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Emergence, Not Bandwidth: Physical Coupling and the Limits of Learned Multi-Agent Communication
Mihir Chauhan, Aniket Bera
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper investigates the limits of learned multi-agent communication in rate-limited Dec-POMDPs, arguing that physical coupling, rather than bandwidth, governs communication value. Through experiments in MuJoCo arenas with zero, partial, and rigid physical coupling, the authors demonstrate that reinforcement learning fails to discover effective communication protocols in partially coupled regimes, whereas engineered senders succeed at the same bit budget. The study highlights that learned protocols are often mutually unintelligible and that the failure is due to optimization/search limitations rather than representational capacity.
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Mihir Chauhan → affiliatedwith → Purdue University
confidence 99% · Mihir Chauhan & Aniket Bera Affiliation: IDEAS Lab, Department of Computer Science Affiliation: Purdue University
Aniket Bera → affiliatedwith → Purdue University
confidence 99% · Mihir Chauhan & Aniket Bera Affiliation: IDEAS Lab, Department of Computer Science Affiliation: Purdue University
Physical Coupling → governs → Communication Value
confidence 95% · Communication value is governed by coupling: under rigid coupling... under partial coupling...
Learned Protocols → are → Mutually Unintelligible
confidence 94% · Cross-play shows learned protocols are individually meaningful but mutually unintelligible
Learned Sender → underperforms → Engineered Sender
confidence 93% · the engineered 2-bit sender reaches an interquartile mean of 1.000 but the learned one reaches 0.482
Partial Coupling → exhibits → Communication Gap
confidence 92% · under partial coupling, the engineered 2-bit sender reaches an interquartile mean of 1.000 but the learned one reaches 0.482
Rigid Coupling → renders → Communication Null
confidence 90% · under rigid coupling through a shared object, no channel beats silence... since proprioception already carries that information
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Abstract
Abstract: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 for a teammate to read. We answer them for rate-limited Dec-POMDPs, then measure how far reinforcement learning falls short of the optimum. Our theorems fix what is achievable independently of any learner, so a gap between an engineered and a learned sender at the same bit budget is an optimization fact, not an information-theoretic one. We instantiate this on three MuJoCo arenas spanning zero, partial and rigid physical coupling, charging every condition exactly 2 bits per decision, and create the discriminating regime by closing a physical side channel within one arena, holding bodies, task and reward fixed. Communication value is governed by coupling: under rigid coupling through a shared object, no channel beats silence (+0.001 +/- 0.001, p = 0.982, n = 25), since proprioception already carries that information; without coupling, every condition solves the task; under partial coupling, the engineered 2-bit sender reaches an interquartile mean of 1.000 but the learned one reaches 0.482, indistinguishable from silence (p = 0.400, n = 25). With a shared alphabet, bandwidth cannot explain the gap. Warm-starting from an engineered receiver localizes the failure: the same channel reaches 0.857 versus 0.562 cold-started (p < 0.001), so it is neither representational nor one of maintenance; reinforcement learning fails to discover the protocol. Cross-play shows learned protocols are individually meaningful but mutually unintelligible: self-play 0.980 collapses to 0.144 across seeds, and our best constructed alignment leaves at least 77% of that gap. All headline results use 25 seeds per arena and seven published baselines at matched rate.
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- Source: https://arxiv.org/abs/2609.34373v1
- Canonical: https://arxiv.org/abs/2609.34373v1
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Emergence, Not Bandwidth: Physical Coupling and the Limits of Learned Multi-Agent Communication Mihir Chauhan & Aniket Bera Affiliation: IDEAS Lab, Department of Computer Science Affiliation: Purdue University Email: chauhanm,aniketbera@purdue.edu Abstract 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 22 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±0.001+0.001± 0.001, p=0.982p=0.982, n=25n=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 22-bit sender reaches an interquartile mean of 1.0001.000 while the learned 22-bit sender reaches 0.4820.482 and is indistinguishable from silence (p=0.400p=0.400, n=25n=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 0.8570.857 against 0.5620.562 cold-started (p<0.001p<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 0.9800.980 collapses to 0.1440.144 across seeds, and the best alignment we can construct leaves at least 77%77\% of that gap standing. Every headline result is reported at 2525 seeds per arena, against seven published baselines reimplemented in our trunk at matched rate. 1 Introduction Any team of robots operating under partial observability, where one agent may not observe what its partner observes, must either communicate or coordinate implicitly through the physics they share. Which of the two happens is usually treated as a property of the task. We argue that it is instead a property of physical coupling, and that coupling can be varied as an independent variable while everything else is held fixed. We use emergence throughout in one narrow, behavioral sense: the event that end-to-end reinforcement learning, starting from a random initialization, arrives at a protocol that the receiver can act on. We do not use it to imply higher-order statistical structure, and we do not infer it from an information-theoretic signature. It is measured by task success at a fixed bit budget against an engineered sender that holds the same alphabet, so the claim that emergence fails is falsifiable by a learner that closes that gap. This is important because emergent-communication literature has, almost entirely, studied only one end of that axis. The continuous-control work we found operated in multi-agent MuJoCo, which decomposes a single robot body into agents controlling disjoint joint subsets. That construction is maximally, rigidly coupled. Our own measurements say it is the exact regime in which communication is a tight null: on a rigidly coupled bi-humanoid arena, an oracle channel, an engineered symbol, and a learned communication code are all within 0.0010.001 of silence. A literature clustered there is measuring the one setting whose answer is known in advance. The regime that differentiates is the partially coupled one, where two agents that share an object but cannot both observe the target are coupled enough that their actions interact and uncoupled enough that communication helps. This is where our central result lies. The theory–experiment relationship. We use theorems to make an empirical point, following the structure of Dorner et al. (2025): establish a bound that holds independently of any particular method, then