Activation manifolds are isometric to behavior manifolds; geodesic steering wins

Wurgaft et al. fit a second manifold over Llama-3.1-8B's output token distributions (in Hellinger/sqrt-probability space) alongside the residual-stream concept manifold [wurgaft-etal-2026-manifold-steering] Geodesic distances between the activation and behavior manifolds correlate at r=0.99 (weekdays), 0.89 (months), 0.999 (letters), and 0.999 (ages), far above the straight-line Euclidean distances (0.89/0.53/0.71/0.36) [wurgaft-etal-2026-manifold-steering] Manifold steering along the activation geodesic produces natural on-manifold behavioral trajectories where linear steering does not, and a pullback test recovers the activation manifold from a target behavior trajectory [wurgaft-etal-2026-manifold-steering] The approach generalizes to synthetic multi-dimensional concept spaces and a Mountain Car visual world-model task [wurgaft-etal-2026-manifold-steering]