Explicit initial Theta-data and the two readings of Step (v) of IUT IV, Theorem 1.10: a catalogue

We collect the explicit initial Theta-data, in the sense of Definition 3.1 of the first paper of Mochizuki's series on inter-universal Teichmüller theory, that the author has constructed over Q(√5) and Q(i): three published earlier and four given here, one of them for fourteen values of ℓ. Each has a split rational prime above which the two places of the field of moduli are either one of good and one of multiplicative reduction, or both of multiplicative reduction with different orders of the q-parameter. For each datum we tabulate the local invariants and compare, in Step (v) of the proof of Theorem 1.10 in the fourth paper, the displayed comparison module with the hull component of the union of possible images. Under the reading in which the theta value stays at the label j, as in Step (v) and in what the public Lean formalization lana-agents/iut calls the reading of IUT IV, every comparison is a containment. Under the reading in which label permutations act before the hull is taken, which the same formalization calls the literal reading, and under stated membership hypotheses, every datum gives a non-containment for a single component; at primes with a good and a bad place the excess grows like j², and for three data the weighted average over the components also exceeds the bound displayed at the prime. No conclusion is drawn about Corollary 3.12, the final inequality of Theorem 1.10, or the abc conjecture. What is new is the data themselves and the numbers: initial Theta-data whose field of moduli is not Q, data realizing both kinds of prime, and exact values of the comparisons of Step (v) on them. The distinctions between the readings, and the mechanism by which they differ, are not new; Section 6 gives the earlier work and the dates. The values for the fourteen levels over Q(i) (at the prime 2357) are conditional on a companion report in preparation, which contains the full verification of Definition 3.1 for those levels. This is a preprint. 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 data and the proofs, 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 initial-theta-catalogue/ (LaTeX source, the programs that compute the tables and check the arithmetic of the new data, and their output), made from tag v0.11.0 of github.com/turahigashi/iut-explicit-initial-theta. The files are licensed under CC BY 4.0.