דרגת ראנק: חוק דרגתי קאוזלי לזיכרונות מטריציים שנלמדו על ידי רכיבה קבוצתית

תקציר מקורי באנגליתarXiv:2609.12259v1 Announce Type: new Abstract: 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 at least $K$; this paper inherits that instrument and extends the rank law from a scalar capacity bound to a representation-theoretic one. We train towar