Activation manifolds and behavior manifolds are approximately isometric across cyclic and sequential concepts.

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

• Activation-Behavior Manifold Isometryconcept0.857
• Representation geometry causally shapes behavior; activation and behavior manifolds are approximately isometric.claim0.853
• The analogous circular structure between representation manifold and behavior manifold is downstream of training data and its implicit cyclic conceptual structure.claim0.824
  The paper's causal explanation for why representation and behavior geometry both appear circular for days of the week.
• Interventions along activation manifold M_h yield behavioral trajectories following behavior manifold M_y, and vice versa — bidirectional relationship demonstrated across language models and video world models.finding0.822
  Central empirical result showing causal coupling between representation and behavior geometry across multiple substrates and modalities.
• Activation Manifoldconcept0.813
  The low-dimensional geometric structure discovered in neural activation space; contrasted with linear/Euclidean geometry.
• behavior manifoldconcept0.802
  One-dimensional curved surface in output probability space; the paper shows this mirrors representation manifold structure.
• Analogous alignment between representation manifold and behavior manifold is found across months, letters, ages, and synthetic in-context learning tasks in language models.finding0.800
  Generalization finding from the full paper extending beyond days-of-week to other structured concepts.
• There is a clear bidirectional relationship between the geometry of behavior and representation: steering along representation manifolds follows behavior manifolds, and vice versa.claim0.792
  The paper's finding that the alignment holds in both directions — from representation to behavior and from behavior back to representation space.