We challenge the common assumption that Large Language Models (LLMs) build confidence gradually during reasoning. Instead, we find that conviction is often reached in a discrete "moment of insight", characterized by a sudden and sharp increase in an answer’s probability-a phenomenon we term a "confidence leap". Leveraging this discovery, we introduce a training-free, model-agnostic early-stopping heuristic that halts generation upon detecting such a leap, significantly reducing the generation length without sacrificing accuracy. We also demonstrate that the reasoning text leading up to this leap is semantically potent and transferable: feeding this partial reasoning to a different model family substantially boosts its performance. This suggests that the "confidence leap" marks a shared, interpretable reasoning milestone, not just a model-specific statistical artifact.