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When the Gradient Sees Rank: Provable Necessity, Causal Recruitment, and Composition in Trained Matrix Memories

arXiv · AI, language, vision and robotics · article · Sep 12, 2026 · UTC

Can gradient-based training learn the rank needed to store and compose associations in a matrix memory? In our earlier study, we used a matrix-augmented reasoner on a task that admits a rank-1 solution, leaving this question open. We train matrix memories on $K$ fresh key-value bindings whose exact linear recovery requires $\mathrm{rank}(Z) \geq K$. A fixed linear readout queries a single matrix state without access to the original bindings. Experiments measure recovery by cosine similarity greater than 0.9, a threshold distinct from mathematical equality. Learned effective rank increases with

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