Galidakis & Extending tetration to non-integers

[Daniel]

Yiannis Galidakis is now supporting my work at math.stackexchange. Feel free to comment on the math.

[MphLee]

Let me start with a provocative note.
The argument is so sharp that I can't see it.

Let me explain. I can't see the logic of the argument and I see at least three critical logic steps that invalidates the proof (both in your and in Galidakis's).
The structure of a proof is a matter of logic. Logic has as one of its main goals that of communicating and checking arguments in a "universal" way. I can not recognize your argument as a proof. The options are two: the argument needs to be improved because it is wrong or, the exposition of the, valid, argument has to be improved.

Since I'm not a specialist in analytic extensions it can be the case that for an expert in the field those three lines are enough and she/he can easily fill the trivial gaps.
I'm not a specialist but I eat a fair amount of logic daily and I can not recognize the structure of the proof as valid.

Can you expand on this filling the missing details?

[Daniel]

Let me start with a provocative note.
The argument is so sharp that I can't see it.

Let me explain. I can't see the logic of the argument and I see at least three critical logic steps that invalidates the proof (both in your and in Galidakis's).
...
Can you expand on this filling the missing details?

MphLee, thank you for your response and your humorous retort. Let me try and provide some clarity as to what is going on. There is a problem with the mingling of private and public correspondence; about a year ago Galidakis contacted me and let me know he felt I had found the correct path to solve the extension of tetration. I didn't go public with his support because Galidikis didn't go public himself. Likewise Galidakis has my support. Also he is the most published researcher with six papers published on tetration. Both Galidakis and I have tried to follow Trappmann and Kouznetsov's work, but couldn't follow and support the conclusions. 

FYI - Stephen Wolfram considers me likely the most important researcher in extending tetration. 

MphLee, can you clarify which proofs you have an issue with? The calculation of the extensions of tetration or convergence of the extension?

[MphLee]

Hi, I'm glad you are willing to help me.

Let's begin with your proof in the MSE link here.
The theorem is for sure not stated with enough care. In fact, as stated, it is false: tration is not bounded on all negative integers \( n\leq -2 \).

In the proof it is not clear how you define the expression \( {}^ba \) for complex numbers. I suspect you are referring to your Taylor series you defined in one of your papers so you agree with me that some context is missing here.
Another point that is not totally clear to me is the following and is probably about understanding your notation: given a \( z\in\mathbb C \) how do you select two integers \( j,k\in\mathbb Z \) such that \( jb\approx k \)? I can't understand the definition of \( \approx \).

Again... maybe that argument works well in the case of your Taylor series but for sure some context is missing. Spelling out explicitly the details can be very helpful imho.


About Galidakis argument in the same MSE link.
The logic goes as follow:
-(claim 1) Exists an unique "method that can guarantee analyticity within some initial region of convergence".
-(claim 2) That unique method is the "di Bruno expressions for the composition of the corresponding functions".

Since the series expansion is different at every point and works only within a small region we have to compare all the expansions at the intersections. If the expansions do agree on the intersections=good if they do not agree=bad and we don't have an everywhere analytic extension of tetration.

Up to here it seems plausible even if Claim 1 and Claim 2 ARE ASSUMED TRUE without a proof. This is a problem... because the proof is not complete or, I'm ignorant and thus important context is missing. But let's assume that c1 and c2 are true.

Galidakis uses a weird argument. He did a "quick numerical checks on some sample expansion" and they "do not seem to agree in the common region." If his numerical check is correct and we have numerical disagreement then it is a good hint that the thesis is true. I don't think that numerical evidence is enough for a proof but a counterexample should be easier to turn into a proof maybe... on this point I will not add more because people like Gottfried, Sheldon and James know alot more about numerical computation.

In fact my real problem is with his next statement:

If I have made a mistake and they agree, then Geisler's solution wins by default and that's the wanted everywhere analytic solution everybody's been looking for

Nope.. How can that be? It is enough to check a finite number of cases to inductively conclude that the expansions agree on all the intersections? Idk how complex analysis works, so maybe it is enough to perform a finite number of checks to conclude that globally something its true. But here Galidakis seems to claim that if he made a mistake, i.e. his computations are wrong, i.e. the computations did agree then...something happens,
This reasoning seems .... well... let's just say that I don't get the logic. The logic is not explicit.

Regards

[JmsNxn]

I have no idea what Daniel is talking about here...