From Traces to Recipes: Constructing and Distilling Reusable Procedural Knowledge into Language Models

Reasoning distillation trains small students on teacher solution traces, but each trace entangles a reusable procedure with the details of one problem, leaving the student to infer what is shared. Existing supervision thus trades reusability against verification: traces solve a concrete problem but are tied to it, whereas abstractions generalize across problems but are never tested on the model that must use them. We introduce DistilRecipe, which represents procedural knowledge as recipes: instance-independent procedures that state when a method applies, which operations to perform in dependency order, and how to check the result. A recipe is admitted only when a small frozen student following it answers correctly and a monitor confirms it applied and was followed. In context, a receipe bank built with one 2B anchor improves 0.8B–4B Qwen3.5 models in all 30 settings, by an average of 9.0% on MATH-500 and 5.6% on GSM8K. Used as answer-free privileged context, recipes also distill into a Qwen3.5-2B student that needs no recipe at test time and outperforms matched chain-of-thought distillation in all four comparisons, by 2.08–3.20%.