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

The abc Gap Series: Part 1 — the gap that resists repair We report on an attempted repair of the key quantitative gap in Corollary 3.12 of Mochizuki's IUTchIII, the central step in the claimed proof of the abc conjecture. The repair partially succeeds: we prove unconditionally that the j-averaged archimedean residual A^tors < 0 (Lemma lem:Aavg) and all non-archimedean residuals are likewise negative. However, the repair reveals a structural obstruction: the IUT theta values q^{j^2} create a positive non-archimedean contribution growing as O(l^2), while direct Tate-curve theta/eta/Jacobi torsion products of the tested class compensate only O(l) (Proposition prop:scaling). No direct Tate-curve theta/eta/Jacobi replacement with standard O(l |log eta|)-product asymptotics at l-torsion can close this gap. The explicit arithmetic translation confirms that R_base^tors < 0 controls the j-averaged archimedean residual structure, while the auxiliary bound |η(τ∞)|^24 < 0.005 follows independently from standard SL_2(Z) fundamental-domain geometry. Neither bound constrains the period-conductor ratio needed for Szpiro and abc. 💡The structural diagnosis: the gap at Corollary 3.12 is not resolved by the direct Tate-curve theta/eta/Jacobi methods of the class tested here. ─────── ✦ ───────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) ✎ 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. ─────── ✦ ───────changelog 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.