Supervised fine‑tuning (SFT) remains the dominant post‑training approach for large language models because fixed demonstrations can directly inject desired reasoning strategies and behavioral patterns. Yet forcing a model to imitate externally provided trajectories may interfere with behaviors already supported by pre‑training, limiting generalization beyond the training distribution.\ \ Recent dynamic SFT methods mitigate this issue by re‑weighting token‑level supervision according to model‑side signals such as predictive probability, prior support, or distributional compatibility. However, model compatibility is not equivalent to learning value: a token weakly supported by the current model may be an undesirable mismatch or precisely the task‑specific decision the demonstration intends to teach.\ \ We propose CIG‑SFT, an information‑theoretic framework that estimates the task‑conditioned information carried by each demonstrated token. Under teacher forcing, we compute the token likelihood from a frozen reference model with and without access to the task input while keeping the gold solution prefix fixed. The difference, called conditional information gain, quantifies how much the task itself reduces uncertainty about the demonstrated decision.\ \ We then convert this information into a KL‑regularized projection of the current policy and derive an equivalent scalar‑weighted SFT objective whose logit gradient exactly matches that of the projected target, achieving fine‑grained, information‑driven supervision.\ \ Across mathematical reasoning and code‑generation benchmarks, CIG‑SFT consistently outperforms standard SFT and recent adaptive fine‑tuning baselines, with especially strong improvements in generalization to challenging evaluation distributions, while requiring neither additional training data nor online rollouts.\ \ Review