A verifiable knowledge graph of tetration

[Lightrunner]

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.

[Lightrunner]

test
sry I messed up smth with editing...

[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?

[Lightrunner]

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.

[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.

[MphLee]

The idea is fire, per se, IMHO. The problem is a personal one: I still don’t know what to make of all the “plastic-fraud” jargon that LLMs inject into anything remotely technical-looking.

I recently read two papers that were clearly AI-generated by free or low-cost models. Everything sounded so good that it couldn’t be true. Some parts of the analysis and the overall inferences seemed fishy, especially in areas of mathematics I wasn’t familiar with. But when the papers got to something I knew I was strong in, I immediately saw how shallow and incomplete the reasoning was, dressed up in perfectly conventional mathematical jargon.

It’s already hard enough to spot errors in formal proofs written by humans. It’s even easier to miss subtle details in “pre-formalized proofs” or discursive proofs, that is, writing that’s half math and half mathematical English and those proofs make 90% of preprints and peer-reviewd literature. I wonder how many errors are hidden behind LLMs’ pristine English grammar, perfectly stochastic style, and good-taste writing practices.

That said, I’m fine with AI tools summarizing and automating semantic classification and tagging tasks, as well as many others tasks. But an expert eye should always check the final result and take responsibility for what it represents and what it’s meant to serve. For example, I’m looking at the fractional-ranks section. I’m sure I’m not smart enough to follow everything, but I can say for sure that the material isn’t explained very well. A more intuition–definition–proposition–proof approach would benefit the “human” reader. Another weak point in the recent use of AI tools to produce mathematical discourse has often been the lack of pedagogical and intuitive friendliness. Just my two cents.

[bo198214]

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

Wow, that's a great idea - actually (long time ago) I was thinking of this kind of undertaking.
But its really a tedious matter and when I wanted to structure it, it became so complex that it seemed not that helpful anymore.
Truth is, that almost all of the numerical methods for the case of no real fixed point have no prove of convergence, however mostly convincing numerical behaviour.
And even under the assumption of convergence it is not clear whether they converge to the same function.
Ok, just wanted to give my 3 cents, though I have not really time to be involved in it

Regards & Greetings

[marcokrt]

Hi Lightrunner,

I have examined the "Congruences" section of your tetration knowledge graph and found a significant problem with the attribution of established results concerning the congruence speed of integer tetration: I would like to request a proper correction, not merely the addition of a few optional references.
In detail, you have included a node named br.satz.ripa, which acknowledges some of my work, ok... however, your node ar.satz.padisch_bewertung describes its result as a refinement of Ripà–Onnis, with the following bibliographical assessment: "Core (constant speed) in Ripà–Onnis; exact form with onset height as order stabilization not found there."

Now, this characterization is seriously incomplete and potentially misleading.

In particular, my 2022 paper (with Luca Onnis) does not merely establish the eventual constancy of the congruence speed: Section 2.1, Equations (4)–(10), already provides exact formulas for the number of frozen digits at individual tower heights for entire infinite families of integer bases.
For example, Equations (8 ) and (9) explicitly give:
--> For \(a=15\): #\(S_5(15,b)\) = \(4b+1\), for \(b\geq 2\).
--> For \(a=25\): #\(S_5(25,b)\) = \(3(b+1)\), for \(b\geq 2\).
These coincide with the corresponding 2-adic height-dependent expressions appearing in your graph (Ripà and Onnis, 2022, Number of stable digits of any integer tetration, NNTDM 28(3), 441–457, https://doi.org/10.7546/nntdm.2022.28.3.441-457).

The distinction matters: your particular characterization of an onset height through multiplicative orders may be a different formulation, but the underlying quantitative phenomenon, its valuation-based analysis, and exact finite-height digit counts were already established in our published work (that's a fact).

Moreover, my recent peer-reviewed paper (with Gabriele Di Pietro), A compact notation for peculiar properties characterizing integer tetration (NNTDM 32(3), 477–496, https://doi.org/10.7546/nntdm.2026.32.3.477-496), received by NNTDM on May 20, 2025, and published online on October 15, 2026, explicitly provides (Appendix B, Equation (3.4)) the following prime-radix congruence speeds for every integer \(a>1\) not divisible by \(p\):
\(V_p(a)=v_p(a^{p-1}-1)\), for odd primes \(p\);
\(V_2(a)=v_2(a-1)+v_2(a+1)-1\), for \(p=2\).

