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

DRAFT version. This is an advanced preprint draft. The paper is published for citation and discussion; further revisions remain possible. 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 Aᵗors < 0 (Lemma lem: Aavg) and all non-archimedean residuals are likewise negative. However, the repair reveals a structural obstruction: the IUT theta values q^j² create a positive non-archimedean contribution growing as O (l²), while any classical theta function at l-torsion compensates only O (l) (Proposition prop: scaling). No classical theta function can close this gap. The explicit arithmetic translation confirms that Rbaseᵗors < 0 controls the j-averaged archimedean residual structure, while the auxiliary bound |η (τ∞) |²4 < 0. 005 follows independently from standard SL₂ (Z) fundamental-domain geometry. Neither bound constrains the period-conductor ratio needed for Szpiro and abc. The structural diagnosis: classical theta families at l-torsion are insufficient for the comparison structure required by IUTchIII. 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.