Multi-agent systems are often sold on a quiet promise: put enough agents in a room, let them debate, critique and vote, and something smarter than any one of them will emerge. Sometimes it does. But there is a hard ceiling on what that emergence can achieve, and it fits in one line: a collective can never know more about the world than its starting data plus what its tools can carry. We call this the Coffee Law, and today we are releasing it — paper, formal proofs, and the scripts that check them.
The law
Write Θ for the truth, D₀ for the starting data and C_t for the Shannon capacity of round t's tool calls. Then the information everything the collective holds carries about the truth is at most min(H(Θ), I(Θ; D₀) + Σ C_t). It holds for any kind of agent — LLMs, people or programs — any number of them, any communication protocol and any query strategy, under four stated assumptions about the environment. Fano's inequality turns the bound into a floor on error. The number of agents does not appear in it: more agents help only if they make more tool calls, or better ones.
Talking moves information; it does not create it
Everything a closed group of agents says is a function of what it knew at the start plus randomness unrelated to the truth. By the data-processing inequality, debate, self-reflection and majority votes cannot add a single bit about the world. Emergence is still real — two agents each holding one random bit know nothing alone about whether the bits match, and everything together — but that bit was in the starting data all along. Communication made it usable. It did not make it exist. Only calls to tools that observe the world raise the budget.
Three ceilings, not one
The paper separates three limits that are usually blurred together. Information: the team cannot know more than its data and tools carry. Reasoning: it cannot use more than it holds, and closing that gap costs computation that ranges from linear to — under standard cryptographic assumptions — infeasible. Communication: when evidence is split across agents, some tasks need about half of the other agent's data exchanged before anyone knows anything. Adding agents, communication, compute and world access are different interventions acting on different ceilings.
Checked, not asserted
The core of the law — the Markov condition for any collective, data processing, and the per-call output-entropy bound — is proved in Lean 4, with no unfinished proof steps. Every finite example in the paper is computed exactly by public Python scripts, and each number in the paper is tied to the script that produces it. In the worked example, a truth with four values and a tool that returns one bit flipped 10% of the time gives 0.531 bits per call; two well-chosen calls reach the 1.062-bit budget exactly, an adaptive strategy that re-asks wastes it at 0.982, and debate afterwards adds zero. We re-ran all five check suites before publishing this note; all pass.
What it does not say
It does not cap intelligence: it caps information about a fixed external truth, and working out consequences of what you know is reasoning. It is not a fixed ceiling: every informative tool call raises it. It does not separate machines from people — the proof never looks inside the agents, so it cannot be used to argue AGI is impossible. And it does not cover outcomes the agents themselves control.
What builders should do with it
Find out which ceiling you are hitting: if the answer is in the data, think harder; if it is not, look at the world. Use debate for reasoning, not facts. Spend effort on tools and starting data, because nothing else raises the information budget. Treat confidence as different from evidence — a message carrying zero information can still flip a correct agent. Sign every message with its author, and check what the collective claims to know against the world, because agreement is cheap.
Why it belongs here
This is a boundary result in the same family as the Defense Trilemma: it says what a design can never deliver, so effort goes where it can actually help. The ingredients are textbook information theory, and the related work is credited in the paper. What is new is the explicit accounting for tools in agent teams — per-call capacity, adaptive queries, an error floor — the three ceilings combined, and a channel account of sandboxing. The paper is being prepared for submission; we welcome counterexamples and replications.