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 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 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 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: classical theta families at l-torsion are insufficient for the comparison structure required by IUTchIII. Changes in Version v5 (May 2026) Two-column layout restoration after the conservative one-column v4. No mathematical claims or bibliography entries were changed. Design: Restored the professional two-column preprint layout while slightly widening the columns relative to the earlier v3 layout. Layout safety: Retained the v4 German hyphenation fix and rebuilt the English and German PDFs with the title and abstract on a full-width first page. DE/EN: English and German sources remain synchronized; the combined EN+GER PDF was regenerated from the final v5 PDFs. Validation: LaTeX logs contain no overfull boxes, undefined references, citation warnings, or rerun warnings after the final build; rendered-page and bounding-box checks show no margin overflow.