How does a single training example trigger phase transitions in reasoning output?

This explores how one training example can trigger a phase transition in reasoning — and the corpus's striking answer is that the example isn't building reasoning, it's *activating* reasoning the model already had. The clearest data point: in RLVR (reinforcement learning with verifiable rewards), a single training example lifts math performance from 36% to 73.6%, and — stranger still — test accuracy keeps climbing for 1,400 steps *after* training accuracy already hit 100% Can a single training example unlock mathematical reasoning?. That post-saturation generalization is the signature of a phase transition: the model isn't memorizing the example, it's being tipped into a different operating regime.

Why would one example be enough? Because the reasoning capacity is latent in the base model, waiting to be elicited. Five independent methods — RL steering, critique fine-tuning, decoding tweaks, sparse-autoencoder feature steering, and RLVR — all unlock reasoning that already exists in base-model activations Do base models already contain hidden reasoning ability?. The bottleneck is elicitation, not acquisition. Seen this way, a single example works like a key, not a curriculum: it selects an existing mode rather than creating a new one. A complementary framing argues RL post-training teaches the model *when* to reason, not *how* — hybrid models recover 91% of the gains by just routing tokens, and reasoning-strategy activation vectors pre-exist before any RL touches the weights Does RL post-training create reasoning or just deploy it?.

If reasoning is being switched on rather than taught, you'd expect the *content* of the training signal to matter less than its role as a trigger — and that's exactly what shows up. Models trained on deliberately corrupted, semantically irrelevant reasoning traces perform comparably to those trained on correct ones, sometimes generalizing *better* out of distribution Do reasoning traces need to be semantically correct?. The traces act as computational scaffolding, not meaningful logic. That fits the broader finding that chain-of-thought is constrained imitation of reasoning *form* rather than genuine inference Does chain-of-thought reasoning reveal genuine inference or pattern matching?, What makes chain-of-thought reasoning actually work?. A trigger that activates a latent pattern doesn't need to be a good example — it just needs to point at the right pattern.

The deeper question the phase-transition framing raises: where did the latent capability come from in the first place? Analysis of 5 million pretraining documents suggests reasoning rides on broad, transferable *procedural* knowledge absorbed across many sources, distinct from the narrow document-specific memorization behind factual recall Does procedural knowledge drive reasoning more than factual retrieval?. So pretraining lays down the reasoning machinery diffusely; a single later example just trips the switch. Some researchers are now trying to move that activation earlier, planting CoT as an exploratory action *during* pretraining with information-gain rewards Can chain-of-thought reasoning be learned during pretraining itself?.

The catch worth knowing: a switch flipped this cheaply has limits the gains can hide. CoT degrades predictably under distribution shift — fluent but logically inconsistent once you move outside the training distribution Does chain-of-thought reasoning actually generalize beyond training data? — and failures track *instance-level novelty*, not task complexity, because models fit instance patterns rather than general algorithms Do language models fail at reasoning due to complexity or novelty?. So a single example can produce a dramatic jump on familiar territory and still leave the underlying reasoning brittle where it counts. The phase transition is real — it's a transition in *deployment*, not in raw capability.