Fourteen initial Θ-data on one curve over Q(i)

Let E: y2 = x(x − 1)(x − a) with a = (41 + 26i)472, a curve over Q(i). We prove that for each prime ℓ in Λ = {89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157} the curve extends to an initial Θ-datum in the sense of Definition 3.1 of Mochizuki's Inter-universal Teichmüller theory I, and that this datum satisfies the hypotheses of Theorem 1.10 of Inter-universal Teichmüller theory IV. The field of moduli is Q(i). Above the split prime 2357 there is one place of multiplicative and one of good reduction, with ramification indices 15ℓ and 1. The curve and the fourteen levels come from §13 of the author's second report (doi:10.5281/zenodo.22601333), which showed that the levels pass five numerical conditions and said explicitly that they had not been verified to be initial Θ-data; the present note gives the verification. The proof uses the individual ℓ only through five integer facts and one Frobenius witness per level. The norm of 1 − a has 1592 digits, and its factorization is not used: the valuation of 1 − a at each split place is bounded through the cyclotomic factor Φd(π), where d is the order of π modulo the place, and the 138,731 primes that this bound does not exclude are tested directly; the inert places and the places above 2357 are treated directly. The fourteen levels are the primes in [√(log q∀), 472/3), and they satisfy condition (C1) of Corollary 2.2 of IUT IV. Nothing is claimed about Corollary 3.12 of IUT III or about abc. 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 finding the example and its proofs, in the computations and in the writing. The author takes full responsibility for the contents of this report. The deposit contains the note and an archive of the folder fourteen-levels/ (the LaTeX source, the program that recomputes the arithmetic of the note and compares it with the printed numbers, and its output), made from tag v0.20.0 of github.com/turahigashi/iut-explicit-initial-theta. The files are licensed under CC BY 4.0.