Gap: linear steering assumes Euclidean geometry and does not account for the actual curved geometry of activation manifolds

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• Linear steering is often mismatched with a model's internal representation geometry, producing noisy, off-target effects.claim0.841
  The paper's critique of the standard linear steering baseline, supported by the days-of-week demo.
• Steering along manifolds provides better control than linear steering when the concept geometry is non-linear.claim0.826
  The central thesis of the paper, motivating the shift from linear to geometry-aware manifold steering.
• Manifold-respecting steering produces smooth natural behavioral trajectories while linear steering teleports between non-adjacent concepts.claim0.817
• Linear steering produces noisy off-target effects; manifold steering cleanly shifts probability mass between sequential concepts.finding0.816
  Core empirical claim comparing steering approaches on cyclic concepts.
• Euclidean geometry assumption in steeringconcept0.813
  Linear steering implicitly assumes a flat, Euclidean activation space, leading to off-manifold excursions.
• Linear steering cuts through off-manifold regions and hence produces unnatural outputs.claim0.807
  Attribution of failure to Euclidean assumption.
• manifold steering produces clean probability shifts along natural behavior structure; linear steering cuts across manifold and produces off-target noisy effectsfinding0.803
  Empirical demonstration on Llama-3.1-8B that steering along representation manifold aligns outputs with behavior manifold, whereas linear steering does not.
• Manifold steering demonstrates bidirectional geometry-behavior link in a video world model on tasks with geometry corresponding to physical dynamicsfinding0.790
  Extension of manifold steering validation to video world models and physical dynamics tasks, demonstrating cross-modal generality