A Unified Per-Token Gating Family for On-Policy Distillation: FKL/RKL Mixing with Multi-Channel and Bias Coefficients

Per-token gating of forward/reverse KL losses has become a standard technique for on-policy knowledge distillation (OPD), but existing methods such as EOPD (Jin et al., 2026) and ToDi (Jung et al., 2025) each fix a single gating signal and a single gating direction, and the two have never been compared directly. We introduce a four-coefficient parameterization lambda_t = sigma(a * h_t + b * u(x) + c + d * gap_t) in which direction-aligned proxies of EOPD and ToDi appear as one-dimensional (1D) restrictions, and which adds multi-channel composition and an explicit bias as further degrees of freedom. On TweetEval (Barbieri et al., 2020) emotion and hate, with a Qwen3-32B teacher and a Qwen3-4B student, configurations in the full family reach higher accuracy than the matched-magnitude single-channel (entropy-only / gap-only) 1D restrictions in 33 of 36 comparable cells, and a 26-cell mean-match isolation experiment places dynamic gating ahead of effective-KL-matched static baselines in 19 of 26 cells. Because cells share training data, models, and parameter substructure, we report both counts as exploratory aggregate directional evidence rather than as independent hypothesis tests. Targeted three-seed paired replications of the nine headline comparisons singled out by that sweep -- including a third task, offensive -- are directionally consistent, but individually smaller than the single-seed estimates and not significant at n=3. We therefore present the parameterization primarily as a shared coordinate system for comparing per-token gating designs in short-output classification OPD.