Do LLMs Know What They Know? Measuring Metacognitive Efficiency with Signal Detection Theory

Standard evaluation of LLM confidence relies on calibration metrics (ECE, Brier score) that conflate two capacities: how much a model knows (Type-1 accuracy) and how well its confidence signal tracks that knowledge (Type-2 metacognitive sensitivity). We apply Signal Detection Theory to decompose them, treating token-level normalised log-probability as a graded confidence variable and answer correctness as the state to be discriminated. We characterise the Type-2 ROC of this signal, including its unequal-variance structure via z-ROC analysis, and -- because the meta-d' efficiency ratio is not well defined for open-ended QA, which lacks a two-alternative Type-1 decision -- quantify efficiency with a model-free information measure, normalised metacognitive information (meta-I_2r). Across four LLMs and 224,000 factual QA trials we find: (1) metacognitive information varies by a factor of 1.98 across models and is not predicted by accuracy, the rank correlation being -0.80 on TriviaQA and +0.00 on Natural Questions; (2) the confidence signal has model-specific unequal-variance structure (z-ROC slopes 0.78 to 1.18) invisible to calibration metrics, the slope ordering replicating on NQ; (3) efficiency is domain-specific, weakest in Science & Technology for every model; (4) temperature dissociates accuracy from metacognitive information, which stays near-flat for three of four models while accuracy falls; and (5) metacognitive information tracks the accuracy gain from confidence-based abstention exactly (rho = +1.00) while accuracy does not. All estimates carry permutation nulls and bootstrap confidence intervals. This version (v3) corrects a differential length bias in the automated correctness scorer, validated against 1,830 human adjudications; the inverse accuracy-efficiency coupling reported in v1 and v2 does not survive relabelling. See the version note on page 1.