A Jet-Order Separation Theorem for Berry Curvature

Abstract We consider the geometric information carried by a smooth rank-one family of orthogonal projections P:M→Proj(H) over a parameter manifold, equipped with the canonical projected Berry connection ∇=P∘d. We prove a Jet-Order Separation Theorem showing that the Berry curvature F P ​ is a first-order differential invariant of the projector family and cannot be recovered from the order-zero (pointwise) data (O,P(x)). An explicit construction in C 2 exhibits two smooth projector families that agree to zeroth order at a point while possessing distinct Berry curvatures, thereby demonstrating that no order-zero operator can determine curvature from pointwise backbone data alone. Conversely, we prove that the full smooth embedded projector family canonically determines the projected Berry connection, its curvature, and its holonomy along every specified loop. The result therefore identifies the precise boundary between pointwise and differential information in Berry geometry: transport is additional structure only relative to reductions that discard first derivatives or the ambient embedding, not relative to the embedded projector family itself. We conclude by discussing the extension to higher-rank bundles and posing the problem of identifying the minimal reduced data required to determine bundle class, characteristic classes, and Berry holonomy.