A Cost That Practice Accuracy Hides: Worked Solutions in Self-Generated Training Data and the Compositional Use of New Rules

A worked solution can make self-generated training targets shorter and cheaper, but accuracy on the practiced operation may hide how well a model can reuse newly learned rules. We study this choice in a 2B-parameter reasoning model learning two invented quantitative rules. The model generates targets from a rule sheet alone or from the sheet plus a program-written worked solution; targets are accepted for final-answer correctness, and the support is removed for fine-tuning and evaluation. Training questions request forward use in one segment; composition requires sequential reuse of the same rule across two segments. After six passes, using a program-checked numerical signature, sheet-only targets lead by 24.3 percentage points (95% item-bootstrap CI 14.6-34.0) on composition in a prospective test. On unseen argument values, observed forward accuracy is identical (87.5% each), but composition accuracy is 58.3% versus 23.6%, a 34.7-point gap (25.7-43.8). An independently sampled second target-bank pair repeats the direction (+14.6 points, 2.1-26.0). Ten passes on the shorter worked-solution targets, approximately matching the domain tokens of six sheet-only passes while changing other training quantities, narrow the gap to 16.3 points (6.3-26.7) in the first bank pair and 6.2 points (-6.3 to 18.8) in the second, where the pre-specified test does not hold. At six passes each, the worked-solution policy uses fewer capture tokens and less update time, yet the trained models use 2.6 times as many generated tokens per composition success under the numerical signature (2.1 times in the second bank pair). Choosing a capture context therefore requires testing how the new rules are reused, alongside the inference cost of those uses, rather than relying on forward accuracy alone.