A verifiable knowledge graph of tetration

[RaeesHarris]

Hey everyone,
I’ve been building a knowledge graph of tetration: an attempt to collect what is known about it in one place and make it accessible to everyone. I did this with a lot of help from AI, so I tried to make everything checkable and structured
Interactive graph: https://lightrunnerwastaken.github.io/tetration-graph/
Repository: https://github.com/Lightrunnerwastaken/tetration-graph
A lot of it comes from this forum, and I’ve tried to attribute results to the threads and people they come from. Mapping the literature also showed me that several things I first thought were new were already here, e.g. Sheldon’s 2017 computation of the critical bases of higher hyperoperations. If you find a result that is attributed wrongly, or missing, please tell me.
A few things where I’d especially value your opinion:

• New values for the TPID 10 sequence: η₇ ≈ 1.8397586607 and η₈ ≈ 1.8681465773 (Heptation/Octation touching y = x). Has anyone computed these independently?
  
• A first attempt at a “half rank” between exponentiation and tetration at base 2, and why perturbation theory around x+1 stops at about 4 digits: the remaining freedom is exactly the 1-periodic θ.
  
• Which papers or threads are still missing from the literature map?
  

Everything is open (code MIT, content CC BY 4.0). I’d be very happy about corrections, missing references, or ideas how to make it more useful.

I am very curious about the half rank between exponentiation and tetration at base 2. How did you define the starting value?

Good question, I had to look it up again myself. I normalize every rank the same way: going from one rank to the next means taking f and building S with S(0) = 1 and S(x+1) = f(S(x)). So from f(x) = 2x you get 2^x, from 2^x you get tetration (with the usual 2↑↑0 = 1), and so on. At base 2 that’s nice because every rank gives H(2,1) = 2 and H(2,2) = 4.
For the half rank I looked at that step itself as a map on functions, T(f) = S. The successor x+1 is a fixed point of it (S(0)=1, S(x+1)=S(x)+1 gives back x+1), so I expanded around it with the family f_ε(x) = x + 1 + ε(2^x − x − 1) and took a “half iterate” of T order by order in ε. The series doesn’t converge (looks Gevrey-1), and there’s the usual periodic θ-freedom, so in the end I only trust about 3–4 digits. Two independent routes gave H_3.5(2, 0.5) ≈ 1.4428 and H_3.5(2, 1.5) ≈ 2.7824, both ± a few 10⁻⁴.
So the starting value is just S(0) = 1 at every rank. The hard part is the interpolation between ranks, and there I don’t have a uniqueness condition yet.

At the first rank for base 2, the function is x+2. So for the first rank, H(2,1) = 3.

Figuring out 2[x]b for \(b \in \mathbb{Z}\) would be quite a great feat even if the starting point function(2[x]0) was interpolated.