Yesterday, 08:33 AM
Hello everyone,
my name is Janis, and I am a high-school student with a strong interest in tetration. Over the last one or two years, I have spent much of my free time experimenting with tetration numerically and trying to understand some of its underlying structure.
The work did not begin as a rigorous research project. It started with computations, observations, and many incomplete ideas. With the help of AI tools, I was able to test these ideas, formulate them more precisely, and develop them into a series of papers. The papers contain a detailed statement explaining the role of AI in the project, and I take responsibility for the final content.
Today I am sharing three connected parts of the project:
1. A five-paper series on mixed-base tetration
The main object I study is the mixed-base height map
F_{b,d}(x) = slog_b(T_d(x)),
where T_d is tetration in base d and slog_b is the superlogarithm in base b.
This provides a type of base change between different tetration functions. Numerically, the difference between this map and the original height does not generally approach a constant. Instead, it appears to approach a small periodic phase function. The papers investigate this phase, its algebraic structure, its interpretation as a circle map, its numerical fine structure, and its dependence on the tetration base.
All five papers are available as preprints on Zenodo, and their PDFs are included in the repository:
https://github.com/Lightrunnerwastaken/g...ain/papers
2. An optimized fork of Sheldonison’s fatou.gp
All computations build on Sheldonison’s original fatou.gp, for which I am very grateful.
I made a number of performance improvements while repeatedly comparing the results against frozen accuracy tests. On my machine, one high-precision base-e computation at approximately 500 digits was reduced from about 5.4 hours to about 25 minutes. The exact benchmarks and methods are documented in the repository.
The repository also contains high-precision reference values with numerical error bounds. The strongest current result is a computation of sexp_e(0.5) with a claimed 972-digit error bound.
3. A base-change calculator
Using the phase functions, I constructed a small “base atlas.” After preparing one anchor tetration function, it approximates sexp and slog for other bases using a short table of Fourier coefficients..
The complete repository is here:
https://github.com/Lightrunnerwastaken/gp-tetration
I would be very grateful for feedback, especially about possible errors, missing assumptions, the claimed error bounds, and the mathematical interpretation of the phase functions. I am still learning and would be happy to correct or improve anything.
I also have a practical question for Sheldonison: I could not find an explicit license in fatou.gp. I have preserved the attribution and documented the origin of the code. Please let me know whether you are comfortable with the fork being distributed in this form and what licensing terms you would prefer. If you would rather have the original file removed from the repository, I will of course do so.
Preprints:
my name is Janis, and I am a high-school student with a strong interest in tetration. Over the last one or two years, I have spent much of my free time experimenting with tetration numerically and trying to understand some of its underlying structure.
The work did not begin as a rigorous research project. It started with computations, observations, and many incomplete ideas. With the help of AI tools, I was able to test these ideas, formulate them more precisely, and develop them into a series of papers. The papers contain a detailed statement explaining the role of AI in the project, and I take responsibility for the final content.
Today I am sharing three connected parts of the project:
1. A five-paper series on mixed-base tetration
The main object I study is the mixed-base height map
F_{b,d}(x) = slog_b(T_d(x)),
where T_d is tetration in base d and slog_b is the superlogarithm in base b.
This provides a type of base change between different tetration functions. Numerically, the difference between this map and the original height does not generally approach a constant. Instead, it appears to approach a small periodic phase function. The papers investigate this phase, its algebraic structure, its interpretation as a circle map, its numerical fine structure, and its dependence on the tetration base.
All five papers are available as preprints on Zenodo, and their PDFs are included in the repository:
https://github.com/Lightrunnerwastaken/g...ain/papers
2. An optimized fork of Sheldonison’s fatou.gp
All computations build on Sheldonison’s original fatou.gp, for which I am very grateful.
I made a number of performance improvements while repeatedly comparing the results against frozen accuracy tests. On my machine, one high-precision base-e computation at approximately 500 digits was reduced from about 5.4 hours to about 25 minutes. The exact benchmarks and methods are documented in the repository.
The repository also contains high-precision reference values with numerical error bounds. The strongest current result is a computation of sexp_e(0.5) with a claimed 972-digit error bound.
3. A base-change calculator
Using the phase functions, I constructed a small “base atlas.” After preparing one anchor tetration function, it approximates sexp and slog for other bases using a short table of Fourier coefficients..
The complete repository is here:
https://github.com/Lightrunnerwastaken/gp-tetration
I would be very grateful for feedback, especially about possible errors, missing assumptions, the claimed error bounds, and the mathematical interpretation of the phase functions. I am still learning and would be happy to correct or improve anything.
I also have a practical question for Sheldonison: I could not find an explicit license in fatou.gp. I have preserved the attribution and documented the origin of the code. Please let me know whether you are comfortable with the fork being distributed in this form and what licensing terms you would prefer. If you would rather have the original file removed from the repository, I will of course do so.
Preprints:
- A Phase Law for Mixed-Base Tetration
https://doi.org/10.5281/zenodo.21346803
- The Mixed-Base Tetration Phase Map Is a Circle Diffeomorphism
https://doi.org/10.5281/zenodo.21361967
- Stopped Channels and Offset Alignment
https://doi.org/10.5281/zenodo.21362206
- Fine Structure of Mixed-Base Tetration Phases
https://doi.org/10.5281/zenodo.21363593
- The Base-Derivative of the Kneser Family
https://doi.org/10.5281/zenodo.21363809