show that practice falls short of even that bound. Our theorems characterize the optimum of a rate-limited Dec-POMDP without reference to a learner. When an engineered sender attains a level at 2 bits that a learned sender does not attain with the same 2 bits, the difference cannot be attributed to channel bandwidth; it is a statement about the optimization problem reinforcement learning has to solve. Contributions: 1. Coupling as an independent variable (Section 3.2). Three continuous-control arenas: 1) uncoupled, 2) partially coupled, 3) rigidly coupled, each with shared networks, trainer, channels and rate accounting. Communication value is ordered by coupling, and the rigid regime is a null. 2. A rate-matched engineered-versus-learned comparison (Section 3.3). Every experiment’s rate — how many bits one agent can send per decision — is derived from its channel configuration, not from realized message statistics. The engineered sender and the learned 44-code channel are both 22 bits per decision, as are three of the published baselines, so the comparison is capacity-controlled. 3. The separation itself (Section 4.1). In the partially coupled arena, the engineered 22-bit sender outperforms silence by +0.441±0.129+0.441± 0.129 (p<0.001p<0.001) while the learned 22-bit sender does not outperform it at all (+0.068±0.163+0.068± 0.163, p=0.400p=0.400), and the direct matched-rate contrast between the two is +0.374±0.125+0.374± 0.125 (p<0.001p<0.001, dz=1.23d_z=1.23). 4. The failure is discovery (Section 4.3). Warm-starting a learned run, where we load the parameters from an engineered receiver, rescues it (+0.295±0.131+0.295± 0.131, p<0.001p<0.001), which rules out representational and maintenance failures and leaves the sender’s search as the bottleneck. 5. Protocol idiosyncrasy (Section 4.5). Learned protocols that each solve the uncoupled arena are mutually unintelligible, with a cross-play gap of 0.8360.836 [0.807,0.862][0.807,0.862] of which the best alignment we can construct recovers at most a quarter, bounded over all 600600 ordered seed pairs. 6. Statistical scale. 2525 seeds per arena and seven published baselines, all under fixed arena, rate and seed set, with seed-paired tests, Holm correction, one-sided bounds in place of bare nulls, and interquartile means with seed-level bootstrap intervals as recommended by Agarwal et al. (2021). 2 Related work Emergent communication. Learned protocols frequently fail to arise at all, or degenerate when they do. Kottur et al. (2017) show that agents solve referential tasks with protocols that are neither compositional nor interpretable. Lowe et al. (2019) show that sender-consistency metrics can be high while the channel has no causal effect on the receiver. Eccles et al. (2019) attribute the failure to a joint exploration problem and propose positive-signaling and positive-listening biases to rescue it. Chaabouni et al. (2019) show learned codes are inefficient without explicit length pressure. Our contribution is not the observation that emergence fails; it is the controlled measurement of what fails, at a fixed bit budget, as a function of physical coupling. The nearest neighbor, and why it does not settle the question. Hill et al. (2025) compare an engineered sender against a learned one in gridworld task allocation and find the engineered protocol far stronger. The comparison is not rate-matched, and the asymmetry in message space is severe: their learned sender uses “a binary message space 0,1\0,1\” (1 bit per step) while their engineered “intention” message is a continuous fixed-length vector produced by attention over an imagined trajectory, with an unknown dimension. The engineered side is therefore unquantified, and a gap measured across that difference is fully consistent with the learned channel simply being too narrow. Our comparison removes that explanation by construction: both senders hold exactly 44 symbols with a fixed 22 bits (Table 1). Discrete channels and bandwidth accounting. Vector quantization (van den Oord et al., 2017) is the standard discrete bottleneck, and a line of work controls communication bandwidth explicitly (Wang et al., 2020; Kapoor et al., 2025). These compare learned variants to other learned variants, but do not introduce a hand-engineered sender at a matched bit count, which is the comparison that differentiates bandwidth capacity from optimization. Vanneste et al. (2022) benchmark discretization estimators (DRU, ST-DRU, straight-through) and find some fail entirely in some environments; we include their estimators as baselines. 3 Method 3.1 Setting and theory We formulate the problem as a Dec-POMDP with two agents. One agent observes a task-relevant variable (while the other does not) and may transmit a message over a channel limited to R bits per decision. We summarize the three results that characterize the optimum. Appendix A gives the full statements, the assumptions each requires, and the exact tabular validation. Why the theory is important. Our experiments show an engineered sender beating a learned one at an identical bit budget, which by itself is consistent with two bits simply not being enough. The engineered sender rules that out by construction, being a witness that two bits suffice to reach its level, but it does not explain why the learned sender stops short. The theory supplies that reason. We state three theorems, T1, T2 and T3. T1 fixes, independently of any learner, what an optimal rate-R message must be a statistic of: the receiver’s advantage, not the sender’s observation. This is what makes a 22-bit sector code a principled reference, and it yields a prediction we can test on the emergent code itself, by tracking the receiver’s steering state and not the fixed sector label (Appendix C). T2 bounds how compression loss scales with horizon, so a shortfall cannot be attributed to accumulated per-step error under filter stability. T3 predicts that independently trained optimal protocols are mutually readable after per-context relabeling, which Section 4.5 tests directly and finds false of the protocols PPO actually produces. The theorems convert a gap between two particular senders into a falsifiable statement about the search problem; two of the three are falsified against our own learners. This structure follows Dorner et al. (2025): establish what binds any method, then show that practice does not reach it. T1: Advantage Sufficiency: The optimal message is a randomized function of the receiver-facing sufficient statistic, which is a statistic of the receiver’s advantage, not of the sender’s observation. The zero-loss rate is the entropy of that statistic and not of the state. Codes that reconstruct the state waste 47%47\% of communication value at a matched rate on our tests and need roughly twice the rate to recover it. A 22-bit engineered sender serves as a meaningful upper reference with respect to these results. T2: Horizon-free Compression Loss: Under filter stability (Dobrushin coefficient ρ<1ρ<1) the value lost to compression is Θ(ϵH) (ε H) compared to Θ(ϵH2) (ε H^2) in worst case, with a matching lower bound, confirmed experimentally with measured exponents 1.081.08 and 2.012.01. T3: Identifiability: Under a binding rate, task diversity, and a generic value-separation condition, the set of optimal protocols collapses to a single orbit of per-context relabelings, which would make decoders consistent and zero-shot interoperation possible. Exhaustive computation shows the genericity condition is load-bearing: identifiability is non-monotone in both rate and task weight and fails in symmetric configurations. We report in Section 4.5 that T3’s empirical leg fails on these arenas, and why the cause is the arenas rather than the theorem. 