Finite-radius gravitational-wave memory from Jacobi holonomy

Gravitational-wave memory is usually formulated either as an asymptotic observable at future null infinity or as the leading displacement of freely falling detectors. We develop a finite-radius detector-side formulation in which memory is encoded in the Jacobi transfer matrix for the relative motion of nearby geodesics. The exact local object is the Sp(6,R) holonomy of the full spatial Jacobi equation. A two-dimensional screen response is then obtained by restricting and projecting this transfer map; its symplectic Sp(4,R) structure is recovered through subleading order once transverse-longitudinal leakage is shown to feed back only beyond the order considered. For asymptotically flat spacetimes in Bondi gauge, we compute the screen tidal matrix through O(r -2 ). The leading term iswhile the subleading coefficient separates into a universal radiative contribution and a detectorframe correction, TThe first two terms arise from the finite part of the Bondi curvature and the orthonormal screen projection; Ξ det AB encodes the residual finite-radius detector tetrad correction. After subtracting the trivial flat-spacetime drift, the reduced Jacobi holonomy has block formwhose entries organize displacement, velocity, drift, and rotation-type persistent observables. At O(r -2 ), the memory contains both the local subleading tidal term and a nonlocal Dyson contribution quadratic in the leading tidal field. The latter is required by symplecticity and can generate rotation-type responses when the leading tidal matrices at different times fail to commute. A radiative-frame diagnostic waveform illustrates the leading displacement memory, the finite-radius local terms, and the commutator-induced rotation response.