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The Rank the Task Demands: A Causal Rank Law for Matrix Memories Trained on Group Composition

arXiv · AI, language, vision and robotics · article · Sep 10, 2026 · UTC

Matrix-valued memories make rank the natural budget of a learned representation: the number of independent directions a state spans bounds what it can bind, compose, and track. We report causal evidence, on a group-composition testbed trained under a hard single-state bottleneck with a fixed decoder that cannot launder rank, that gradient descent recruits precisely the rank the task's algebra demands. A companion paper [Larson, 2026a] establishes the analogous recruitment and causal necessity pattern on a $K$-pair associative-binding testbed, where exact recovery provably requires state rank a

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