Post-training updates change what large language models (LLMs) can answer, but may leave their answerability judgments misaligned. We study this knowledge-judgment desynchronization through functional feeling of knowing (FoK), which supports selective generation, retrieval triggering, and model routing without implying subjective experience. We propose two trainable channels: a knowledge channel that changes answerability and a FoK channel that judges the current knowledge state. Three stages train them: on-policy supervised fine-tuning (OP-SFT) updates knowledge from corrected on-policy answers; knowledge-delta FoK training (K$Δ$-FT) adapts judgments across intermediate knowledge states; and FoK-guided policy optimization (FGPO) trains the subsequent answer or explanation. Factual-QA experiments, cross-dataset evaluation, and three sequential updates show that explicit FoK alignment improves judgment quality on updated knowledge. MMLU and ARC-Easy separately assess capability preservation.

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