A local non-containment in Step (v) of IUT IV, Theorem 1.10
Abstract
Step (v) of the proof of Theorem 1.10 in Mochizuki's Inter-universal Teichmüller theory IV bounds, for each tuple of places, the log-volume of the corresponding component of the holomorphic hull of the union of possible images of a Θ-pilot object defined in Corollary 3.12 of the third paper. This note gives an explicit initial Θ-datum, with two places of multiplicative reduction above the prime 30965771, and an element of the projection of that union to one summand, obtained from a theta value by a permutation of labels allowed by the indeterminacy (Ind1), that does not lie in the module through which Step (v) bounds that component; the corresponding component of the hull therefore has larger log-volume than the bound obtained in Step (v). No assertion is made about Corollary 3.12 itself, the final global inequality of Theorem 1.10, or the abc conjecture. This is a preprint. It has not been refereed. The first report in the same GitHub repository (10.5281/zenodo.22537342) compared a different datum, at the prime 211, with the local estimates of IUT IV; the source application of that comparison was withdrawn in its version 0.1.7. This note is a separate paper, not a revision of the first report: its datum has two places above the prime, both of multiplicative reduction, and it neither cites nor relies on the first report. Use of generative AI. Generative AI tools (large language models: Claude by Anthropic, and ChatGPT and Codex by OpenAI) were used extensively in this work, under the author's direction: in finding the example and its proofs, in the computations, in checking the statements against the cited sources, and in writing the text. They affected all parts of the note. The author takes full responsibility for the contents of this note. The deposit contains the paper and an archive of its LaTeX source, made from tag v0.8.0 of github.com/turahigashi/iut-explicit-initial-theta (folder stepv-local-noncontainment/). The files are licensed under CC BY 4.0.
