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

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 at the level of number-field element valuation scales. An elementary derivation from quoted statements — carried out under the explicitly recorded reading assumptions (F1)–(F2) — conditionally indicates movement of the product-formula hyperplane under Frobenius. This does not establish the indexed middle logarithmic-volume equality (E2). Frobenius stability of a collated adelic set is a source-reported candidate contribution; matching the compared ambient spaces, integral structures, theta-set images, hulls, measures and weights remains an independent obligation. Under residual place-selective freedom, the tautology verdict does not follow. The paper states this alternative as an explicit diagnostic trilemma and retains an elementary lemma that constrains its product-formula branch: in the stated positive real-exponent setting, the product formula determines its multiplier only up to a single global scalar (via Dirichlet's unit theorem and finiteness of the class number). It cannot by itself supply a place-selective normalization rule. The basis conversion is displayed before the lemma; this lemma concerns absolute values, not Haar measures. The cross-frame measure comparison stays open in all three branches. The standard identity m(pZp)=p−1 for normalized additive Haar measure is separated from a norm-power convention |p|pc=p−c. Corrections in v1.2: ATS IV v2 Section 1.13 and Remark 1.13.2 use the corrected domain L̅*; the separate ATS II(1/2) Theorem 8.7.1 remains on L*, and the height passage F-NEU-1 remains open. The outer absolute-value bars in ATS IV Theorem 6.10.1 are restored in both presentations of E2. The generic identification of algebraic hull with convex hull is retracted: the example in an unramified quadratic extension has conv(Zp·1)=Zp·1 strictly inside its algebraic hull. This local counterexample does not refute the special tensor-region claim in ATS III Proposition 9.10.8.1(3), Theorem 9.11.1, Mochizuki's specific regions or abc. A named compatible transport is presented only as a possible stronger sufficient route to E2, not as a necessary condition. Abstract, roadmap, classification tables and conclusions carry the same limits in English and German. Claim discipline: The repaired claims concern this paper's own quotations and classifications. No error in the original sources, no complete refutation of the ATS/IUT programmes, and no claim on abc is asserted. Open selection, ordering, transport, hull and measure obligations remain open. Independent review accepts the corrective manuscript delta and bilingual synchronization, not a proof of E2 or a whole-source mathematical certification. Positive imports from later series parts require their separate source-bound review. Files in v1.2: English version (24 pp.) and German version (26 pp.). The older v1.1 record retains its original three-file set; this follow-up contains two separate language PDFs. ─────── ✦ ───────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. Changes in Version 1.2 (2026-10-02) Corrects the ATS IV domain and restores the outer absolute-value bars in the E2 quotation; the separate height statement and F-NEU-1 are kept distinct. Retracts the generic algebraic-hull/convex-hull identification, with a local counterexample and explicit limits on what that counterexample establishes. Withdraws the positive cross-frame measure pinning; all diagnostic branches retain the open measure comparison. Absolute-value powers, Haar measure and module are separated. Presents named transport as one possible sufficient route; E2, transport compatibility, selection and ordering remain open. The elementary multiplier lemma and conditional F1/F2 derivation remain. Synchronizes all dependent statements in English and German, implements the basis and transport structure corrections, and supplies two separately checked PDFs (24/26 pages). Preserves the preceding record and the user-designed description/series layout. No claim is upgraded to a proof of E2 or abc.