Cobordism Atlas of Extremal ABC Families: Macro-Moves, Categorical Probes, Quantum Heights, and Barrier Morphology

We propose a "Cobordism Atlas" for extremal ABC behaviour, organised around families of triples rather than individual solutions. On this atlas, territories correspond to Diophantine families, borders are implemented by cobordism-type local move systems, and travel is constrained by geometric barriers, categorical probes, and height-like invariants. We develop three structural ingredients for such an atlas: family-level macro-moves between hostile regimes, categorical "sniffer" functors built from ascending chains of rationally-augmented ideals, and the topology of safe-to-monster barrier spaces arising from height–radial parametrisations. We also sketch a quantum-mechanical analogue in which inner products of states behave as height-like quantities, reinforcing the view that extremal ABC behaviour should be controlled by obstructions in a hybrid geometric, categorical, and probabilistic landscape rather than attacked directly. The manuscript concludes with a viscous cobordism flow that yields a monotonic energy dissipation principle and a tightening inequality as a toy model for how extremal behaviour contracts under controlled transport.