Multi-turn jailbreaks turn dialogue safety into a trace property: a violation may become identifiable only when user intent and assistant behavior are considered across multiple turns. Existing guard models largely classify individual messages or exchanges and return risk scores or labels, without executable safety semantics or a formally checkable decision trail. We present SpecGuard, a training-free, specification-driven framework for monitoring multi-turn LLM interactions. SpecGuard uses a fixed atomic-proposition contract to map natural-language observations into causally scoped, evidence-backed facts, and expresses forbidden interaction patterns as Linear Temporal Logic specifications. Each specification is compiled into a symbolic finite automaton that incrementally consumes the resulting dialogue trace. The same monitoring mechanism supports buffered runtime checking and retrospective auditing, while accepting runs produce replayable witnesses that identify the triggered specification, supporting facts, and relevant dialogue turns. Experiments on multi-turn safety benchmarks show that SpecGuard consistently outperforms representative open-source guard models while maintaining controlled false-positive rates. These results demonstrate a practical path from heuristic risk classification toward formally specified, verifiable, and auditable LLM safety monitoring, without learned detector parameters or access to the protected model.

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