Classification of localizing ideals in the stable homotopy category

Classify the localizing ideals of the stable homotopy category of spectra, thereby resolving the outstanding classification problem for this prototypical tensor-triangulated category.

Background

The paper begins by framing the classification of localizing ideals as a fundamental structural problem for tensor-triangulated categories. It notes that although this classification is known in several important examples, the corresponding problem for the stable homotopy category remains unresolved. The statement is included because the authors explicitly describe the problem as widely open, rather than merely suggesting it as a direction for future research.