The above-mentioned article distinguishes these established prime-radix results from the genuinely conjectural assertions concerning non-squarefree radices and composite squarefree radices (see https://doi.org/10.7546/nntdm.2026.32.3.477-496).

There is also a rather striking bibliographical issue here: I have been presenting and discussing this research on the Tetration Forum itself since December 2011, including in my December 2024 thread "A very special set of tetration bases" (https://tetrationforum.org/showthread.php?tid=1796).
Some other examples are https://tetrationforum.org/showthread.php?tid=720 (December 2011), https://tetrationforum.org/showthread.php?tid=1621 (August 2022), https://tetrationforum.org/showthread.php?tid=1815 (December 2025).
My earlier publications were also referenced in the forum's "Major references" discussion.
Consequently, these are not obscure, unpublished results that would have required an extensive literature search to discover... they have been publicly available, peer-reviewed, and repeatedly discussed in this very community.
Your bibliography currently lists my 2020 paper, "On the constant congruence speed of tetration" and my 2022 paper with Luca Onnis, "Number of stable digits of any integer tetration",as checked only through secondary sources, explicitly stating that you consulted the abstract of the latter. With all due respect, consulting an abstract is not sufficient to assess whether a mathematical result is new, particularly when the purported refinement concerns exactly the subject of that paper.
IMO, a knowledge graph presented as "verifiable" should not leave readers with the impression that established results are new merely because the earlier proofs were not examined.

I therefore kindly request that you:
1) Correct the attribution and description of ar.satz.padisch_bewertung, explicitly acknowledging the earlier exact finite-height results of Ripà–Onnis (2022).
2) Clearly distinguish any genuinely new characterization of the onset height from previously published formulas and results. The same distinction should be maintained in ar.satz.padisch_koenigs, wherever it relies on established congruence-speed or digit-stabilization properties;
3) Update the relevant nodes and their visible references, not merely the bibliography stored in the repository;
4) Include the principal references needed to represent the development of this theory accurately:
- Ripà, M. (2021). The congruence speed formula. NNTDM, 27(4), 43–61 (also available at https://arxiv.org/abs/2208.02622 ), https://doi.org/10.7546/nntdm.2021.27.4.43-61);
- Ripà, M. and Onnis, L. (2022). Number of stable digits of any integer tetration. NNTDM, 28(3), 441–457, https://doi.org/10.7546/nntdm.2022.28.3.441-457;
- Ripà, M. and Di Pietro, G. (2026), A compact notation for peculiar properties characterizing integer tetration. NNTDM, 32(3), 477–496 https://doi.org/10.7546/nntdm.2026.32.3.477-496.

For completeness, two other peer-reviewed papers address related results and applications:
- "On the relation between perfect powers and tetration frozen digits" (2024): https://doi.org/10.59400/jam1771
- "Graham's number stable digits: An exact solution" (2025): https://doi.org/10.7546/nntdm.2025.31.3.607-616
Finally, there is also a recent, this time primarily GPT-6-generated preprint developed under my research guidance, proposing further results on the nonconstant congruence speeds of non-squarefree radices: https://hexagonmath.org/2610.00589v1 (again, consider that this last work has not yet undergone comprehensive independent proof verification and its Lean formalization is in progress, so it should be distinguished from all the established, peer-reviewed results listed above).

Just let me clarify that I am not claiming priority over every possible \(p\)-adic reformulation or over the classical tools used in their derivation, I am asking for something much more specific: accurate attribution of the mathematical results that my collaborators and I have already established and published in peer-reviewed journals.
I have no objection to genuinely new developments building on my work: on the contrary, I welcome them (but the boundary between previously established results and new contributions must be clearly identified).

In conclusion, I think that scientific credit is not an optional detail in a reference project of this kind… rather, it is an essential part of its accuracy and reliability.
I am confident that these attribution issues can be easily fixed, so "thanks in advance".

Regards,
Marco Ripà