Why a room full of agents can’t talk its way to new knowledge, and exactly how much its tools can buy.
By Manish Bhatt · October 2026 · Paper (PDF)
Multi-agent systems are often pitched with 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 can be stated 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.
That isn’t a hunch or an empirical trend. It’s a theorem, and it holds for any kind of agent — LLMs, people, or programs — under four assumptions, listed below. This post explains what it says, why it’s true, and what it does and doesn’t rule out. The full proofs and the scripts that check them are in the formal proofs.
The Coffee Law is one of three ceilings on what a team of agents can do:
A few terms. Bits measure information: one bit settles a yes/no question. Mutual information is how many bits one thing tells you about another. A tool’s capacity is the most bits a single call can ever reveal; a pass/fail test has at most one. Usable information is the part of what you hold that you can actually turn into answers with the computation you have.
Start with a group of agents that can only talk to each other and think. No web search, no experiments, no new data after the conversation starts.
Everything such a group will ever say is a function of two things: what it knew at the start, and randomness (sampling noise, who speaks first) that has nothing to do with the truth. The data-processing inequality, one of the most basic facts in information theory, then says:
the information the whole conversation holds about the truth is at most the information the starting data held.
So debate, self-reflection, critique rounds and majority votes cannot add a single bit about the world. They can only re-express what was already there (Theorem 1). Re-expressing can still be very valuable, as the next two sections show, but it adds no facts.
That doesn’t make collectives useless. Information can be present jointly while no individual can use it.
Here is the cleanest example. Two agents each hold one random bit, B₁ and
B₂. The question is whether the bits are equal (formally, B₁ ⊕ B₂). Alone,
each agent knows exactly nothing about the answer: whatever its bit is, the
answer is still a coin flip. Together, they know it with certainty.
Zero plus zero equals one bit. That is genuine emergence, and it’s the strongest argument for shared boards and agent communication (Theorem 2).
But notice where that bit came from: it was in the starting data all along, split across two agents. Communication made it usable. It didn’t make it exist. Emergence rearranges the budget; it doesn’t enlarge it. That difference between information that exists and information you can use turns out to be a second ceiling of its own.
The moment an agent calls a tool — runs a search, executes code against real data, queries a sensor, runs an experiment — something different happens. The tool’s answer depends on the actual state of the world, not just on what the agents already believed. That breaks the chain that capped the closed group, and total information can grow (Theorem 3).
Choosing which tool call to make adds nothing by itself: the choice is made from what the group already knows. All new information arrives in the results (Theorem 4).
Every tool call is a communication channel from the world to the collective, and every channel has a maximum rate — its Shannon capacity. A call that returns one of 16 possible answers can carry at most 4 bits. A noisy call carries less.
Put the pieces together and you get the Coffee Law (Theorem 7):
information about the truth ≤ min( total uncertainty,
starting information + Σ capacity of each round's tool calls )
This holds for:
The number of agents doesn’t even appear in the bound. More agents help only if they make more tool calls, or better ones.
And because information limits accuracy, the bound becomes a floor on error.
Fano’s inequality says the chance of a wrong answer is at least roughly
(remaining uncertainty − 1) / log₂(number of possible answers). For a question
with 2^k equally likely answers, getting the error down to ε takes at least
((1 − ε)·k − 1) / C rounds of tool use, where C is the capacity per round —
however clever the agents are.
Suppose the truth is one of four values (2 bits), and each tool call returns
one bit of it, flipped by noise 10% of the time. That gives a capacity of about
0.531 bits per call. With two calls, the budget is 1.062 bits.
verify.py
computes the exact information gained:
| Strategy | Information gained |
|---|---|
| Ask for bit 0, then bit 1 | 1.062 bits — exactly the budget |
| “Adaptive” strategy that sometimes re-asks for the same bit | 0.982 bits — below the budget |
| Any amount of debate afterwards | +0 bits |
| Ask for bit 0, then bit 1, with a noise-free tool | 2 bits — the whole truth |
Here the bound is tight: a good strategy reaches it exactly. (That isn’t guaranteed in general; sometimes no strategy can use a tool at full capacity.) A clever-looking strategy can waste the budget. Nothing exceeds it.
Having information isn’t the same as being able to use it. Give a collective a
large number N and ask for its prime factors: the answer is completely
determined by N, so every bit of it is already in the data and the Coffee Law
is satisfied with room to spare. Yet finding the factors can take an enormous
amount of computation.
