No reading avoids the height inequality: 36 readings of Corollary 3.12 and the conservation of the height load

Corollary 3.12 of inter-universal Teichmüller theory (IUT) III compares the log-volume −|log q| of the q-pilot object with the log-volume −|log Θ| of the holomorphic hull of the union of the possible images of the Θ-pilot object. Theorem 1.10 of IUT IV bounds −|log Θ| from above, and the two together give the height inequality. The value of −|log Θ| depends on how the statement is read, along five axes: the range of the label permutations of (Ind1), container or iteration for (Ind3), the group of (Ind2), whether the hull is taken, and the archimedean container. Combining the axes found so far gives 36 readings that change the value. We show that in each of the 36 readings, also with a bounded Kummer distortion added, Corollary 3.12 and the upper bound of IV together give the same height inequality (k − 1)|log q| ≤ C. The reason is that |log q|, k and C are computed by one procedure that takes no reading; only the middle term −|log Θ| depends on the reading. Moreover, a reading decides only how much of the wall (k − 1)|log q| is carried by Corollary 3.12 and how much by IV. This split is decided by the (Ind1)-axis alone; the other axes change only a bounded quantity that does not involve the q-orders. No reading makes both loads nonpositive. In the family obtained by replacing q with qN, the statement that holds for every N carries no positive load, and the other carries at least the whole wall. We prove these statements in Lean 4 for a component model (96 declarations, standard axioms only), and compute the 36 values for seven explicit initial Θ-data. That each axis other than (Ind1) changes only a bounded remainder rests on a paper argument and on the programs; Lean uses it as a hypothesis. The 36 readings are the combinations of the axes found so far. The fact that proofs through the IUT package use the wall itself is already in the public formalization lana-agents/iut for one reading; what is new here is the form over the family of readings. No claim is made about the truth of Corollary 3.12, Theorem 1.10 of IUT IV, or the abc conjecture. This is a preprint. Use of generative AI. The question studied in this note — whether choosing the readings differently avoids the height inequality or makes the proof cheaper — arose in the author's discussions with generative AI tools; the author judged it important and decided to make it the subject of this note. Its ultimate aim, to understand exactly the algorithm by which IUT computes the height inequality, is the author's research direction. Generative AI tools (large language models) were used extensively, under the author's direction, in finding the proofs, in the computations, in the verification in Lean and in the writing. The author takes full responsibility for the contents of this note. The deposit contains the note and an archive of the folder reading-family/ (the LaTeX source, the Lean files with their logs, and the programs with their outputs, including a copy of version 1.1.0 of the program of the author's walkthrough), made from tag v0.17.0 of github.com/turahigashi/iut-explicit-initial-theta. The files are licensed under CC BY 4.0.