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 self-reported confidence. Accuracy is computed via LLM-judge semantic similarity to gold objectives, converted to binary correctness by a human-aligned threshold calibrated on N=300 items (tau = 0.66; F1 = 0.891). Metacognition is evaluated with ECE, Brier, Wrong at 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 three datasets, kimi-k2 attains the highest objective-extraction accuracy (0.612), with claude-sonnet-4 (0.603) and deepseek-v3.1 (0.599) statistically comparable. claude-sonnet-4 yields the best selective risk and calibration (AURC 0.242; ECE 0.206; Brier 0.254). Dataset heterogeneity (16-82 percent accuracy variance) reveals that automated obfuscation poses fundamental challenges beyond model choice. High-confidence errors persist: Wrong at 0.90 ranges from 14.9 percent (claude-sonnet-4) to 47.7 percent (Qwen3-235B-A22B-FP8). ObjexMT provides 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. Data at this https URL.