So there are two ceilings, one stacked on the other (Theorem 8):
usable knowledge ≤ available information ≤ starting data + tool capacity
↑ ↑
raised by reasoning raised by tools
This changes one claim from earlier. Debate adds no facts, but it can add
reasoning: an agent that posts an intermediate result hands the others a
computation they couldn’t do themselves. In the XOR example, an agent that can
only look at one bit at a time gets nothing from (B₁, B₂); once someone posts
B₁ ⊕ B₂, it gets the full bit. The information never changed; its usability
did.
In the noisy worked example above, the transcript holds 1.062 bits about the answer, but an agent that can only read one entry of it at a time can use just 0.266 of them. The rest is a reasoning gap. More observations won’t close it; more computation will.
That splits tasks into two kinds:
Most real tasks are a mix, and the useful question is which ceiling you are hitting.
Closing a reasoning gap costs computation, and that cost has bounds of its own (Theorems 9–10):
n coin flips came up heads, you must look at all n. Read
n − 1 of them, cleverly or not, and you know nothing; read the last one and
you know everything. Ten agents can split the reading and finish faster, but
together they still read all n.So the reasoning ceiling is real, but how hard it is to reach depends on the task, from cheap to effectively impossible. No general shortcut exists: predicting the cost for every task would settle open questions like whether P = NP.
The budget is a ceiling for every agent. Ideal reasoners can reach it; real agents, and LLMs in particular, fall short of it by losing information along the way. Three losses are worth naming (LLM specialization):
I(Θ;data) − I(Θ;message)
(B2).These combine into one upper bound on how far a collective’s decisions can fall short (B4). It also explains a familiar failure: agreement grows while knowledge doesn’t. Perfect consensus on a random claim carries zero information about the truth (Theorem 6).
The bound needs exactly these, and nothing about the agents themselves (Theorem 7):
The four assumptions keep the core result clean. Relaxing them one at a time gives the rest of the picture (extensions):
It’s tempting to stretch a result like this. Here is where it stops.
If you’re building agent systems, the bound turns into practical advice:
When different agents hold different pieces, the team as a whole may hold the answer while no single agent can use it. Some pieces combine cheaply: if each agent only needs to report one summary bit, a short message does it. Others don’t: for some questions, an agent must receive about half of the other agent’s data before it knows anything at all, however clever both are (Theorem 14). Experiments find exactly this failure: groups of LLM agents that each hold part of the facts reach 30% where one agent with all the facts reaches 81%, because they don’t seek out what others know (Li, Naito and Shirado, ICML 2026).
Mostly no, and that’s a strength. The ingredients — the chain rule, the data-processing inequality, channel capacity and Fano’s inequality — are textbook information theory. The closed-group case has been written up for LLM debate by Shin (arXiv:2605.01704) and Choi et al. (arXiv:2508.17536), and Yang et al. (arXiv:2602.03794) give a bound of the same shape with agents as the channels. The reasoning ceiling is “usable information” in the sense of Xu et al. (arXiv:2002.10689), who also show computation can create it. Related bounds exist for single settings: Fano-type limits on adversarial queries (Kaneko and Baldwin, NeurIPS 2025) and on single-pass multi-hop reasoning (Wan et al., ICLR 2026). What these notes add, as far as we found, 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 lists twelve published findings the theory explains, and one prediction nobody has tested yet.
The published experiments line up with both ceilings. On closed-book knowledge questions, giving reasoning models more thinking time doesn’t reliably help and often increases hallucination (arXiv:2509.06861): an information-bound task. On math and strategy, debate helps (Du et al., arXiv:2305.14325): a compute-bound task. And debate tends to do no better than majority voting or self-consistency at equal effort (Smit et al., PMLR 235; Choi et al.), which is what you’d expect if its value is extra reasoning. Even Du et al.’s finding that debate improves factuality fits: for a model in the real world, the question already determines a factual answer, so the facts are something to compute from what the model holds, not something new to learn. Debate helped the model use what it already knew (Proposition 18). (These comparisons are based on the papers’ abstracts and published summaries.)
The short version, and the Coffee Law in one sentence: a collective’s intelligence can be emergent, but its knowledge has a budget, and only the world can raise it.
Proofs: formal proofs ·
information geometry ·
LLM specialization.
Checks: verify.py,
verify_llm.py,
verify_geometry.py.