DERIVED SURGERY OBSTRUCTIONS: PRISMATIC PONTRYAGIN CLASSES OF NON-ORIENTABLE STACKS IN VOID-MANIFOLD METRIC DECAY

Mathematical Applications of Science Fiction We construct a canonical obstruction theory for the derived surgery of non-orientable Artin stacks subject to metric decay toward a void state. By lifting the classical Pontryagin classes to the prismatic site, we define the prismatic Pontryagin classes for a perfect complex on a derived stack. We show that the nonvanishing of these classes in syntomic cohomology constitutes the primary obstruction to the existence of a void-metric structure. Furthermore, we establish a spectral sequence connecting the metric decay filtration to the algebraic L-theory spectrum, proving that the void-manifold limit can be achieved if and only if the Frobenius-twisted trace map degenerates at the second page. This framework unifies the Bhatt–Scholze prismatic formalism with Wall’s surgery obstruction theory in the setting of singular metric collapse.