The abc Landscape: Multi-Route Cartography of an Ongoing Expedition

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, 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. The abc Gap Series — series directory (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 about this (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. Version 1.2 changes (2026-08-16): The paper is retitled From Landscape to Atlas: Multi-Route Cartography of an Ongoing Expedition Toward the abc Conjecture (formerly: The abc Landscape: Obstructions, Failed Routes, and HCT Diagnostics) to reflect its role as the growing map of an ongoing expedition. Content changes: (1) the bridge-search section is rewritten as a documented candidate triage — five candidate classes excluded against the stated obligations, one level-exact class outside those five retained, and the aggregation step typed into two sharp open quantities; (2) a case history applies the bridge-contract form audit of Parts 2/3 of the abc Gap Series to the paper's own route programme (first self-application; no verdict on external programmes); (3) the rigidity theorem of Part 6 is adopted as a new route-map diagnostic (a filter for routes requiring place-selective weight deformations under a valid product formula); (4) the witness-degree reduction of the aggregation obligation is recorded with its two proven lemmata and the remaining open repulsion target; (5) the route map takes up the published state of the abc Gap Series (Parts 1-6 and the GNC tool supplement, with DOIs), including the openly recorded recoding of Part 3 (v0.2); (6) new verified references (a.o. Hamidi, arXiv:2607.23244). No claim upgrade anywhere: the landscape documents, the series proves. From this version the record carries the English and German manuscripts as separate files, without a combined PDF. Version 1.1 changes (2026-08) 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).