Multi-agent LLM systems pair a sender with broad context and an executor with a limited local view.
We study when a short message improves the executor's next decision, when raw context is preferable, and when a stronger sender helps.
Our framework, receiver-relative bounded coordination, expresses message utility as receiver gain minus protocol tax.
Compression beats raw context when tax savings exceed losses from omitted information and decoder mismatch.
Even Bayes-sufficient compression can fail when a bounded executor cannot use its surface form.
A three-stage decomposition separates externalization, absorption, and action closure, explaining how errors remain after the correct content reaches the receiver.
Under a single-crossing condition, sender upgrades help above a receiver-burden threshold.
Across six benchmarks, the same Qwen protocol raises ContextBench joint accuracy from $0.633$ to $0.775$ but lowers ToolSandbox from $0.889$ to $0.653$.
Fixed-message replay reveals closure failures despite correct artifact recovery.
These results guide an inference-time selector that improves the accuracy-cost frontier on the evaluated communication regimes.