Split bad primes and the tuple structure of inter-universal Teichmüller theory: what the source states, and what follows (A fourth report)

Abstract

Version note. The first report has withdrawn, in its version 0.1.7, the source application under which the values 8/7 and 88/7 were read as an incompatibility with IUT IV; see that report's §7 for the locators and for a proposition limiting the scope of the withdrawal. No statement of the present report used that application, no numerical value changes, and nothing here is withdrawn. A fourth report in the series begun by 10.5281/zenodo.22537342. This report collects statements about inter-universal Teichmüller theory that are either quoted from a source or proved outright, together with the computations that accompany them. It records: that a tensor packet admits place tuples of mixed reduction type, and that selecting only nonsplit bad primes does not remove them; that the general-stage parameter is λ = j²·ord(q) and that the admissibility condition of [IUT IV, Prop. 1.4(iii)] yields the stage margin j²s − (j+2); where the source places the evaluation, quoted verbatim from the proof of [IUT III, Cor. 3.12]; two constraints on the indeterminacies, at a fixed nonarchimedean tuple and a fixed stage; and an identity between the two forms in which [IUT IV, Thm. 1.10] prints its procession-normalized coefficient. No verdict on the correctness of the theory or on abc is asserted, and no claim is made about [IUT III, Cor. 3.12] or about the objections of Scholze–Stix. Formulas printed in the source are marked [S] and deductions from them [P]; writing S(R) for “the source asserts R”, the report notes that S(R) does not imply R, and states the scope of each proposition rather than offering a blanket guarantee. All forty quoted passages ship with a checker and a manifest recording the SHA-256 of each source consulted.