Coordinate patching on circular manifolds guarantees off-manifold representations for boundary point pairs with orthogonal deviations

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• Steering Llama-3.1 8B along the circular representation manifold produces outputs that follow the natural circle of the behavior manifold, cleanly shifting probability mass from Monday through successive days.finding0.725
  Core empirical result demonstrating that manifold steering produces on-target, behavior-aligned outputs.
• manifold steering produces clean probability shifts along natural behavior structure; linear steering cuts across manifold and produces off-target noisy effectsfinding0.715
  Empirical demonstration on Llama-3.1-8B that steering along representation manifold aligns outputs with behavior manifold, whereas linear steering does not.
• Off-manifold divergences can activate hidden pathways that produce misleadingly confirmatory behavior while the true mechanism is never exercisedclaim0.714
  Core claim about why pernicious divergence undermines mechanistic conclusions
• Manifold geometry provides a practical steering blueprint in an image-action model predicting car position on a hill, extending results across modalities.finding0.699
  Cross-modality result from the full paper demonstrating that representation-behavior geometry alignment is not limited to language models.
• There is a clear bidirectional relationship between the geometry of behavior and representation: steering along representation manifolds follows behavior manifolds, and vice versa.claim0.697
  The paper's finding that the alignment holds in both directions — from representation to behavior and from behavior back to representation space.
• Representation geometry causally shapes behavior; activation and behavior manifolds are approximately isometric.claim0.695
• The analogous circular structure between representation manifold and behavior manifold is downstream of training data and its implicit cyclic conceptual structure.claim0.695
  The paper's causal explanation for why representation and behavior geometry both appear circular for days of the week.
• Model stitching can use the behavioral null space of the source model when mapping to the target, making successful stitching insufficient evidence of representational similarityclaim0.694
  Formal analysis showing the theoretical limitation of model stitching as a similarity measure.