3.2 Arenas: coupling as an independent variable All three arenas are MuJoCo (Todorov et al., 2012) continuous-control tasks that are compiled through MJX, trained with PPO (Schulman et al., 2017) under matched networks, trainer, channel implementations and rate accounting. In each, one agent observes a hidden variable taking one of four values and the other must act on it. Comparing across the three is a between-arena design, and the arenas differ in embodiment, task and, most importantly, physical coupling, so a difference between them is not by itself attributable to coupling. We therefore also vary coupling within a single arena. On the bimanual arena, for example, the two arms hold one bar, and the bar’s pose already reveals where the partner is steering: an agent that can see the object can follow it and never needs a message. For the blind agent, we set partner_blind, which removes the bar pose, the bar velocity, the tilt and the partner’s hand from one agent’s observation while leaving its own proprioception, the clock, the bodies, the physics, the reward and the rate accounting unvaried. The flag closes a physical sensing channel without changing the task, which makes coupling a manipulated variable on the same hardware. Arena specifications are in Appendix F. Uncoupled (UAV–UGV). An aerial observer can view which of four sectors holds the goal; a ground vehicle must drive there. The agents cannot physically interact, so coupling is zero and the message is the only information path. Partially coupled (bimanual, blind). Two arms manipulate a shared object and the target is hidden from one of them. The arms interact through the object by friction alone, and contact forces do not disclose the target. Rigidly coupled (bi-humanoid, blind). Two humanoids carry an object through fixed handles, which effectively constrain each body to the other continuously, so proprioception is itself an information channel. 3.3 Rate accounting Rates are computed from each channel’s configuration and not from realized message statistics. An entropy-coded charge would award a collapsed codebook a lower apparent rate, rewarding the exact failure under study, and would not be available to the engineered sender. Table 1: Communication rate on every arena, derived from each channel’s configuration and never from realized message statistics. Every finite charge is a fixed-length cost over the alphabet the channel actually uses. The engineered sender and the learned vq_k4 channel are exactly matched, as are three published baselines. condition bits/decision engineered (scripted sector symbol) 2.000 vq_k4 (learned, ours) 2.000 cat_k4 (Gumbel straight-through) 2.000 dru_b2 / stdru_b2 2.000 vq_k2 / vq_k8 / vq_k16 1.000 / 3.000 / 4.000 silence 0.000 oracle (unquantized) excluded from the rate axis Figure 1: Communication value is set by physical coupling. Left: each condition’s success against silence, n=25n=25 seeds per condition, 95%95\% t-intervals. Right: the contrast of engineered minus learned, with both senders charged exactly 22 bits per decision. The matched-rate gap is negligible where the agents are rigidly coupled (proprioception already carries the message), small where they are uncoupled (any channel suffices), and large in between. Since the alphabets are identical by construction, the middle bar is not a bandwidth effect. 3.4 Controls and nulls A gap between two senders can be caused by a weak baseline, a broken channel or a favorable statistic, but we rule each out by our construction. • The floor is a strategy, not a failed learner. The score under silence 0.0000.000 on the uncoupled arena is a shaping equilibrium, so gaps there are reported against the strongest scripted no-channel controller (Appendix D). • Rate is matched by configuration. Both senders are allowed only 22 bits per decision, so a gap cannot be bandwidth (Table 1). • The channel is not the weak link. RIAL and ST-DRU beat our channel where coupling is zero, but where it binds, neither closes the matched-rate gap. • Physical coupling is manipulated. The discriminating regime comes from closing a physical sensing channel within one arena, holding bodies, task, reward and rate fixed. • Alignment is tested against a destroying null, which is bounded. A shuffled control recovers 0.0860.086 [0.039,0.137][0.039,0.137], so recovery above zero is not itself evidence of alignment, and every non-significant contrast carries a one-sided 95%95\% upper bound. • Summaries resist outliers. Means with t-intervals are reported beside interquartile means with seed-level bootstraps (Agarwal et al., 2021). 4 Results 4.1 Communication value by coupling regime Figure 1 shows the whole result; Table 2 reports all three arenas. Seeds are paired across conditions, so each contrast is a paired test on the same seeds; p-values are Holm-corrected within each arena. We report the mean with a 95%95\% t-interval and, following Agarwal et al. (2021), the interquartile mean with a seed-level bootstrap interval. The two disagree on the partially coupled arena because its seed distribution is bimodal rather than merely wide; Figure 2 shows every seed behind that row. Table 2: Success against silence on three arenas, n seeds per condition. IQM is the interquartile mean with a seed-level bootstrap interval. arena condition n success IQM [95% CI] vs silence p uncoupled oracle 25 1.000±0.0001.000± 0.000 1.000 [1.000, 1.000] +1.000±0.000+1.000± 0.000 — engineered 25 0.999±0.0010.999± 0.001 1.000 [1.000, 1.000] +0.999±0.001+0.999± 0.001 <0.001<0.001 vq_k4 25 0.983±0.0100.983± 0.010 0.992 [0.977, 0.999] +0.983±0.010+0.983± 0.010 <0.001<0.001 silence 25 0.000±0.0000.000± 0.000 0.000 [0.000, 0.000] reference partial oracle 25 0.984±0.0320.984± 0.032 1.000 [1.000, 1.000] +0.490±0.124+0.490± 0.124 <0.001<0.001 engineered 25 0.936±0.0760.936± 0.076 1.000 [1.000, 1.000] +0.441±0.129+0.441± 0.129 <0.001<0.001 vq_k4 25 0.562±0.1270.562± 0.127 0.482 [0.321, 0.719] +0.068±0.163+0.068± 0.163 0.4000.400 silence 25 0.494±0.1240.494± 0.124 0.368 [0.297, 0.583] reference rigid oracle 25 1.000±0.0001.000± 0.000 1.000 [1.000, 1.000] +0.001±0.001+0.001± 0.001 0.982 engineered 25 1.000±0.0001.000± 0.000 1.000 [1.000, 1.000] +0.001±0.001+0.001± 0.001 0.982 vq_k4 25 1.000±0.0001.000± 0.000 1.000 [1.000, 1.000] +0.001±0.001+0.001± 0.001 0.982 silence 25 0.999±0.0010.999± 0.001 1.000 [1.000, 1.000] reference Coupling orders communication value. In the rigidly coupled arena every channel is within 0.0010.001 of silence and nothing is significant; the handles already carry the information. In the uncoupled arena every channel solves the task, and against the scripted floor of 0.242±0.0370.242± 0.037 the learned sender gains +0.740±0.038+0.740± 0.038. Communication