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.
A transformer's answer lives on one axis: the direction its unembedding reads. Its intermediate states largely do not, and that off-axis position is usually treated as an obstacle to interpretation. We show it is functional. A 12-layer model computes in two phases. Through the first, every sublayer writes into a subspace held near-orthogonal to the read-out, attention 75 to 96 degrees off it at every depth. Moving attention's values onto the read-out is 64 to 84 times more damaging than a matched random rotation, and the damage is entirely in cross-token mixing: the subspace insulates composition from the vocabulary. Beneath it the frame itself turns rigidly with depth. In the second phase the answer arrives on-axis, late, and by addition rather than by turning accumulated content onto the read-out. Pressing every layer onto the read-out instead, as training for early exit does, matches the baseline on perplexity, LAMBADA and BLiMP while cutting the concept-phase workspace from about twenty-five effective dimensions to fourteen, a change none of those benchmarks register. The geometry can also be imposed, though not by asking for it. Prescribing it through the loss is a lottery: six of eight seeds collapse, because a model told to null its read-out projection obeys most cheaply by discarding dimensions. Inserting one fixed rotation at the phase boundary lands it instead, at baseline quality. A sparse rotation the surrounding weights can absorb converges on all nine seeds, against five of nine for ordinary training. Which rotation is immaterial: twenty-five runs across thirteen distinct ones reach the same quality, and two baselines from different seeds hold their concepts in near-orthogonal frames while agreeing on their read-outs. That freedom is usable: a basis drawn at random and prescribed before training is adopted across the concept phase, with quality unchanged.
Fine-tuning LLMs to inject new knowledge faces a critical challenge: LLMs can quickly memorize new facts, yet fail to use them for downstream reasoning tasks. We formalize this failure as the \textit{\textbf{Knowing--Using Gap}}, characterized by an accuracy gap and a temporal lag between memorization and generalization. To understand this phenomenon, we fine-tune LLMs with unseen knowledge and monitor the spatial permeation dynamics of the knowledge internally using a novel intervention technique called self-patching. Self-patching identifies activation locations where relocating representations substantially improves failed generalization cases. These results are consistent with a knowledge-circuit misalignment hypothesis: memorized representations can exist internally but may not be routed to computation-effective layers. To demonstrate the practicality of this diagnostic finding, we design a simple heuristic strategy which recovers 58--75\% of the oracle headroom in generalization failure. Experiments are done cross-domain for the robustness of this finding.
Language models (LMs) capture large amounts of factual knowledge applicable to a wide range of tasks, motivating the view of their parameters as a knowledge base. An important property of knowledge bases is that different queries for the same fact return consistent results, drawing on a single source of truth. We investigate whether LMs satisfy this property through behavioral and mechanistic analyses. Our results suggest that they encode knowledge in a task-specific manner. Behaviorally, facts acquired on one task frequently fail to co-emerge on others during training. Parameter localization experiments suggest a mechanistic explanation, revealing distinct parameter subsets underlying different tasks for the same fact. Finally, we show that chain-of-thought reasoning draws part of its effectiveness from engaging task-specific parameters beyond those tied to the evaluation task. Our findings suggest that what the model knows and how it is asked are intertwined in parameter space, undermining the "knowledge base" analogy and carrying implications for the reliability and controllability of factual knowledge in LMs.
Modern language models increasingly adopt hybrid architectures that combine full attention with efficient attention modules, such as sliding-window attention (SWA) and recurrent sequence mixers. However, how these efficient modules shape model capabilities remains poorly understood. To address this gap, we conduct a systematic analysis across hybrid architectures from three perspectives: scaling behavior, mechanism analysis, and architecture design. First, from a scaling perspective, we find that efficient-attention design primarily affects how fast long-context capability emerges, while different hybrids eventually converge to comparable long-context performance under sufficient training. Second, mechanistically, we show that long-range retrieval is mainly carried by full attention, whereas efficient attention shapes its optimization trajectory. This explains a counter-intuitive phenomenon we call Large-Window Laziness: larger SWA windows can delay the formation of retrieval heads in full-attention layers. Third, guided by this mechanism, we show that applying NoPE to only the full-attention layers of a small-window SWA hybrid substantially improves long-context performance with negligible impact on short-context performance.