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

Large language models (LLMs) are increasingly used as judges of other models, yet it is unclear whether a judge can reliably infer the latent objective of the conversation it evaluates, especially when the goal is distributed across noisy, adversarial, multi-turn jailbreaks. We introduce OBJEX(MT), a benchmark that requires a model to (i) distill a transcript into a single-sentence base objective and (ii) report its own confidence. Accuracy is scored by an LLM judge using semantic similarity between extracted and gold objectives; correctness uses a single human-aligned threshold calibrated once on N=100 items (tau* = 0.61); and metacognition is evaluated with ECE, Brier score, Wrong@High-Conf, and risk-coverage curves. We evaluate gpt-4.1, claude-sonnet-4, and Qwen3-235B-A22B-FP8 on SafeMT Attack_600, SafeMTData_1K, MHJ, and CoSafe. claude-sonnet-4 attains the highest objective-extraction accuracy (0.515) and the best calibration (ECE 0.296; Brier 0.324), while gpt-4.1 and Qwen3 tie at 0.441 accuracy yet show marked overconfidence (mean confidence approx. 0.88 vs. accuracy approx. 0.44; Wrong@0.90 approx. 48-52%). Performance varies sharply across datasets (approx. 0.167-0.865), with MHJ comparatively easy and Attack_600/CoSafe harder. These results indicate that LLM judges often misinfer objectives with high confidence in multi-turn jailbreaks and suggest operational guidance: provide judges with explicit objectives when possible and use selective prediction or abstention to manage risk. We release prompts, scoring templates, and complete logs to facilitate replication and analysis.