Training on recovered failures makes one-shot decoding catch up
GapFT raises single-sample accuracy without search by training on problems an LLM misses once but solves within K tries, giving deployed models more of the benefit of test-time sampling. On LogiQA 2.0 and ReClor, Llama-3.1-8B gained 14.4 and 13.9 percentage points over the source model, matched verifier-selected Pass@4 accuracy with one decode, and matched full-pool fine-tuning using one third of the data.