value is therefore not a property of the task alone; it is set by how much the physics already couples the agents. The matched-rate contrast. We highlight the comparison between two senders at the same rate. Paired on the 2525 seeds both conditions ran, the engineered sender exceeds the learned one by +0.374±0.125+0.374± 0.125 (p<0.001p<0.001, Cohen’s dz=1.23d_z=1.23) in the partially coupled arena. The same contrast is +0.017±0.010+0.017± 0.010 (p=0.002p=0.002, dz=0.68d_z=0.68) where the agents are uncoupled, and exactly +0.000±0.000+0.000± 0.000 where they are rigidly coupled. These three tests ask one question of three arenas, so the two with defined p-values are Holm-corrected as a family; the third is exactly zero with zero variance and admits no test. Correction leaves both significant and the matched-rate gap is therefore not a fixed property of our engineered protocol. It is zero under rigid coupling, small under no coupling, and large in between. What the null excludes. The one-sided 95%95\% upper bound on the learned channel’s advantage over silence is +0.202+0.202. The engineered sender’s measured advantage on the same seeds is +0.441+0.441. The learned sender’s entire plausible range therefore lies below the engineered sender’s point estimate, and hence, a bound on the effect. The separation. The partially coupled arena is where the two senders come apart at equal bandwidth. The engineered 22-bit sender clears silence by +0.441±0.129+0.441± 0.129; the learned 22-bit sender, holding the same alphabet under the same charge, does not clear it (p=0.400p=0.400). The interquartile means sharpen this, as the engineered seed is at ceiling (1.0001.000), while the typical learned seed sits at 0.4820.482 against silence’s 0.3680.368. The seed distribution here is bimodal, which is why the IQM is the more faithful summary: the engineered mean of 0.9360.936 is depressed by a few failing seeds, while the modal engineered seed solves the task outright. 4.2 Published learners at the same rate Table 3 places seven baseline communication learners on the uncoupled arena under our trunk, arena, rate and seeds, differing only where their respective papers differ. All clear silence decisively. Two estimators, RIAL and ST-DRU, outperformed our channel by a small but significant margin, while IB-VQ, Eccles biases and AE-grounding fall below it. Our channel is therefore not the strongest discrete estimator available, and the claim does not depend on it being so. This arena instead shows that a rate-matched discrete channel of any of these kinds succeeds where coupling is zero. Table 3: Communication learners on the uncoupled arena. Same trunk, arena, rate and seeds; paired on seeds, Holm-corrected across the table. This is a positive control, not a ranking: all seven clear silence decisively and the whole field sits within 0.070.07 at the top of a saturated scale, which establishes that a rate-matched discrete channel of any of these kinds succeeds where coupling is zero. The two that beat our channel here are carried forward below to the arena that discriminates. method source n success vs ours (p) DIAL + VQ (ours) Foerster et al. (2016) 25 0.983±0.0100.983± 0.010 — RIAL Foerster et al. (2016) 25 0.998±0.0030.998± 0.003 +0.016±0.011+0.016± 0.011 (0.024) ST-DRU Vanneste et al. (2022) 25 0.998±0.0020.998± 0.002 +0.016±0.009+0.016± 0.009 (0.010) DRU Foerster et al. (2016) 25 0.991±0.0050.991± 0.005 +0.009±0.011+0.009± 0.011 (0.105) Eccles biases Eccles et al. (2019) 25 0.931±0.0410.931± 0.041 −0.052±0.040-0.052± 0.040 (0.041) AE-grounding Lin et al. (2021) 25 0.931±0.0430.931± 0.043 −0.052±0.045-0.052± 0.045 (0.050) IB-VQ Farooq and Iqbal (2026) 25 0.930±0.0360.930± 0.036 −0.052±0.033-0.052± 0.033 (0.015) The separation is not specific to our channel. We suspected that our VQ channel may simply be a weak estimator, so we test the two that outperformed ours on the uncoupled arena, with the same 22 bits. The two behave differently: RIAL reaches 0.691±0.1290.691± 0.129 (n=25n=25) and clears silence by +0.196±0.150+0.196± 0.150 (p=0.013p=0.013), making it the better estimator here, where ours does not clear silence at all. ST-DRU reaches 0.486±0.1170.486± 0.117 (n=25n=25) and does not clear it (−0.008±0.138-0.008± 0.138, p=0.904p=0.904). However, neither closes the matched-rate gap. The engineered sender exceeds RIAL by +0.245±0.121+0.245± 0.121 and ST-DRU by +0.449±0.142+0.449± 0.142, both p<0.001p<0.001 after Holm correction. A stronger estimator therefore recovers part of the channel’s value and still leaves most of the gap standing, which is what we expect if the obstacle is search rather than how the channel is written down. 4.3 Discovery, not representation or maintenance The separation in Section 4.1 shows that a learned sender does not reach what an engineered sender reaches at the same rate. However, this does not say which part of the problem defeats it. Three failures are consistent with the evidence so far: the channel may be unable to represent a useful protocol, the optimizer may be unable to discover one, or it may discover one and fail to maintain it. We separate them by initializing a learned run from an engineered run’s final parameters. The engineered run spends its whole budget training a receiver to act on a 22-bit sector code and never trains a message head, because its symbol is scripted. A warm-started run therefore begins with a receiver that already listens and a sender that knows nothing. It asks the sender-side question directly: given a competent listener, can PPO find a speaker? Parameter trees are identical by construction and the codebook is always reinitialized, so nothing about the channel itself is inherited. The transfer is not confined to the receiver, however: it carries both agents’ motor policies and the critic, and its 1515M steps come on top of the engineered run’s 1515M. The contrast below therefore bounds listener, motor initialization and budget together. The answer is that it can on most seeds, and we see this on the paired 2525 seeds, where it improves on 16, is worse on 2 and ties on 7. The warm-started learned channel reaches 0.857±0.0960.857± 0.096 against the cold-started 0.562±0.1270.562± 0.127, a difference of +0.295±0.131+0.295± 0.131 (p<0.001p<0.001, dz=0.93d_z=0.93); Figure 4 joins the two conditions seed by seed, and the gain is spread across a majority of seeds rather than produced by a handful. It clears silence by +0.362±0.134+0.362± 0.134 (p<0.001p<0.001, dz=1.11d_z=1.11), where the cold-started run does not clear silence at all; it sits at 0.5620.562 against silence’s 0.494±0.1240.494± 0.124. The engineered sender, at 0.936±0.0760.936± 0.076, exceeds it by +0.079±0.092+0.079± 0.092 (p=0.090p=0.090, dz=0.35d_z=0.35). This is not an equivalence, since a non-significant difference is weak evidence either way, but the one-sided 95%95\% bound puts the engineered advantage over the warm-started run at no more than +0.155+0.155, against +0.374+0.374 over the cold-started one. Most of the gap closes when the sender is given a listener. From this it follows that: (1) the channel can represent a protocol that solves this task at two bits, so the failure is not representational, and (2) the protocol also survives