We hypothesize that interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally

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• 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.finding0.838
  Central empirical result showing causal coupling between representation and behavior geometry across multiple substrates and modalities.
• Optimizing interventions in activation space to produce paths along M_y recovers activation trajectories that trace the curvature of M_h.finding0.828
  Demonstrates bidirectional causal link: behavior manifold geometry can be recovered by optimizing in representation space.
• Representation geometry causally shapes behavior; activation and behavior manifolds are approximately isometric.claim0.827
• Does the geometric structure of activation space causally shape neural network behavior?question0.816
  Central research question driving the work.
• Neural representation geometry causally shapes behavior; interventions respecting that geometry will yield natural trajectories.hypothesis0.795
  Central hypothesis tested via manifold steering experiments across language models and video world models.
• Linear interventions across voids in activation space produce incoherent output, while following the manifold curve produces smooth control.claim0.792
  General principle derived from the Mountain Car experiment: curved manifold-following yields coherent manipulation, linear shortcuts fail.
• optimization of interventions to follow behavior manifold M_ymethod0.786
  Method that optimizes activation interventions so that resulting behaviors trace M_y, recovering activation paths that follow M_h curvature.
• There is a bidirectional relationship between the geometry of representation and behavior across tasks and modalities.claim0.778
  Author’s interpretive claim that the shared geometry is general and robust.