When a language model sees a document contradicting its training knowledge, it must choose: follow the document or trust itself. Prior work proved this choice depends on how well-known the fact is. We ask: does the model's chain-of-thought (CoT) reasoning faithfully report this mechanism? We introduce introspective faithfulness and test it across 200 questions, 8 models, and 4 prompt conditions. We find CoT reasoning is highly stable across opposite decisions: flip pairs retain 96% of same-answer similarity (d=0.34; confirmed by ROUGE-L, d=0.45). Yet self-rated confidence carries a faint genuine signal: for obscure facts where entity fame is uninformative, confidence still predicts decisions (p<0.001) and tracks item-level knowledge (r=0.134). GPT-4o is the only model with statistically reliable reasoning-decision coupling. Claude Sonnet 4.6 shows the widest confidence range (SD=1.39) but near-zero pooled correlation because the confidence-decision relationship reverses between conditions; a temperature ablation confirms this is model-specific. Internal thinking tokens show greater decision-sensitivity than user-facing CoT (p=0.033). CoT decomposes into a decision-invariant knowledge display (~96%) and a thin confidence layer with weak but real signal. For monitoring: read confidence, not the argument.