Spectral Starvation in On Policy Distillation

On-policy distillation can improve from a small selected query set while absorbing teacher supervision progressively more slowly. We study this phenomenon through the spectrum induced by student-generated prefixes. Specifically, for every smooth $f$-divergence over tokens, the local OPD operator is an occupancy-weighted pullback of categorical Fisher geometry. The exact cross-entropy Hessian separates this Gauss-Newton operator from a teacher-gap-weighted network-curvature remainder. Under this local geometry, the linearized dynamics show that the step-to-step decline in absorption equals a normalized variance of active contraction factors, while the continuous log-rate derivative equals minus four times the active eigenvalue variance. Fast modes disappear first and leave the residual concentrated on weak directions. Kernel components form an invariant floor, and the smallest active eigenvalue gives the finite-time envelope and limiting rate. Consequently, equal-budget query choice is horizon dependent, while changing-operator bounds characterize the validity window of a frozen spectral forecast. Experimentally, in 160 unclipped Transformer runs, the initial student-Fisher predictor explains short-horizon gap reduction with $R^2=0.906$ and rank correlation 0.925. Across 64 matched replays, tangent drift tracks frozen-forecast error with rank correlation 0.504 and seed-cluster interval $[0.392,0.623]$. The broader 1,088-run study spans GSM8K and ARC-Challenge; all eight ARC selector and training-mode conditions exhibit positive starvation strength with intervals bounded away from zero. Consistent with this local-validity picture, direct measurements find order-one occupancy-induced operator replacement by update 64. Finally, after a common 32-candidate screen and at equal 256 online teacher calls, four selected Residual E prompts absorb more maximum teacher gap than sixteen selected random prompts. Explicit accounting specifies screening, online-trajectory, and token-position cost.