continued training rather than decaying, so it is not a maintenance failure. Neither depends on the confounded inherited motor policies or the doubled step budget, and what end-to-end reinforcement learning cannot do here is find the protocol, which we show by elimination from (1) and (2), not from the +0.295+0.295 contrast, whose magnitude those two confounds inflate. This is the joint-exploration account of Eccles et al. (2019), measured here at a fixed bit budget against an engineered upper reference rather than inferred from a rescue method. This condition now runs the same 2525 seeds as every other, so its cold, engineered, and silent references are the main table’s rows rather than a subset. The discovery contrast softened as seeds accumulated and then held — +0.372+0.372 (dz=1.18d_z=1.18) at n=12n=12, +0.272+0.272 (dz=0.83d_z=0.83) at n=16n=16, +0.295±0.131+0.295± 0.131 (dz=0.93d_z=0.93) at n=25n=25 — the ordinary behavior of an estimate early in a sweep, and the reason we reported it throughout at the n it had reached rather than the n that flattered it. 4.4 What the learned code carries On the uncoupled arena the learned protocol succeeds, so we can understand what messages mean. Rolling out trained checkpoints under greedy actions and comparing the transmitted code against candidate meanings, over 2525 seeds and 4848 episodes each, the code carries the goal sector (lift +0.176+0.176 [+0.141,+0.213][+0.141,+0.213], p<0.001p<0.001) and the required steering bearing (lift +0.167+0.167 [+0.137,+0.197][+0.137,+0.197], p<0.001p<0.001) in roughly equal measure, with smaller but significant information about range (+0.021+0.021) and elapsed time (+0.011+0.011), and nothing about whether the vehicle is already aligned (+0.003+0.003, n.s.). The emergent code is therefore neither a pure sector name nor a pure control signal but a mixture of both. This is consistent with T1, under which the optimal message is a statistic of the receiver’s advantage rather than of the sender’s observation. 4.5 Cross-play, and the limits of the identifiability test T3 predicts that optimal protocols form a single orbit of per-context relabelings, which would make protocols from independent seeds mutually readable after alignment. We tested this on the uncoupled arena across 2525 seeds, evaluating every one of the 600600 ordered pairs of distinct seeds on held-out episodes, with seed-level cluster-bootstrap intervals. The protocols are individually excellent and jointly unreadable. Self-play success is 0.9800.980 [0.967,0.991][0.967,0.991], while cross-play with no alignment is 0.1440.144 [0.123,0.171][0.123,0.171], a gap of 0.8360.836 [0.807,0.862][0.807,0.862]. The question is whether any relabeling recovers that gap, and none comes close. A global relabeling recovers only 0.1730.173 [0.126,0.226][0.126,0.226] of it, and the per-context G G alignment recovers 0.1470.147 [0.102,0.196][0.102,0.196]. These intervals exclude zero at this sample size, so we state the result as a bound rather than a null: with 95%95\% confidence the best alignment we can construct leaves at least 77%77\% of the cross-play gap standing. Figure 3 draws every ordered pair, one panel per alignment map: a bright diagonal against a dark field in all of them. A bound is the right form for this claim because a shuffled control, which should destroy alignment structure entirely, itself recovers 0.0860.086 [0.039,0.137][0.039,0.137]. Recovery above zero is therefore not evidence of alignment. Some of it is an artifact of relabeling into a better-occupied part of the codebook, and the shuffled interval overlaps both real alignments from below. What separates the conditions is small, and it does not point the way T3 predicts. Why T3’s empirical leg fails. The direct test is G G minus global: −0.022-0.022 [−0.043,−0.001][-0.043,-0.001], P(≤0)=0.983P(≤ 0)=0.983. The per-context alignment is significantly worse than the global one. At n=8n=8 this same contrast was +0.005+0.005 [−0.045,0.047][-0.045,0.047] and we reported it as underdetermined; at n=25n=25 it resolves against the prediction. The reason for this is occupancy; the four context cells hold [1270.5,484.5,8.8,1.6][1270.5,484.5,8.8,1.6] time-paired steps on average because episodes terminate on success and almost nothing survives into the later cells. G G is granted one permutation per cell, but two of those four permutations are estimated from ∼9 9 and ∼2 2 steps respectively, leaving 2.12.1 distinct per-cell permutations per pair. These extra degrees of freedom are hence fitted to noise, which costs variance and buys no signal — exactly the sign and the size of the deficit we measure. We tried a second context partition and obtained the same degeneracy. The result is therefore not evidence against T3 as a theorem, which we verify exactly at the tabular level; it is evidence that these arenas cannot test it, and with enough power to say that the per-context refinement actively hurts rather than merely failing to help. Individually meaningful, collectively unreadable. A decoder from code to sector fitted within a single seed reads 0.5350.535 against a majority-class rate of 0.4450.445. Transferred across seeds, the same decoder reads 0.2710.271 unaligned and 0.4160.416 after alignment, which is at or below chance. Each seed’s code genuinely carries the sector, and no other seed can read it. This is a stronger statement than protocol sub-optimality: the learned solutions are idiosyncratic in a way that no relabeling we can construct repairs. 5 Discussion and limitations Three findings organize the picture: coupling sets what a channel is worth, the matched-rate shortfall belongs to the sender’s search rather than to the channel, and the protocols that do emerge are mutually unreadable. We state the scope of each plainly. • One optimizer family. Every result uses PPO at 1515M environment steps per seed. Within that family we do vary credit assignment: MAPPO’s centralized critic also fails to close the matched-rate gap (Appendix E, n=11n=11). No non-PPO learner, optimizer setting or longer budget was tested, and that remains the most important open item. • T3’s empirical leg fails on these arenas. The per-context alignment is significantly worse than the global one, and the cause we identify — degenerate context occupancy, two of four cells nearly empty — is consistent with the measurement but not proved by it. Testing T3 properly needs an arena with balanced occupancy, which we have not built. • Our channel is not the strongest estimator. RIAL and ST-DRU significantly outperform our VQ channel on the uncoupled arena. We report this rather than promote the strongest condition to “ours”. • The open-channel condition is unmeasured. The side channel is closed by a flag, so the arena can be run with it open; currently the manipulation is a design control, not a measured contrast. • Three arenas from one family. All three are MJX continuous-control tasks. They dissociate along coupling, the axis of interest, but do not span benchmark suites. Readings below 1515M environment steps are excluded throughout: the codebook sits at perplexity 1.001.00 until roughly 22M steps, and a channel that looks like flat failure at 0.380.38 reaches 0.980.98 by 1313M. 6 