Recent work suggests that some large language model representations have content or reference. Grounding can secure either without supplying live routes for correction. This paper asks what follows from that gap. An output is answerable when discrepancies can affect what a target- and task-specific arrangement produces, accepts, or withdraws. The arrangement has corrective control only when live, sufficiently independent routes can detect and repair fresh discrepancies. A route profile records which routes constrain the arrangement and how they are related. Those profiles support analysis of truth-tracking: patterned support for representational success. Language models are the pressure case; text-only arrangements provide a task-relative limiting case. Text-trained models inherit patterns of testimony, coherence, and prior correction. Where target-sensitive correction survives training, these can supply derivative answerability (inherited constraint); live answerability is the relation supplied by a current route for fresh discrepancies. Fluent failures should follow when a task requires independently informative access to the facts. Self-consistency, retrieval, tools, code execution, multimodal input, and feedback should help selectively. Route-by-task interactions test the distinctions. The decomposition's empirical burden is to predict held-out route--task combinations or improve intervention choice without conceptual refitting. Surface improvement and truth-tracking improvement can come apart.
Tables are a critical knowledge source in retrieval-augmented generation (RAG), but a retrieved table may lack sufficient evidence to answer a query, a property we call answerability. While answerability broadly concerns whether a source or collection of sources contains sufficient evidence, retrieval models optimized for semantic relevance do not guarantee it even in the single-source case, creating a fundamental mismatch. To study this, we introduce TCR-Bench, a diagnostic benchmark for Table Content-level Answerability in RAG, built around sibling tables, i.e., tables with highly similar schemas but subtle content differences. On TCR-Bench, the dense retrievers we evaluate persistently exhibit a Semantic-Answerability Gap: they often retrieve the correct sibling group yet struggle to pinpoint the uniquely answerable table within it, dropping QA performance from 0.755 (oracle) to 0.330 (top-5 retrieved). Our analysis suggests this gap is associated with semantic accumulation, schema-level cue dependence, and weak row-column binding. As a diagnostic probe into the source of this gap, we test whether a lightweight two-stage pipeline, Answerability-Aware Reranking (AAR), applying direct query-table answerability judgment, can recover performance: it raises top-1 target retrieval from 18.2% to 57.4%, and this large gain is itself evidence that much of the observed failure reflects a missing answerability verification step, rather than an inherent limitation of model capacity alone.
A model should refuse two different things: answers it would get wrong, and questions it should not answer at all, such as unanswerable ones or ones resting on a false premise. The usual recipe thresholds a single confidence score, which cannot tell these apart. Across five instruction-tuned models from three families (2B to 14B), we find they are separate axes. Ordinary answer-confidence tracks whether an answer is right but is nearly blind to whether the question is answerable; a linear probe on hidden states does the reverse. The blind spot does not shrink with scale. It is worst on naturally occurring false-premise questions (CREPE). There, answer-confidence, P(IK), P(True), and even asking the model outright whether a premise is false all stay near chance, while a hidden-state probe reaches 0.69 to 0.77 AUROC: the model represents a problem it will not report. This turns out to be fixable. Instructing a model to check premises backfires, because it then disputes sound and false premises alike (57% false challenges), unable to tell them apart; routing the same instruction with the probe roughly triples challenge precision. We turn the two axes into a calibrated policy that answers only when an answerability score and a correctness score each clear a separately certifies behave differently: the unanswerable-answer rate is controllable at every scale, while the wrong-answer rate is capped by model accuracy, so the guarantee tightens as threshold policy certifies both budgets at 0.75 coverage of correct answers, against 0.31 for a single threshold; at 14B it is the only policy that certifies at all.