Second-Order Detection by Boundary Balls in a Source–Readout Architecture

Can the same boundary balls that miss a gravitational perturbation at first order detect it at second order? This paper constructs an explicit transverse perturbation of four-dimensional anti-de Sitter spacetime for which the first variations of extremal surface area and gravitational energy vanish for every rest-frame ball on a fixed boundary time slice. The canonical energy of the corresponding bulk wedge is nevertheless strictly positive for every ball radius.

The paper derives the exact response in closed form, verifies it independently by covariant phase space at two radii, and proves a positive lower bound without solving for the extremal surface displacement. A separate bounded oscillatory example gives positive numerical lower-bound estimates at four frequency–radius products. Within Shadow Theory’s source–readout architecture, these results show how increasing the order of a readout can reveal information while keeping the measurement regions fixed. Under the holographic dictionary and for an admissible boundary state family, the second-order response is the relative-entropy Hessian, a form of quantum Fisher information.