A programmer's walkthrough of inter-universal Teichmüller theory: running the multiradial algorithm on explicit initial Theta-data
Year: 2026
Type: preprint
Abstract: We describe a program, with an interactive web page, that runs on explicit numbers those operations of inter-universal Teichmüller theory (IUT) that can be written as procedures. Its input is one of seven explicit initial Θ-data from an earlier catalogue (doi:10.5281/zenodo.23870448). The program computes the Tate parameters and the local data p-adically, arranges the labels and processions, and keeps the two Hodge theaters of a Θ-link as copies of the local fields separated by type, so that only a "cable" (the unit-group data and one unlabelled value-group object) passes between them, after an analogy that the source itself draws. It then computes log-shells, Kummer read-outs, the containers of the multiradial representation and the possible images of the theta values, and evaluates the log-volume of the holomorphic hull of their union, as an exact combination of logarithms, in a model in which the indeterminacy (Ind2) acts by units, for three readings of the label indeterminacy. Finally it places side by side the two quantities that Step (xi) of the proof of Corollary 3.12 of IUT III compares. Infinite objects are computed in finite windows, and the program prints how its output stabilizes as the windows grow. The properties IPL, SHE and APT of Remark 3.11.1 of IUT III appear as the shape of the code and as records of its execution; the inference (xi-f) is displayed, not evaluated. The purpose is expository. No conclusion is drawn about Corollary 3.12, Theorem 1.10 of IUT IV, or the abc conjecture. What is not new: turning the objects and processes of IUT into data types, with IPL and APT proved for a Lean model, has been done earlier and on a broader scale in the toolkit motizukipdf, whose C code also evaluates a numerical form of the comparison; the formulas for the containers and the distinction between the readings are taken from the Lean formalization lana-agents/iut. Section 8 of the note gives the related work. This is a preprint. Use of generative AI. Generative AI tools (large language models) were used extensively in this work, under the author's direction, in writing the program and the web page, in the computations and in the writing. The author takes full responsibility for the contents of this note. The deposit contains the paper and an archive of the folder iut-walkthrough/ (the LaTeX source and figures, the program with its outputs, and the interactive page, which is one HTML file), made from tag v0.12.0 of github.com/turahigashi/iut-explicit-initial-theta. The output of the program and the page are in Japanese. The composition of the animation tab of the page follows the animation The Multiradial Representation of Inter-universal Teichmüller Theory by Etienne Farcot, licensed under CC BY 4.0; this adaptation is not endorsed by the authors of the original. The files are licensed under CC BY 4.0.
Author: Urahigashi, Toshihisa
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225101354