Global Normalization in Joshi's Arithmetic Teichmüller Theory: What the Identification Carries — and What Carries the Theorem

The Global Normalization Contract Lemma Tool supplement of the abc Gap Series — a connected sequence of papers that locate, formalize, verify, and repair the central gap in the Mochizuki/Joshi approaches to the abc conjecture (Part 1: Forensic Diagnosis, DOI 10.5281/zenodo.21899747). Parts appear individually as they are completed; this lemma is the normalization tool used by the repair programme. This paper asks a single question about the identification that carries Theorem 6.10.1 of Joshi's ATS IV (arXiv:2403.10430): does "identically normalized" assert an isometry between two valuation scales, or does it establish one by definition? The answer is given against Joshi's own definition of the normalized arithmeticoid arith(L)nor — and it is conditional, because we did not locate, in the passages examined, a rule that selects the normalization coordinate λy of a moved arithmeticoid. Under the branch in which λy is pinned place by place to the Artin–Whaples standard, the identification is definitional and not load-bearing: an elementary derivation from quoted statements — carried out under two explicitly recorded reading assumptions (F1)–(F2) — indicates that the product-formula hyperplane moves under the Frobenius, the substance of the middle equality lying instead in the by-construction Frobenius stability of the collated adelic set and in a measure computation that factors over number-field data. Under the branch that Joshi's own use of the normalization elsewhere appears to require — a residual, place-selective freedom — the tautology verdict does not follow. The paper states this alternative as an explicit trilemma, marks every verdict for the branches under which it holds, and proves a lemma that constrains the middle branch: the product formula determines its multiplier only up to a single global scalar (via Dirichlet's unit theorem and finiteness of the class number), so no place-selective rule can be extracted from the product formula alone. A concluding section records the precise correspondence question that decides the branch. Claim discipline: No error in the sources is asserted, and no claim on abc is made. Every verdict is branch-marked; the open selection rule is stated as an open question, not as a defect. Files: English version (24 pp.), German version (25 pp.), and a combined bilingual PDF. ─────── ✦ ───────seriesdirectory (each part lists the whole series; planned parts are updated with title and DOI upon publication): ✔ Part 1: The Gap in IUTchIII, Corollary 3.12 — DOI 10.5281/zenodo.19960781 (latest version)✔ Part 2: The Bridge-Contract Criterion: A Form Audit for Transport Arguments, with an Instantiation in the abc Literature — DOI 10.5281/zenodo.21960476 (latest version)✔ Part 3: The Bridge Certificate — DOI 10.5281/zenodo.21925803 (latest version)✔ Part 4: The Bridge Certificate Applied to the Joshi ATS Series: A Ledger Audit — DOI 10.5281/zenodo.21951155 (latest version)✔ Part 5: The Joshi ATS Series under the Bridge-Certificate Standard: A Targeted Substance Audit and an All-Place Rigidity Classification of Effective Normalization Weights — DOI 10.5281/zenodo.21956398 (latest version)✔ Part 6: Repairing the ATS Chain? A Rigidity Theorem for Product-Formula Weights and an Effective-Weight Test for the Height Subchain — DOI 10.5281/zenodo.21952486 (latest version) ↳ coming soon - Part 7: The bridge certificate applied to our own programme (self-application; working title) — (planned; appears in sync with the landscape paper) ✎ This Paper: The Global Normalization Contract Lemma — DOI 10.5281/zenodo.21924253 (latest version) ℹ Companion (meta) paper: From Landscape to Atlas: Multi-Route Cartography of an Ongoing Expedition Toward the abc Conjecture (formerly: The abc Landscape) — DOI 10.5281/zenodo.21916900 (latest version). The atlas paper is the complete memory of all routes; the series parts and capsule papers present selected results as independently citable units.