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

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 toward chosen minimal faithful reference representations embedded in larger matrices. On group-composition state tracking over five finite groups spanning the solvable/non-solvable divide, the recruited rank equals the group's minimal faithful real representation dimension d (Spearman ρ= 0.9747, the design's tie-capped maximum), the dimension-matched solvable/non-solvable pair S 4 / A 5 is statistically equivalent under a pre-registered test, and a pre-registered force-rank test separates a guaranteed similarity ceiling from empirical recovery at the target dimension: one rank below d, cosine similarity is capped by the target's tied unit spectrum at sqrt (d - 1)/d 0.894, below the 0.9 threshold in every group by construction, with observed cells at 86-95% (mean 91%) of that ceiling; at d, not guaranteed a priori, recovery clears the pre-registered anchor-relative bar at four seeds per group in all five groups. Within this testbed, measured effective rank tracks representation dimension; the matched-dimension S 4 / A 5 comparison establishes equivalence within the pre-registered tolerance.