Conclusion Whether a team benefits from an explicit channel is decided by how strongly its members are already coupled through physics, and the field has worked mostly where the answer is “not at all”. Where it discriminates, two senders holding the same two bits diverge sharply: the obstacle is not capacity but what reinforcement learning can find. Matching the rate is what licenses that reading. The engineered sender is charged the same two bits and still reaches the task, so the budget demonstrably suffices and the shortfall cannot be a bandwidth effect. Warm starting narrows it further, to discovery rather than representation or maintenance, while the theorems fix the reference point independently of any learner, which leaves the residue as a fact about optimization. Work in this area should therefore report where it sits on the coupling axis: a result obtained where the message is the only information path need not survive where the bodies already talk. LLM Disclosure We used a large language model for: research execution and writing assistance. An LLM-based coding assistant was used to aid in debugging and implementing parts of the experimental infrastructure, such as the job supervisor, parallelization of experimentation, and to aid in designing the controls, such as working out that a rate charged from realized message statistics would reward codebook collapse, and that the cross-play test must align per context rather than globally. The research question, arena design, theoretical results, and every claim are the authors’ own. For writing assistance, an LLM was used to refine the manuscript: assist with mathematical proofs in Appendix B, improving transitions, and catching grammatical errors. 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Appendix A Theorem statements and exact validation The assumption labels below are the registry the three theorems share. assumption content A1 finite spaces State, action, observation and message spaces are finite. A2 filter stability ρ=supa(1−(Ta))(2−(Za))<1ρ= _a(1- d(T_a))(2- d(Z_a))<1. Used by T2’s upper bound; its lower bound constructs families with ρ→1ρ→ 1. A3 binding rate R<R0:=infR′:δ⋆(R′)=0R<R_0:= \R :δ (R )=0\: the budget is strictly below what reaching the envelope needs, so the constraint is active at every optimum. A4 κ-diversity The task family E is κ-diverse at rate R for some κ>0κ>0. A5 bounded rewards ‖r‖∞≤rmax<∞\|r\|_∞≤ r_ <∞, with H<∞H<∞ whenever γ=1γ=1. A6 common design The protocol, environment and prior are common knowledge, and common randomness is available to stochastic encoders. A7 reachability Every history in the stated spaces has positive probability under some feasible pair, so the conditional laws are defined everywhere. A8 channel synchrony The channel is memoryless and per-step, and delivers within the step. A9 reward genericity Rewards avoid the measure-zero tie sets on which distinct semantic partitions achieve equal value. A9′ is the strengthened no-mixing variant, assumed rather than derived. A10 finite alphabet, finite horizon H<∞H<∞ and the channel is a finite alphabet over noiseless broadcast with |i|≤2R|M^i|≤ 2^R — the setting of every experiment reported here. T1.B (advantage sufficiency) Fix an optimal (π⋆,μ⋆)(π ,μ ) at rate R. Define the receiver-facing statistic Ψti:=σ(κti,u~ti) ^i_t:=σ(κ^i_t, u^i_t), where κtiκ^i_t is the receiver-posterior map and u~ti u^i_t is the response-profile map built from the continuation payoff net of downstream information costs at the dual optimizer. Then for every dual optimizer λ⋆λ and every (i,t)(i,t) with λi,t⋆>0λ _i,t>0, I(Hti;Mti∣Ψti,U)=0.I (H^i_t\,;\,M^i_t ^i_t,\,U )=0. The optimal message is hence a randomized function of a statistic of the receiver’s advantage, not of the sender’s observation, and the zero-loss rate is the entropy of that statistic not the state. Corollary T1.C makes the state reconstruction strictly suboptimal: in the tabular family used for this check, the Hamming-optimal (state-reconstructing) code achieves value 0.90640.9064 against an optimum of 1.01.0, so δ=0.0936δ=0.0936 of a total communication value of 0.200.20 — 47%47\% wasted at a matched rate. This percentage, we show, is a property of that family’s parameters, not a universal constant. T2 (horizon-free compression loss) With ρ=supa(1−(Ta))(2−(Za))ρ= _a(1- d(T_a))(2- d(Z_a)) the composite Dobrushin coefficient, and under A1, A2, A5–A8 with broadcast topology: if ρ<1ρ<1, the total loss is uniformly bounded in H (including H=∞H=∞) by δstep/((1−γ)(1−γρ)) _step/((1-γ)(1-γρ)), giving Θ(ϵH) (ε H) rather than the Θ(ϵH2) (ε H^2) obtained by chaining per-step errors naively. A matching lower-bound family shows that the quadratic rate is attained in the worst case, so the dichotomy is tight rather than an artifact of the proof. The exponent is fitted numerically on tabular families. In the healing family an absorbing failure flag instead repairs with probability λ per step, so ρ=2(1−λ)ρ=2(1-λ) sweeps across 11 as λ varies, and the fitted exponent moves with it (the flag rows are the same construction at λ=0λ=0, where the flag never repairs): family ρ exponent healing 0.200 1.079 healing 0.500 1.097 healing 1.000 1.152 healing 1.600 1.356 healing 1.900 1.732 healing 1.980 1.946 family ρ exponent chain 0.088 1.071 chain 0.875 1.082 chain 1.137 1.086 chain 1.662 1.096 flag 2.000 2.013 The stable regime sits at ≈1.08≈ 1.08 and the worst-case family reaches 2.0132.013. Note that ρ>1ρ>1 is necessary but not sufficient for the quadratic regime — chain, a tracking task in which a target teleports to a uniform cell with probability λ and the sender’s view of it is erased with probability η, has ρ above 11 at some settings and stays near 1.081.08 — which is what a one-directional bound predicts. T3.1 (identifiability). Assume A1, A5, A6, A7, A10 (model (a), H<∞H<∞), a task family E, A3 (binding rate R<R0R<R_0), A4 (κ-diversity), and A9/A9′ (generic value separation and strict tie-breaking). Then after trimming, Opt¯(R)=G^⋅μ⋆ Opt(R)= G·μ : a single orbit of the common-information-adapted relabeling group G G. All elements share one semantic protocol, and the identifiability margin η is strictly positive. The distinction between G G and the constant relabeling group G is load-bearing and is why we align per context in Section 4.5: with G the statement is false in general sequential problems. At H=1H=1 the two coincide, which is why referential-game prior work does not see the difference. Because T3.1 is a claim about the argmax set of an optimization problem, it can be computed exactly on small families rather than estimated — no seed noise, and no dependence on whether RL finds the optimum. Doing so shows A9 is not removable: identifiability is non-monotone in the rate. K bits R<R0R<R_0? V⋆V #orbits identifiable 2 1.000 yes 0.9500 1 yes 3 1.585 yes 0.9750 2 no 4 2.000 no (=R0=R_0) 1.0000 1 yes At K=3K=3 the rate still binds (1.585<R0=21.585<R_0=2) yet Opt(R)Opt(R) splits into two orbits. A binding rate constraint is therefore not by itself sufficient for identifiability: the failure is an A9 violation, and identifiability fails exactly at symmetric configurations. The checker also verifies orbit-invariance (every relabeling of a protocol shares its value) for every