Two computations of the height inequality and the multiradial algorithm: reading IUT from the viewpoint of computer science

This note contains almost no new mathematical results. It is an essay that proposes a viewpoint and an idea: to read inter-universal Teichmüller theory (IUT) from the side of computer science. It presents the tools for doing so — two programs that read the same input, and a mockup of the multiradial algorithm in which some parts are left as oracles — and a little analysis made with them. It proves no conclusion. The view that IUT computes the same quantity in two ways is Mochizuki's own and is also recorded in the interim report of Project LANA, and the explicit computation of the IUT side is in Dupuy–Hilado and in Zhou. What this note adds is to place the two computations side by side as programs that read the same input, the mockup with oracles, and a computer-science reading together with a position. In each of the 36 readings listed by the author and by lana-agents/iut, Corollary 3.12 of IUT III and Theorem 1.10 of IUT IV together give the same height inequality (k − 1)|log q| ≤ C. This inequality is written in classical quantities only, and for a given curve both sides can be computed without IUT; what IUT supplies is a different method of computing the two sides. The key is universality: Corollary 3.12 claims to hold for every initial Θ-datum, and what is supposed to supply this is the multiradial algorithm of Theorem 3.11. The note regards this algorithm as the essence of IUT, and takes the position that one has not understood IUT essentially unless it has been elucidated and implemented. Two C programs compute the classical side and the side of IUT (in the reading of IUT IV) from the same local data. A C++ mockup of the multiradial algorithm implements what can be implemented at present — the local invariants from the abstract group GKvab, the log-shells, Kummer classes at a finite level, cyclotomic rigidity from the commutator in a theta group, the labels and tuples of the tensor packets, and the hull of the possible images — and leaves the rest as named oracles (the anabelian reconstructions, the Kummer classes of the theta values, the global data, the containment of (Ind3), and Step (xi)). The values of the oracles are reverse-computed from the ring structure of a theater or from the conclusion, and the output says so. On seven explicit initial Θ-data the mockup agrees with the C programs on 204 items. The implemented part is separated by types so that it cannot read the ring structure of a theater, and its output does not depend on the coordinates that a theater hides. In the family q → qN, the implemented part runs the same steps for every N, while the reverse-computed Step (xi) answers "no" from N = 3 (from N = 4 for one datum) on the six completely factored data. lana-agents/iut has shown that Step (xi) does not follow from the formalized content of Theorem 3.11: over that content Step (xi) is equivalent to the Scholze–Stix inequality, and the missing input is of ABC strength. This is the result of taking the content of Theorem 3.11, as currently written, as a specification; it does not show that IUT cannot be repaired, as lana-agents/iut itself states. In the view of this note, the key lies in the implementation of the multiradial algorithm; the future task is to implement the oracles as actual algorithms, or as approximations to them. No claim is made about the truth of IUT, of Corollary 3.12, or of the abc conjecture. This is a preprint. Use of generative AI. The question studied in this note — whether placing the computation of IUT and the classical computation side by side as programs lets one approach the content of IUT from the viewpoint of computer science — and the approach of regarding the multiradial algorithm as the essence and placing it in a mockup as oracles arose in the author's discussions with generative AI tools; the author judged them important and decided to make them the subject of this note. Its ultimate aim, to understand exactly the algorithm by which IUT computes the height inequality, is the author's research direction. Generative AI tools (large language models) were used extensively, under the author's direction, in the computations, in writing and checking the programs, and in the writing. The author takes full responsibility for the contents of this note. The deposit contains the note and an archive of the folder two-computations/ (the LaTeX source, the two C programs, the C++ mockup, the local data, the scripts and their outputs), made from tag v0.18.0 of github.com/turahigashi/iut-explicit-initial-theta. The files are licensed under CC BY 4.0.