Large Language Models often generate incorrect factual answers, yet it remains unclear how they internally distinguish facts they know from those they do not. Existing hallucination-detection methods largely rely on behavioral signals or opaque scalar scores, leaving the structure of factual knowledge in hidden state space poorly understood. We address this gap by introducing a coarse-to-fine framework to discover a knowledge boundary: a decision surface in transformer hidden states that separates representations leading to correct factual answers from those leading to errors. Across multiple datasets and model families, we show that factual correctness is predictable from hidden states, and that this signal is concentrated in specific layers and a small number of influential dimensions. Using Qwen3-8B as a case study, we further identify a compact "knob" subspace that retains near-full boundary-prediction performance and supports targeted interventions that bidirectionally modulate the likelihood of the full reference answer under teacher forcing.