Teacher-Free Self-Training Amplifies but Does Not Compound: A Pass@$K$ Crossover on a Free-Verifier Domain

Abstract

When a language model trains on its own verified outputs, does it acquire capability beyond its base, or merely get better at expressing capability the base already had? We make the question decidable with a teacher-free "constellation" -- a generator, a learned critic, and a free exact verifier -- on a FlashFill-style "trapdoor" DSL, where verified (problem, solution) pairs are cheap to synthesize, hard to invert, and free to check exactly. Everything runs on one 4-bit Qwen3-4B on a single 24 GB GPU, with no model in the loop larger than the base. We report three findings. (i) Critic-guided selection beats verifier-filtered best-of-$k$ by $+9.1$ pp ($6/6$ seeds), with the entire gain localized to tasks where candidates disagree on held-out inputs. (ii) Per-round STaR self-training raises the ceiling but never accelerates -- the gain tracks remaining headroom and decelerates across $K=4$ independent training trajectories. (iii) The domain has no clean zero-capability frontier, so the usual "$0\% \to$ climb $=$ emergence" test is invalid here. A measured pass@$K$ crossover settles the diagnosis: the trained model wins at the operating budget (pass@$8$) but the base overtakes it at a large budget (pass@$64$) on every trajectory, so self-training concentrates probability mass rather than expanding reach. This is amplification, not compounding. ($K=4$ is indicative, not yet a robust across-trajectory CI.)