Task-trained RNN manifolds warp with sign-changing curvature and eigenvalue collapse

Pellegrino & Chadwick derive a Riemannian pullback metric on the state-space manifolds of two task-trained RNNs: a contextual evidence-integration network and a sequential working-memory network whose activity forms a hyper-torus [pellegrino-chadwick-2025-rnn-dynamic-warping] The manifold warps so as to compress irrelevant input information [pellegrino-chadwick-2025-rnn-dynamic-warping] In the working-memory network, the torus's Gaussian curvature is non-flat with both positive and negative curvature that varies spatially over the torus (not over time) [pellegrino-chadwick-2025-rnn-dynamic-warping] In the context-integration network, two metric eigenvalues (the time and irrelevant-input components) decay to zero by decision time, so that manifold becomes effectively one-dimensional [pellegrino-chadwick-2025-rnn-dynamic-warping] The study is purely geometric measurement, with no ablation of the warping mechanism [pellegrino-chadwick-2025-rnn-dynamic-warping]