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2.2. Proof of [IUTT-3, Corollary 3.12]. Now let us try to unravel what happens in the criticalstep in the series of papers, namely towards the end of Step (xi) in the proof of [IUTT-3,Corollary 3.12]: “If one interprets the above discussion in terms of the notation introduced inthe statement of Corollary 3.12, then one concludes [...] that -|log(q)| ≦ -|log(Θ)| ∈ R.”
An extremely rough outline of what happens in this step is the following. One considers twoHodge theaters HT1 and HT2 linked by a Θ-link, so in particular the abstract Θ-pilot objectfrom HT1 is mapped to the abstract q-pilot object belonging to HT2.
As we indicated earlier,there is no clear distinction between abstract and concrete pilot objects in Mochizuki’s work,so it is argued in [IUTT-3, Corollary 3.12] that the multiradial algorithm [IUTT-3, Theorem3.11]*12
implies that up to certain indeterminacies, e.g. (Ind 1,2,3) (without which the conclusionwould be obviously false), this becomes an identification of concrete Θ-pilot objects and concreteq-pilot objects (encoded via their action on processions of tensor packets of log-shells), and thenthe inequality follows directly.

*12 We pause to observe that with the simplifications outlined above, such as identifying identical copies ofobjects along the identity, the critical [IUTT-3, Theorem 3.11] does not become false, but trivial.

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