protocol enumerated, and reproduces mechanically each value computed by hand for the parity family of Appendix B.3. Appendix B Proof overviews The three results are proven in full in the supplementary material, with assistance from an LLM, as with the proofs in this section. This appendix gives the argument of each at the level of its important steps and every overview names the lemmas it compresses; the statements are checked exactly by enumeration on small tabular families (Appendix A). B.1 T1: optimal messages are sufficient statistics for receiver advantage Two preliminary repairs to the rate-limited model make the optimization well posed. First, the rate functional must condition on the common randomness U: without that conditioning the achievable region fails to be convex, and time sharing between two feasible protocols can appear to violate the budget. Second, with the conditioned functional, the region is convex and compact, so an optimum is attained and strong duality holds. The consequence we use throughout is an exact Lagrangian bridge: there is a multiplier vector λ⋆λ with zero duality gap, so a maximizer of the unconstrained Lλ⋆L_λ that is also feasible is a constrained optimum. The variational (Donsker–Varadhan) form of mutual information turns the rate term into a sum of per-history stage costs D(μ(⋅∣h)∥q⋆)D(μ(· h)\,\|\,q ) against a fixed reference measure, which is what lets the information charge be treated as part of the stage reward rather than as a global constraint. T1.A (belief sufficiency). Four steps. (1) Lagrangianization: replace the constrained problem by Lλ⋆L_λ , in which the rate appears as a per-history cost. (2) Coordinator reduction: under broadcast the common information is shared, so the common-information approach of Nayyar et al. (2013) applies and an optimal solution exists in which each agent’s stage kernel is a function of the common belief and its own private information. (3) Belief quotient: two private histories inducing the same private belief are bisimilar for the Lagrangianized problem — they generate the same distribution over futures and the same stage costs — so the optimal kernel factors through the belief. (4) Assembly: the maximizer constructed in (1)–(3) is feasible, so by zero gap it is a constrained optimum. Broadcast is where this is fragile, and the fragility is specific. Step (2) needs every agent to see the same public record; with point-to-point channels the common information shrinks, the private belief no longer determines the common belief, and the statement must carry the pair rather than the private belief alone. The quotient in step (3) still goes through formally, so the failure is in the reduction, not the bisimulation. T1.B (advantage sufficiency). Fix an optimum (π⋆,μ⋆)(π ,μ ) and a stage (i,t)(i,t) whose multiplier is strictly positive. Define Ψti:=σ(κti,u~ti) ^i_t:=σ(κ^i_t, u^i_t) from the receiver-posterior map and the response-profile map, the latter built from the continuation payoff net of downstream information costs at the dual optimizer. Suppose, for contradiction, that I(Hti;Mti∣Ψti,U)>0I(H^i_t;M^i_t ^i_t,U)>0. Now garble: replace the stage kernel by μ′(⋅∣h):=ℙ(Mti∈⋅∣Ψti=Ψti(h))μ (· h):=P(M^i_t∈· ^i_t= ^i_t(h)), drawing the message from its conditional law given the atom of the statistic and nothing else. Two facts make this an improvement. The receiver-facing joint law of (Htj,u~ti,Mti)(H^j_t, u^i_t,M^i_t) is exactly preserved, because on an atom of Ψti ^i_t the posterior map is constant, which gives Htj⟂Hti|ΨtiH^j_t H^i_t ^i_t; every value term of the decomposition is therefore unchanged, not merely bounded. And the information cost weakly drops, since the chain rule gives I(Hti,Mti)=I(Ψti,Mti)+I(Hti;Mti∣Ψti)I(H^i_t;M^i_t)=I( ^i_t;M^i_t)+I(H^i_t;M^i_t ^i_t), with the assumed strict positivity of the last term making the drop strict. So (π⋆,μ′)(π ,μ ) has the same value at strictly lower Lagrangian cost, contradicting optimality. Hence the conditional information vanishes: the optimal message is a randomized function of the receiver’s advantage statistic, and the zero-loss rate is the entropy of that statistic rather than of the state. This is self-contained because the information costs are carried inside the stage payoff by the variational representation, so preserving the receiver-facing law preserves them in expectation, which is all the Lagrangian charges. This is why no assumption about the continuation being already optimal is needed, and so why nothing is assumed that the conclusion supplies. T1.C (state reconstruction is strictly suboptimal). A code that is optimal for reconstructing the sender’s observation spends rate on distinctions the receiver’s advantage statistic does not separate. On the tabular family used for the exact check this is 47%47\% of the available communication value at matched rate. The percentage is a property of that family; T1.C asserts only the direction, and the direction follows from T1.B because any state-reconstructing code with I(Hti;Mti∣Ψti)>0I(H^i_t;M^i_t ^i_t)>0 is strictly improvable by the same garble. B.2 T2: a horizon-free bound when the filter contracts, and a quadratic one when it does not The object being bounded is the gap between the omniscient belief βt _t, which conditions on everything, and the deployed common belief β^t β_t reconstructed from what the rate-limited channel actually carried. Write et:=‖βt−β^t‖TVe_t:=E\| _t- β_t\|_TV and let εt _t be the per-step message distortion injected at t, with ε¯:=suptεt := _t _t. The recursion. The proof rests on a controlled one-step filter contraction. Inserting the truth-updated deployed belief and applying the triangle inequality splits the step-t error into a propagated part and a freshly injected part: ‖βt−β^t‖≤‖F(βt−1,At−1,Ot)−F(β^t−1,At−1,Ot)‖+‖F(β^t−1,At−1,Ot)−β^t‖.\| _t- β_t\|≤ \|F( _t-1,A_t-1,O_t)-F( β_t-1,A_t-1,O_t) \|+ \|F( β_t-1,A_t-1,O_t)- β_t \|. Conditioning on t−1:=σ(O1:t−1,U)G_t-1:=σ(O_1:t-1,U), the observation OtO_t is drawn from the βt−1 _t-1-side predictive, so the contraction lemma bounds the first term in conditional expectation by ρ‖βt−1−β^t−1‖ρ\| _t-1- β_t-1\|, where ρ=supa(1−(Ta))(2−(Za))ρ= _a(1- d(T_a))(2- d(Z_a)) is the composite Dobrushin coefficient of transition and observation kernels. The second term is εt _t by definition. Hence et≤ρet−1+εte_t≤ρ\,e_t-1+ _t with e1≤ε1e_1≤ _1, and unrolling gives et≤∑k≤tρt−kεk≤ε¯min(t,1/(1−ρ))e_t≤ _k≤ tρ^\,t-k _k≤ \! (t,1/(1-ρ) ). The dichotomy in the theorem is already visible here: the minimum is 1/(1−ρ)1/(1-ρ) when ρ<1ρ<1 and t when it is not. From belief error to value loss. A performance-difference argument converts ete_t into regret, with the per-step sensitivity κt _t supplied by a Lipschitz and greedy-mismatch bound. In the discounted case κt≤rmax/(1−γ) _t≤ r_ /(1-γ) and summing the geometric double sum gives J⋆−J≤2κ1−γρ∑kγk−1εkJ -J≤ 2κ1-γρ _kγ^k-1 _k, which is bounded uniformly in H — including H=∞H=∞. In the undiscounted finite-horizon case κt=(H−t+1)rmax _t=(H-t+1)r_ , and the same sum yields rmaxε¯H(H+1)/(1−ρ)r_ H(H+1)/(1-ρ): the extra factor of H comes from the sensitivity, not from the belief error, which is why a bounded-advantage condition removes it and restores a linear rate. Tightness. The