Autonomous LLM agents act through tools under per-action authorization: a local guard screens each tool call in isolation. We show that local authorization does not compose into global authorization. A compositional authorization failure (CAF) is a locally authorized execution, or a pair of locally authorized executions, that violates a global policy spanning intent, purpose, provenance, consent, and delegation. We characterize CAFs along two axes, expressiveness (single-trace safety versus relational k-safety) and observability (a ladder of monitor profiles). Our central result is an impossibility: relational purpose violations form a 2-safety hyperproperty with no per-trace locus of blame, so a per-action monitor that never raises a false alarm detects none of them, independent of the underlying model or decision procedure; they become visible only under cross-execution monitoring, which none of the six guardrails we survey implements. We make this constructive with AL-SYNTH, a SAT/SMT synthesizer (using no language model) that emits minimal CAF witnesses, each checked against an exhaustive oracle. The empirical signature is sharp: across nine LLM judges the per-action view approves the synthesized witness almost without exception, blocking it at a rate comparable to a benign control, yet the same models flag it when shown the two executions together. Per-action authorization is therefore structurally insufficient for relational policy conjuncts not reducible to single-trace safety: to bound them, a monitor must compare executions or accept false alarms.

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