Large language models often answer structurally unanswerable questions, such as computing cot(-540°) or evaluating (1).startswith("1"), instead of abstaining. We ask whether this failure reflects missing recognition or failed routing from recognition to abstention. Across instruction-tuned models from 1.7B to 70B parameters, a single linear direction in the hidden state separates answerable from structurally impossible math and code prompts, showing that models represent impossibility before generation. Yet this recognition direction is nearly orthogonal to the canonical safety-refusal direction that mediates trained harmful-content refusal. An in-domain behavior-defined invalidity-aware direction is closer to recognition, but only partially aligned with it, and remains near-orthogonal to safety refusal. Generation-time steering along the recognition direction changes invalidity-aware behavior bidirectionally and dose-responsively on structural math and code cells, while random directions do not. Base/instruct comparisons further show that the low-cosine geometry is already present at the pretraining endpoint. The confident-on-impossible failure is therefore better explained as a routing failure than as an encoding failure: the model has a usable "no admissible answer" signal, but the safety-refusal pathway is not aligned to use it.
Reliable handling of unanswerable questions (UAQs) is critical for trustworthy LLM-based agents. Although memory is widely used in agent systems, its role in reliable UAQ handling remains unclear. We present a systematic study of agent memory for UAQ handling under a unified agentic RAG framework, evaluating four representative memory methods across three UAQ-related datasets and two base models. We find that memory can improve UAQ performance in some settings, but such gains are selective rather than universal and remain fragile under dataset shift. Interestingly, cross-model memory reuse is often more feasible than cross-dataset transfer, suggesting that shifts in answerability patterns pose a greater challenge to memory reuse than changes in the base model itself. We further find that UAQ gains are more strongly preserved through decision guidance than through trajectory shaping, and that memory effectiveness depends strongly on representation. In particular, procedural and rule-based memories often provide the most reliable support for UAQ handling, while memory composition is most effective when procedural guidance is combined with complementary behavioral signals. Overall, our findings suggest that reliable UAQ memory depends less on storing larger amounts of experience and more on preserving transferable behavioral guidance.