ObjexMT: Objective Extraction and Metacognitive Calibration for LLM‑as‑a‑Judge under Multi‑Turn Jailbreaks

LLM-as-a-Judge (LLMaaJ) now underpins scalable evaluation, yet we lack a decisive test of a judge’s qualification: can it recover a conversation’s latent objective and know when that inference is trustworthy? LLMs degrade under irrelevant or long context; multi-turn jailbreaks further hide goals across turns. We introduce **ObjexMT**, a benchmark for objective extraction and metacognition. Given a multi-turn transcript, a model must return a one-sentence base objective and a self-reported confidence. Accuracy is computed via LLM-judge semantic similarity to gold objectives, converted to binary correctness by a single human-aligned threshold calibrated once on **N = 300** items ($\tau^\star = \mathbf{0.66}$; $F_1@\tau^\star = 0.891$). Metacognition is evaluated with ECE, Brier, *Wrong@High-Confidence* (0.80 / 0.90 / 0.95), and risk–coverage. Across six models (`gpt-4.1`, `claude-sonnet-4`, `Qwen3-235B-A22B-FP8`, `kimi-k2`, `deepseek-v3.1`, `gemini-2.5-flash`) on *SafeMTData_Attack600*, *SafeMTData_1K*, and *MHJ*, `kimi-k2` attains the highest objective-extraction accuracy (**0.612**; 95 % CI [0.594, 0.630]), with `claude-sonnet-4` (**0.603**) and `deepseek-v3.1` (**0.599**) not statistically distinguishable from it by paired tests. `claude-sonnet-4` yields the best selective risk and calibration (AURC **0.242**; ECE **0.206**; Brier **0.254**).**Striking dataset heterogeneity (16–82 % accuracy variance) reveals that automated obfuscation poses fundamental challenges beyond model choice.**Despite improvements, high-confidence errors remain: Wrong@0.90 ranges from **14.9 %** (`claude-sonnet-4`) to **47.7 %** (`Qwen3-235B-A22B-FP8`). ObjexMT thus supplies an actionable test for LLM judges: when objectives are not explicit, judges often misinfer them; we recommend exposing objectives when feasible and gating decisions by confidence otherwise. **All experimental data are provided in the Supplementary Material and at [https://github.com/hyunjun1121/ObjexMT_dataset](https://github.com/hyunjun1121/ObjexMT_dataset).**