Disentangling Computation in Multi-Task Neural Networks with the Green’s Operator

We study how computation is organized and reused across tasks and time in recurrent neural networks through the finite-horizon Green's operator, a global source-to-destination map of first-order perturbation response. Unlike activity geometry, which describes where trajectories lie, or Lyapunov analysis, which summarizes perturbation growth and stability, the Green's operator directly records how influence propagates through a task-conditioned computation. In a flexible 15-task RNN, simple task-level reductions of this operator recover much of the known organization of shared dynamical motifs and provide a response-based view complementary to hidden-state representations. Reducing the same operator over hidden-state dimensions instead reveals temporal routing: training transforms initially local response into task-specific, delay-spanning pathways for computations requiring persistent information, while reactive computations remain comparatively local. Together, these results suggest that global perturbation-response geometry provides a compact, matrix-free representation for studying how learned recurrent computations are composed, routed, and reused.