Beyond Embedding Transfer: Component Roles in Grokking Transfer and Stability

Warm-start transfer can make algorithmic tasks generalize rapidly, yet it is unclear which model components provide the gain and whether that gain remains stable under continued optimization. We study cross-operator transfer on modular arithmetic and separate *efficacy* (early velocity) from *stability* (post-reach drawdown). In a scale-matched 108-run battery across 12 seed blocks (a 96-run $2^3$ factorial plus a 12-run scale control), transferring internal attention/MLP weight matrices ($B$) in addition to token embeddings and readout ($E+U$) improves early mean accuracy by 5.46 percentage points (Holm $p=0.0039$) and reduces confirmation latency by 558 steps (Holm $p=0.0088$). While readout plus internal-block transfer satisfies the pre-specified $\pm 500$-step mean-latency equivalence criterion in one-layer models (TOST $p=0.0011$, although Full is modestly faster in 11/12 paired seeds), a prospective two-layer replication confirms the internal-block acquisition advantage (12/12 seeds, $+704.67$ integral units, $p=4.88\times 10^{-4}$) while revealing an architecture-dependent boundary: omitting donor embeddings falls $4475.6$ integral units below Full transfer, far outside the pre-specified $\pm 250$-unit equivalence margin. Continued target training, however, frequently triggers severe post-grokking relapse. Freezing transferred representation carriers ($E,U$) nearly eliminates offline relapse ($19.40\%\to 0.07\%$, Holm $p=0.005859$). Online validation-triggered gating slashes True Max Drawdown from $22.06\%$ to $0.60\%$ on $2a+b$ ($p=0.000488$), with prospective confirmations extending protection across affine, nonlinear quadratic, and two-layer targets (10.94–23.47 pp reductions), while distinguishing continual stabilization from static early stopping. In non-abelian $S_5$, unshielded transfer surges transiently ($95.4\%$ peak), but a prospective shielding cohort yields no confirmed benefit ($+0.15\pm 1.14$ pp). These results establish a component-level dissociation between transfer acceleration and trajectory stability, and expose the empirical boundaries of parameter shielding.