Emergence, Not Bandwidth: Physical Coupling and the Limits of Learned Multi-Agent Communication

Rate-limited multi-agent teams raise three questions the emergent-communication literature has answered only empirically: what an optimal message should encode, what compression costs over a horizon, and when a learned protocol is unique enough to be read by a teammate. We answer them for rate-limited Dec-POMDPs, and then measure how far reinforcement learning falls short of the optimum the theory locates. Our theorems fix what is achievable independently of any learner, so a gap between an engineered sender and a learned one at the same bit budget is an optimization fact rather than an information-theoretic one. We instantiate this on three MuJoCo continuous-control arenas spanning zero, partial and rigid physical coupling, with every condition charged exactly 2 bits per decision by construction, and the discriminating regime is produced by closing a physical side channel within one arena, holding bodies, task and reward fixed. Communication value is governed by coupling: where the agents are rigidly coupled through a shared object, no channel beats silence ($+0.001\pm 0.001$, $p=0.982$, $n=25$), because proprioception already carries what a message would say; where they are uncoupled, every condition solves the task; and in the partially coupled regime the engineered $2$-bit sender reaches an interquartile mean of $1.000$ while the learned $2$-bit sender reaches $0.482$ and is indistinguishable from silence ($p=0.400$, $n=25$). Because the two senders hold the same alphabet, bandwidth cannot explain the gap. Initializing a learned run from an engineered run's receiver localizes the failure: the same channel then reaches $0.857$ against $0.562$ cold-started ($p<0.001$), so the failure is neither representational nor a matter of maintenance: what reinforcement learning cannot do here is discover the protocol in the first place. Cross-play shows the learned protocols are individually meaningful and mutually unintelligible: self-play $0.980$ collapses to $0.144$ across seeds, and the best alignment we can construct leaves at least $77\%$ of that gap standing. Every headline result is reported at $25$ seeds per arena, against seven published baselines reimplemented in our trunk at matched rate.