The Comparison in IUTchIII, Corollary 3.12: From Attempted Repair to Structural Diagnosis

The abc Gap Series: Part 1 — a corrected classical comparison Corrective revision of the proposed comparison at Corollary 3.12. We revisit an attempted classical repair of the comparison at Corollary 3.12 of IUTchIII. For a real analytic parameter q in (0,1), torsion-normalized Jacobi products give a strictly negative archimedean mean. Finite-place residuals are negative away from l; above l, negativity requires an additional normalized valuation hypothesis. Weighted residuals do not descend to conjugacy classes solely from theta symmetry. The covolume-to-degree correspondence requires measures, weights and pilot-object identifications, and a rank-one kernel calculation supplies no residual sign. The scaling comparison uses separately fixed archimedean and local parameters and gives no uniform arithmetic impossibility theorem. Normalization and transfer back to the original IUT comparison remain open. This paper provides analytic partial results and a conditional diagnostic framework; it establishes neither abc nor a complete refutation of IUT. Wir überprüfen einen klassischen Reparaturversuch für den Vergleich bei Korollar 3.12 von IUTchIII. Das negative archimedische Mittel gilt für reelle Parameter q in (0,1); oberhalb von l erfordern negative lokale Residuen eine zusätzliche Bewertungsvoraussetzung. Konjugierte Theta-Normen identifizieren keine gewichteten Residuen. Die globale Gradbrücke sowie die Rückübertragung auf die ursprünglichen IUT-Objekte bleiben offen. Der Skalierungsvergleich gilt für getrennt fixierte Parameter. Der Beitrag liefert analytische Teilresultate und eine bedingte Diagnose, keinen abc-Beweis und keine vollständige IUT-Widerlegung. ─────── ✦ ───────seriesdirectory (each part lists the whole series; planned parts are updated with title and DOI upon publication): ☞ Part 1: The Comparison 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) ✎ Tool supplement: 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 v6.4 (October 2026) Corrective revision following the R23 inventory. Mathematical claim scope has changed; no claim is promoted to PROVED. Restricts the finite-place baseline to explicit valuation hypotheses, including the l=5, valuation 1/8 counterexample to the previous negative-summand argument; handles the empty finite-place set. Corrects the l-adic theta norm to l^(-1/(l-1)), distinguishes conjugate theta norms from weighted residuals, and treats global averaging and Arakelov identification as open requirements. Separates the rank-one kernel lemma from any residual-sign inference, and preserves its independent secular hypothesis. Replaces the overbroad scaling no-go with separately fixed archimedean/local parameters; corrects the claim that finite residuals contain no discriminant information. States the original comparison transfer as conditional with an explicit obstruction upper-bound assumption. Normalization, pilot-object identification and arithmetic translation remain open. Softens the title to "The Comparison in IUTchIII". Condenses unsupported route and equivalence claims while preserving the valid analytic product proofs. This is a partial correction, not a complete withdrawal or refutation of the paper. Provides synchronized English and German PDFs as separate files; the redundant combined edition is omitted. Independent corrective-manuscript review accepted both languages, without L11 approval or claim-status promotion. Changes in Version v6.3 (July 2026) Citation and source-boundary correction release. No mathematical claim is upgraded. Primary-text attribution: Corrected the initial-theta-data reference to IUTchI, Definition 3.1.Source-bound framing: Narrowed the discussion of Dupuy--Hilado to the result actually established by the cited source.Frey-curve invariants: Replaced an over-uniform minimal-discriminant/conductor formulation with the model discriminant and an explicit caveat for the 2-adic distinctions.Bibliographic metadata: Updated the Dupuy--Hilado journal metadata and DOI, the Watkins DOI, and the Yamashita revision date.Claim boundary: The open M3 bridge and the structural diagnosis remain unchanged; this release does not claim a proof upgrade.DE/EN: English and German versions were rebuilt and checked in parallel; the combined PDF contains English followed by German. Changes in Version v6.2 (June 2026) Structural-design maintenance release. This version adds an early diagnostic claim/gate ledger to make the contribution, evidence status, and claim boundaries visible before the detailed proof spine. Diagnostic ledger: Added an introductory claim/gate table in both the English and German manuscripts, covering S7 localization, the torsion-normalized archimedean mean, non-archimedean residuals, the M1--M5 partial-repair route, arithmetic translation, and the scaling obstruction.Claim boundaries: The new ledger explicitly states that the paper does not derive a Szpiro/abc consequence, does not provide an unconditional rebinding to the original IUT branch, and does not exclude non-classical multiradial mechanisms.RNT preparation: The release prepares a clearer basis for the Research in Number Theory transfer by making the diagnostic sequence easier to cite and audit.Validation: English, German, and combined PDFs were rebuilt; hard LaTeX log scans, reference checks, PDF size/date checks, and the German umlaut text layer were verified.DE/EN: English and German versions remain synchronized; the combined PDF is generated from English followed by German. Changes in Version v6.1 (June 2026) Post-v6 maintenance release. This version publishes the local citation cleanup, proof/paper synchronization, strict-review claim hygiene, and German-style pass accumulated after the v6 source-check release. Citation cleanup: Activated the local reference to the companion Frey-curve programme in both language versions; EN and GER remain synchronized with 17 cited keys and 17 bibliography entries each.Proof/paper synchronization: Brought the scaling-degree lens from the proof notes into the paper, including M_l(F)=|log|F_l||, product degree 1 versus 3, and the averaged O(l) versus O(l^2) residual comparison.Claim hygiene: Narrowed M4 to its classical replacement-branch content, restricted the no-go statement to direct Tate-curve theta/eta/Jacobi torsion products of the tested class, and marked M3/conditional-core language as a bridge contract.Disclosure and style: Updated the AI/KI disclosure to L5+ and smoothed the German version without changing the mathematical claim level.DE/EN: English, German, and combined PDFs were rebuilt locally; the combined PDF follows English first, then German. Changes in Version v6 (May 2026) Bibliography source-check revision. No mathematical claims were changed. Source check: The bibliography was checked against EMS Press/PRIMS, Annals of Mathematics, Springer, arXiv, EuDML and DOI metadata.Corrections: Added missing DOI/URL metadata for Arakelov, Silverman, Wiles, Taylor--Wiles, Watkins, Köhler and Ribet; updated the Dupuy--Hilado arXiv entry to the current v2 title.DE/EN: English and German sources remain synchronized; the combined EN+GER PDF was regenerated from the final v6 PDFs.Validation: Final LaTeX logs contain no undefined references, citation warnings or rerun warnings; the German TeX and extracted PDF text contain real umlauts.