Language-model agents increasingly coordinate in large groups, and current agent protocols focus on delivering messages. For these agents, however, the scarce resource is not delivery but reading: every message costs tokens, and the context is finite. Current infrastructure works by push: the sender decides when a message enters the recipient's context, so the recipient reads in arrival order and drops the oldest messages once its context is full. We study the alternative, pull, in which messages wait in an inbox outside the context and the recipient's own rules decide what it reads and keeps. We argue that the inbox is a coordination primitive: each agent sets its rules privately, and those rules decide whether other agents' requests reach it. In simulated societies of up to 16,000 agents, with every decision an LLM judgment, we report three findings. First, push fails as an agent takes part in more coordination at once: an agent leading 20 negotiations with a full memory proposes options already rejected 97% of the time, because it has dropped the oldest replies. Second, an inbox is necessary under heavy, noisy traffic. Where first come, first read misses every request, an inbox that first reads messages carrying a refundable deposit still finds about 60% of the agents a request needs, at every size from 1,000 to 16,000. On a real email network, losses compound: when half the agents adopt such an inbox, forwarded search recovers only 5% of full adoption, against 42% for one-hop search. Third, an inbox comprises four decisions (what to read, what to keep, what a sender pays, and what a reply may say), and the best rule for each depends on the traffic and the agent's role. Multi-agent coordination under load requires an inbox layer designed for the traffic each agent receives.

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