dense set--which is the real challenge of the paper. Then, we can write,
\(
e \uparrow \uparrow s = \lim_{n\to\infty}\exp \exp \cdots(n\,\text{times})\cdots \exp \phi(s+\omega-n) + \tau(s+\omega-n)\\
\)
Again, almost everywhere. But this converges geometrically to,
\(
\lim_{n\to\infty}\exp \exp \cdots(n\,\text{times})\cdots \exp \phi(s+\omega-n) + L\\
\)
And it's fairly routine to show that this function converges uniformly on compact sets for \( \Re(s) < -T \) and \( 0 < \Im(s) < 2\pi \) as \( n\to\infty \). So in an essence, to get this construction to work, we want to look at the recursion from the opposite end to fill in the blanks that are missing. Now this form is definitely not ideal for the real line, but it works well on strips way off in the left hand plane. It also clearly shows why our construction CANNOT be holomorphic in a strip larger than \( 2\pi \)--it would be periodic if it were and tetration can't be periodic. So branch cuts arise and cluster in the iteration at the lines \( \mathbb{R} + 2\pi i k \) because otherwise we'd get a periodic tetration (which is absolute nonsense). And with it different \( L \) appear, as otherwise we'd achieve periodicity.
Some quick notes on \( L \); not too sure which \( L \) we converge to, but each strip \( 2 \pi k < \Im(s) < 2\pi(k+1) \) should garner a different \( L \). The key fact being, there is some \( L \) and \( \overline{L} \) in which this iteration will be real valued on the real line; which would correspond to the strips \( -2\pi < \Im(s) < 0 \) and \( 0 < \Im(s) < 2\pi \).
I'll set a deadline for myself on when I'll post this updated version. I'll release it on wednesday whether I feel it's perfect or not. If I keep hacking at this sooner or later it'll end up being a 50 page paper saying every possible thing I can say; including alternative representations... that may be a bit much for a quick update...
EDIT:
I think a zoom call for next saturday will be best. I'll send the request later--would 1pm be okay? Not to do it too late, but our 9am zoom call isn't optimal. Even though I live in toronto, I am on australia time, Sheldon. Lol.
Now, as we've done it, we've written our solution:
\(
\lim_{n\to\infty}\log \log \cdots(n\,\text{times})\cdots \log \phi(s+n+\omega) = e \uparrow \uparrow s = \phi(s+\omega) + \tau(s + \omega)\\
\)
Now, as \( \Re(s) \to -\infty \) we can write \( \phi(s) \to 0 \) and \( \tau(s) \to L \)--both geometrically--and \( e^L = L \). This can be done, given that \( \tau(s) \) is holomorphic on \( 0 < \Im(s) < 2\pi \) upto a nowhere dense set--which is the real challenge of the paper. Then, we can write,
\(
e \uparrow \uparrow s = \lim_{n\to\infty}\exp \exp \cdots(n\,\text{times})\cdots \exp \phi(s+\omega-n) + \tau(s+\omega-n)\\
\)
Again, almost everywhere. But this converges geometrically to,
\(
\lim_{n\to\infty}\exp \exp \cdots(n\,\text{times})\cdots \exp \phi(s+\omega-n) + L\\
\)
And it's fairly routine to show that this function converges uniformly on compact sets for \( \Re(s) < -T \) and \( 0 < \Im(s) < 2\pi \) as \( n\to\infty \). So in an essence, to get this construction to work, we want to look at the recursion from the opposite end to fill in the blanks that are missing. Now this form is definitely not ideal for the real line, but it works well on strips way off in the left hand plane. It also clearly shows why our construction CANNOT be holomorphic in a strip larger than \( 2\pi \)--it would be periodic if it were and tetration can't be periodic. So branch cuts arise and cluster in the iteration at the lines \( \mathbb{R} + 2\pi i k \) because otherwise we'd get a periodic tetration (which is absolute nonsense). And with it different \( L \) appear, as otherwise we'd achieve periodicity.
Some quick notes on \( L \); not too sure which \( L \) we converge to, but each strip \( 2 \pi k < \Im(s) < 2\pi(k+1) \) should garner a different \( L \). The key fact being, there is some \( L \) and \( \overline{L} \) in which this iteration will be real valued on the real line; which would correspond to the strips \( -2\pi < \Im(s) < 0 \) and \( 0 < \Im(s) < 2\pi \).
I'll set a deadline for myself on when I'll post this updated version. I'll release it on wednesday whether I feel it's perfect or not. If I keep hacking at this sooner or later it'll end up being a 50 page paper saying every possible thing I can say; including alternative representations... that may be a bit much for a quick update...
EDIT:
I think a zoom call for next saturday will be best. I'll send the request later--would 1pm be okay? Not to do it too late, but our 9am zoom call isn't optimal. Even though I live in toronto, I am on australia time, Sheldon. Lol.