A physical theory of experiments carried out in a space-time region can accommodate a detector localized in another space-like separated region, in three, not necessarily exclusive, ways: (1) the detector formally collapses physical states across space-like separations, (2) the detector enables superluminal signals, and (3) the theory becomes logically inconsistent. If such a theory admits autonomous evolving states, the space-like collapse must be instantaneous. Time-like separation does not allow such conclusions. We also prove some simple results on structural stability: within (...) the set of all possible theories, under a weak empirical topology, the set of all theories with superluminal signals and the set of all theories with retrograde signals are both open and dense. (shrink)