LCPO (Length-Controlled Policy Optimization / L1)

Trains reasoning models via GRPO to obey explicit user-specified token budgets $n_{\text{target}}$ passed in the prompt: $R_{\text{LCPO}}(x, y, n_{\text{target}}) = \mathbb{I}[\text{correct}(y)] - \alpha \cdot \max(0, |y| - n_{\text{target}})$ (or exact-length adherence $-\alpha | |y| - n_{\text{target}} |$), enabling smooth test-time compute scaling curves from 256 to 16,384 tokens with a single model.