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.
Multimodal Large Language Models (MLLMs) have made substantial progress in 3D scene understanding from multi-view RGB observations. Yet a fixed object and query can elicit inconsistent target-related responses across camera viewpoints, a phenomenon we characterize as viewpoint-induced semantic drift. We introduce LieVLM, a Lie routing framework for view-robust 3D understanding. Motivated by the rotational component of viewpoint change, Lie routing adapts visual representations by applying local $\mathrm{SO}(3)$ transformations to projected features within learned three-dimensional subspaces. Geometry tokens extracted by VGGT predict local rotation parameters and channel-wise routing coefficients. A closed-form Padé--Cayley map transforms projected visual features, whose identity-relative corrections are projected back and gated into the original visual and geometry features. This design retains the pretrained semantic and direct spatial context while learning view-dependent feature adjustments. Experiments on 3D captioning, grounding, detection, and spatial reasoning show consistent improvements over additive fusion across training scales and backbones. Controlled ablations examine the roles of geometric conditioning and transformation structure. Viewpoint analyses further reveal systematic variation in the predicted routing states alongside more localized target responses and stronger task performance across viewpoint changes.