Three tiers of indeterminacy: the exact hull at small places and the container of Step (v) of IUT IV

In Step (v) of the proof of inter-universal Teichmüller theory (IUT) IV, Theorem 1.10, the holomorphic hull of the union of the possible images of the Θ-pilot object is bounded by a container that contains the region pλ·(RI)∼; the indeterminacies (Ind1) and (Ind2) are accounted for by the arbitrary automorphisms φ of IUT IV, Proposition 1.2, and (Ind3) by taking upper bounds. At places that are small in the sense of Proposition 1.2(iii) (p odd and every ramification index at most p − 2), this note compares the container with the hulls reached by four classes of maps: unit scalars and field automorphisms; the maps induced factor by factor by the actual (Ind1) and (Ind2); all factorwise automorphisms of the log-shells; and all automorphisms φ of Proposition 1.2. For the last two classes the hull is exactly p⌈λ⌉−𝔡I·(RI)∼, where 𝔡I is the sum of the orders of the differents, whereas the container of Step (v) has hull p⌊λ⌋−𝔡I·(RI)∼. So when λ is not an integer, which is always the case at bad places, the container is larger than anything these automorphisms reach by exactly a factor p (log p in log-volume). The difference comes from the proof of Proposition 1.2(ii), which uses Proposition 1.1 with the different of every factor; Proposition 1.1 applied to the factor that carries pλ removes it. Unit scalars and field automorphisms do not enlarge the hull; the actual (Ind1), (Ind2) reach the exact value when λ is an integer, provided that automorphisms of the local Galois groups carry a unit to the boundary of the log-shell, which at small places reduces to one condition on the transvections of Hoshi–Nishio and Kondo. The difference is a term of conductor type and does not change the shape of the inequality of IUT IV. An example at a bad place of an explicit initial Θ-datum is computed. The note credits the earlier statements it builds on: IUT IV, Proposition 1.1, is stated with a free choice of the factor that does not pay its different; Dupuy and Hilado choose the factor that makes the inclusion tightest (arXiv:2004.13108); the public repository lana-agents/iut of Project LANA states that for λ = 0 the hull is exactly the container of IUT IV; Mochizuki states that automorphisms of the log-shells give only bounded discrepancies. It is the local, numerical counterpart of the author's note Where Theorem 3.11 sits, whose Theorem 22 uses only an upper bound for the same quantity. No claim is made about the truth of Corollary 3.12 of IUT III or of the abc conjecture. The proofs are on paper and the computations are exact; versions of the statements about the actual indeterminacies were also checked in Lean 4 in a development of the author that is not part of this deposit, and the proofs do not rely on it. The comparison studied here arose in the author's discussions with generative AI tools, and the author decided to make it the subject of this note. The proofs were found with generative AI tools (large language models) under the author's direction, and these tools were also used for the computations, the formalization, the literature search and the writing. The author's own reading of the proofs is still in progress. The author takes full responsibility for the note. The deposit contains the note and an archive of the folder three-tiers/ (LaTeX source, Python scripts and their outputs), made from tag v0.25.0 of github.com/turahigashi/iut-explicit-initial-theta.