The abc Landscape: Obstructions, Failed Routes, and HCT Diagnostics (DRAFT)

Abstract

This paper systematizes a sequence of Frey-curve approaches to the abc conjecture and shows why they stop short — deliberately negative, but structural. The routes examined arose from theta products, Tate uniformisation, Tamagawa budgets, period lattices, modular degrees, level lowering, and Hecke congruence diagnostics. Several plausible routes are shown to reduce to known obstructions or to a single remaining nonlocal problem: the local p-adic theta route is valuation-trivial; the archimedean eta factor is uniformly bounded after SL2(ℤ)-reduction; Tamagawa/component budgets, root-number tests, ordinary level lowering, and standard analytic averages do not supply the missing Frey period bound. The resulting landscape isolates the main unresolved component as a Frey-specific new-level Hecke/congruence sparsity statement (FAQS/M* in the HCT program). The Reyssat-3863 no-Magma Hecke quotient is recorded as a diagnostic calibrator and as an entry point into a separate proof route (a companion paper in preparation), not as a standalone proof of abc. A concluding survey closes the structural routes that would generalize an individual diagnostic to a universal bound — self-reduction, Buium delta-character depth, an unavoidable congruence trace, and measure rigidity — each either reducing to the individual-to-family step or lacking a quality-sensing realisation, supported by a self-averaging diagnostic showing the high-quality tail is non-self-averaging. Claim discipline: No proof of the abc conjecture is claimed. The contribution is a classified map of obstructions, failed routes, reductions, and the precise location of the remaining hard problem, together with the HCT diagnostic toolkit used to chart it. Files: English version (15 pp.), German version (16 pp.), and a combined bilingual PDF. Changes in Version 1.1 (August 2026) Route map: The paper now works as an ongoing multi-route map of the abc landscape (in the spirit of the RH landscape series). A new Route Map section with a consolidated status table — closed, diagnostic, reduced, living, dormant, open, plus a few further labels for entries that fit none of these — indexes the routes the investigation took seriously, cut by families rather than by entries of the internal register. 16 routes added from the project registers that were missing in v1.0, including the adjoint Fitting/Selmer channel (living), the family-wide uniformity bridge (dormant), the rectangle route/D-cap, the K2 trace route, the CCM-Shimura and Gross-Zagier/ANC closures, and the generalization forcer candidates. Companion-paper transition removed: the former "Transition to Paper B" section and all references to an unpublished companion were removed or softened to "a companion paper in preparation". AI disclosure replaced by the pipeline-standard bilingual disclosure; paper date updated to August 2026. Bibliography unchanged (26 verified entries; no new references added).