Teaching Is More Than Solving: Compiling Executable Lessons for Agent-to-Agent Transfer

Language-agent systems routinely ask one agent to summarize experience for another, yet they evaluate the sender as a solver rather than as a teacher. We separate these capabilities by defining teaching utility: the improvement that a budgeted lesson causes in a frozen student's decisions on disjoint held-out states. We introduce TeachWorld, a controlled testbed of 144 latent decision policies spanning six semantic domains, three rule structures, and 124 distinct family/structure signatures, and evaluate 432 teacher-concept artifacts across three independently served model families. We then present LessonSmith, a training-free lesson compiler that pairs a teacher's candidate rule with branch-covering executions from an exact verifier and makes every confirmation or correction explicit. Agent-written lessons improve frozen students by 44.6 percentage points over no lesson. Given an executable verifier and a branch-coverage mechanism, LessonSmith adds 4.3 points (95% concept-clustered CI [2.8, 5.9]). Two full-matrix controls identify why. Given the same executed evidence, explicit receipts outperform an execution-informed rule rewrite by 2.7 points ([1.2, 4.2]). When teachers instead choose four probes from eight fresh unlabeled cases, they cover only 2.40 of four policy branches and trail branch-covered lessons by 6.6 points ([4.8, 8.3]). A 12-item solving battery predicts teaching ($\rho =0.56$) but still misorders 27.5% of comparable teacher pairs. Teaching is therefore a measurable transfer capability: solving quality matters, while executable coverage and the form in which evidence is communicated determine what the next agent can use.