An initial Θ-datum over an imaginary quadratic field, and numerical values of the local quantities of inter-universal Teichmüller theory IV (A second report)

A second report in the series begun by 10.5281/zenodo.22537342. We exhibit an initial Θ-datum in the sense of Mochizuki, Inter-universal Teichmüller Theory I, Definition 3.1, whose field of moduli is the imaginary quadratic field Q(i): the curve E0: y² = x(x−1)(x−a) with a = (10+3i)29 and ℓ = 7, for which the rational prime 109 splits, so that one place of multiplicative and one place of good reduction lie above it. All conditions (a)–(f) are verified. The local invariants are eg = 1, eb = 105 and ord109(q) = 29/7, so the ramification invariants agree with those of the datum over Q(√5) in the first report; the residual data do not (fg = 12, fb = 6 here against 24 and 2 there). Sections 8–17 tabulate the local quantities of Inter-universal Teichmüller Theory IV, Theorem 1.10 and Proposition 1.4(iii) at these explicit inputs. Every entry is either a formula printed in the source or an exact computation from it, and each table names the locator of the quantity it tabulates. Scope. This report constructs data and evaluates local quantities on them. It does not compute the hull of the union of the possible images of the actual Θ-pilot or compare the two sides of [IUT III, Corollary 3.12]; no claim is made about Corollary 3.12, about the objections of Scholze–Stix, or about the abc conjecture. Artifact. The archive contains the paper, the reproduction scripts (all numbers regenerate from the standard library), an executable 89th-power-free certificate with its own negative control, and a Lean 4 development of fourteen files — 263 stated theorems, 768 declarations, no sorryAx and no Classical.choice — together with the audit log. Seven of the fourteen files are pinned by SHA-256 to the bytes released with the first report. What the kernel does not carry is stated explicitly in the paper. Version 1.0.1 (2026-09-07). Corrects one mathematical error in the comparison table of §9: the ramification indices printed for the curves 11a1 and 37a1 of Dupuy–Hilado carried over the torsion level 105 of the datum of this paper, which those curves do not have. Since [IUT I, Def. 3.1(b)] makes the 2·3-torsion rational over F, at the reference level ℓ = 7 the field K contains E[42], and both primes are prime to 42 with Tate orders 5 and 1, so eb ≥ 42 at both. The exclusions those rows state are unchanged (42 > 9 and 42 > 35), and nothing about the datum itself, the tables computed at it, Proposition 2, the fourteen candidate levels or the Lean development is affected. Also aligns the description of the 89th-power-free certificate with the fact that it now ships and runs. The manuscript was revised through nineteen rounds of adversarial review.