The Berry-Bordism Correspondence in 3 + 1 Dimensions via Anomaly Inflow

We establish the Berry–Bordism correspondence in 3+1 dimensions, linking the global Pin⁺ bordism classification Ω4Pin+≅Z16 to the local Berry phase defined on the flip surface Σflip=PD(w1T). The central statement takes the form of a two-part theorem: Part 1 (unconditional) proves θF=θB via the group isomorphism Hom(Z2,Z16)≅Z2 together with nonvanishing w1T=0, a conclusion cross-validated by invertible SPT field theory classifications. Part 2 (conditional on the existence of a Pin⁺ tangent structure) fixes θF=π through an anomaly inflow sequence relating the η-invariant to the Berry phase via the Witten–Yonekura formula; the homomorphism argument uniquely singles out the order-2 element 8∈Z16. The novelty of the present work lies in explicitly constructing and validating this anomaly inflow chain, while all required mathematical building blocks are pre-existing established results. We also isolate and address two subtle technical fine points: the inheritance of Pin⁺ structures under cut-and-paste gluing operations, and the spatial localization property of the Berry phase.