Donor-Derived Token Organization Changes What a Neural Network Can Reach

Learned neural representations can alter what a new learner subsequently reaches, but identifying which properties of a transferred representation carry that effect requires interventions that preserve some properties while replacing others. We study this problem by transplanting a rank-4 carrier learned on modular addition into independently initialized recipients. In a prospectively specified experiment, donor carriers reach a fixed generalization criterion in 45 of 64 donor–recipient cells, whereas matched substitutes preserving the donor spectrum, right basis, norm, rank, and recipient coupling reach it in 0 of 64. The effect replicates on a fresh recipient cohort (47/64 versus 0/64). Severe projection onto only the donor’s dominant Fourier modes retains substantial reachability (35/64). Moreover, synthetic carriers matched to these projections in rank, Fourier-support complexity, dominant-mode concentration, and singular spectrum are themselves active (36/64 versus 0/64 for norm-matched random carriers), although the experiment does not identify which matched property carries the effect. Conversely, randomly permuting a donor carrier across token identities reduces reachability from 47/64 to 0/64 while leaving it in exactly the same row-permutation orbit. Thus, descriptors invariant to row-to-token assignment cannot determine learning reachability at this operating point. Together, these interventions show that compact representational organization can causally alter finite-horizon learning reachability, while progressively narrowing—but not yet identifying—the structural properties sufficient to carry the effect.