On-policy distillation (OPD) supplies dense token-level supervision by aligning a student to the teacher's next-token distribution on its own generated rollouts. Existing works mostly implement top-$k$ OPD, applying reverse Kullback-Leibler ($\RKL$) divergence between the teacher's and the student's renormalized distributions over teacher top-$k$ support. However, the teacher scores student-generated rollouts that are inherently off-policy for itself, leaving the reliability of its next-token guidance unclear. Restricting the divergence to the teacher top-$k$ support also discards the remaining mass outside that support. These gaps leave two questions implicit: 1) which tokens to align, and 2) how to align those tokens. To answer these questions, we propose Overlap-gated On-Policy Distillation (O$^{2}$PD). As for 1), we quantitatively analyze the reliability of teacher supervision and find that alignment on the student's trajectory prefix is more reliable than alignment on the full context. Therefore, our O$^{2}$PD is applied only to the 20\% prefix tokens of each student rollout. As for 2), we theoretically characterize two problems of top-$k$ OPD: the \emph{Teacher-support Uncovering} and the \emph{Dominant-Mode Mismatch}. Based on these analyses, we propose two support strategies: the \emph{Reset Support} strategy attaches a reset atom to every token, pooling mass outside the teacher top-$k$, and the \emph{Support Convergence} strategy converges the support to the already aligned intersection. Our O$^{2}$PD employs the designed overlap-gated $α$-divergence to interpolate between mode-covering and mode-seeking for per-token alignment, where the overlap is used as a gate to switch the support and adaptively modulate $α$.

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