A No-Go Theorem for Trapped Surfaces in Scalar Theories with CPC Foliation

We derive a two-derivative, shift-symmetric action for a complex scalar–gauge system under three physical assumptions: Phase Current Conservation (CPC), Null Energy Condition (NEC), and Velocity Normalization Condition (VNC). The scalar phase gradient \(\partial_a S\) dynamically sources an emergent Lorentzian geometry \(g^{\text{eff}}_{ab}\) via a Kerr–Schild–like deformation of Euclidean space, while the gauge field \(A_a\) (coupled via \(\partial_aS\to\partial_aS - eA_a\)) stabilizes the foliation against topological singularities. We prove that hypersurface-orthogonal foliations within this framework exclude closed trapped surfaces, thereby greatly narrowing the search-space of viable models—regarded as a feature rather than a bug. Crucially, this no-go result requires no exotic stress–energy or higher-derivative couplings: it holds strictly under the classical NEC and standard two-derivative dynamics. To our knowledge, CPC + VNC slicing is the only purely conservative mechanism that guarantees the absence of trapped surfaces in a scalar–gauge theory.