quadratic rate is attained, not merely permitted by the bound. A flag family in which a rate deficit prevents the deployed belief from ever re-synchronizing gives a matching Ω(min(εH2,H)) ( ( H^2,H)) lower bound, so the dichotomy is a property of the problem rather than slack in the analysis. The fitted exponents in Appendix A are the numerical counterpart: the stable regime sits near 1.081.08 and the flag family reaches 2.0132.013. One caveat the numbers make visible is that ρ>1ρ>1 is necessary but not sufficient for the quadratic regime — the chain family exceeds 11 at some settings and stays near 1.081.08 — which is exactly what a one-directional bound predicts, and is a place where a reader could otherwise suspect the fit rather than the theory. Two honest limitations. The decoder must not rule out observations the true system can produce; this is a support condition, and η-smoothing the deployed belief enforces it at a cost of 2η2η in the distortion, so it is a technical rather than a substantive hypothesis. And the claim that the deployed pair is admissible-and-feasible is automatic in the models we use but is a hypothesis in general. B.3 T3: optimal protocols form a single orbit, under the right group T3.1 is a statement about the argmax set of an optimization problem, which is why it can be verified exactly on small families rather than estimated: there is no seed noise and no dependence on whether a learner finds the optimum. The argument. Five lemmas compose. Blockwise convexity and purification give a deterministic optimum, and strict tie-breaking (A9′) upgrades this to: every optimum is deterministic on reachable histories. Trimming then leaves finitely many deterministic protocols, hence finitely many semantic protocols σ(μ)σ(μ) — the decision-relevant partition each protocol induces. Value factors through σ, so the problem reduces to a finite maximization over semantic protocols; generic value separation (A9) makes the maximizer σ⋆σ unique. Finally, two deterministic trimmed protocols share a semantic protocol exactly when they lie in one orbit of the common-information-adapted relabeling group G G. Combining: every optimum has semantics σ⋆σ and therefore lies in G^⋅μ⋆ G·μ , and the whole orbit is optimal by G G-invariance. The margin η>0η>0 follows from finitely many semantic values with a unique maximum; against cell-merging deviations, κ-diversity (A4) gives the stronger bound η≥κη≥κ directly. Why the group must be G G. This is the load-bearing distinction and the reason the paper aligns per context in Section 4.5. With the constant relabeling group G the theorem is false in general sequential problems: the optimum set is then a disjoint union of many G-orbits, because a protocol may permute its labels differently at different common-information states without changing anything a receiver can act on. At H=1H=1 the two groups coincide, which is why work on referential games does not encounter the difference — and why a cross-play experiment that aligns with a single global permutation is testing a weaker statement than the one T3 makes. Necessity, and what the exact sweep adds. A3 (a binding rate) and A4 are close to necessary rather than convenient: if the rate does not bind, distinct protocols achieve the optimum and identifiability fails; if κ-diversity fails, the margin collapses. A9 is the assumption a reader is most likely to suspect of being unfalsifiable, so the enumeration in Appendix A is aimed at it: at K=3K=3 the rate still binds (1.585<R0=21.585<R_0=2) and yet the optimum set splits into two orbits. A binding rate is therefore not sufficient for identifiability, the failure occurs precisely at symmetric configurations where A9 is violated, and identifiability is non-monotone in the rate. The same enumeration checks orbit-invariance for every protocol it encounters and reproduces the hand-computed values of the parity family, which is the construction underlying the interpretability tax lower bound. Appendix C What the emergent code carries Trained checkpoints on the uncoupled arena are rolled out under greedy actions and the transmitted code is compared against candidate meanings, over 2525 seeds and 4848 episodes each. Lift is the accuracy of the best code-to-candidate lookup minus the majority-class rate; intervals are bootstrap over seeds and p-values are Holm-corrected across candidates. NMI is the mutual information between the code and the candidate, normalized by the smaller of their entropies; because that denominator depends on the candidate’s own cardinality, NMI is not comparable across rows, and the claims rest on lift. Table 4: The code carries the goal sector and the required steering bearing in roughly equal measure. aligned — whether the vehicle is already pointed correctly — is the one candidate it does not track. candidate meaning NMI lift 95% CI p (Holm) sector 0.232 +0.176+0.176 [+0.141,+0.213][+0.141,+0.213] <0.001<0.001 bearing (4 buckets) 0.298 +0.167+0.167 [+0.137,+0.197][+0.137,+0.197] <0.001<0.001 turn sign 0.070 +0.058+0.058 [+0.040,+0.077][+0.040,+0.077] <0.001<0.001 range (4 buckets) 0.029 +0.021+0.021 [+0.011,+0.033][+0.011,+0.033] 0.002 elapsed time (4 buckets) 0.035 +0.011+0.011 [+0.006,+0.017][+0.006,+0.017] 0.002 aligned 0.218 +0.003+0.003 [+0.000,+0.010][+0.000,+0.010] 0.322 Appendix D The scripted floor on the uncoupled arena The silent condition scores 0.0000.000, which is a shaping equilibrium rather than an information bound: the policy learns to stand still and returns exactly −0.640=160×0.004-0.640=160× 0.004, the horizon times the step cost. The honest floor is what a strategy achieves with no channel, so we script one. It drives to a sector center and, because the in-sector jitter (0.452=0.6360.45 2=0.636 m) is smaller than the sensing radius (0.70.7 m), arriving reveals the goal whenever the guess was right. Over 512512 episodes it reaches 0.242±0.0370.242± 0.037. Time caps it: the horizon buys 160×5×0.005=4.0160× 5× 0.005=4.0 s at 1.61.6 m/s, so 6.46.4 m of travel, while the ring hop between adjacent sectors is 1.92=2.6871.9 2=2.687 m. A second sector is reached in 0%0\% of episodes, which is a real property of the arena and the reason it discriminates at all. All gaps on this arena are reported against 0.2420.242 instead of silence. Figure 2: Per-seed evaluation success on the partially coupled arena, one point per seed, with the mean (bar) and interquartile mean (triangle) marked. The right panel histograms the learned channel alone. The paper reports an interquartile mean beside the mean on this arena because the distribution is bimodal; this is the evidence for that, and it also shows why the learned channel and silence are hard to separate: they have the same two-cluster shape, and the learned channel merely moves a few more seeds into the upper cluster. Values are those of Table 2. Figure 3: Cross-play success for every ordered pair of seeds on the uncoupled arena, one panel per alignment map. The bright diagonal in the leftmost panel is self-play; the dark field is every other pairing. Diagonal cells are gray in the alignment panels because a map from a seed to itself is not a pairing that was run. Panel labels give the cross-play mean and the share of the self-play gap the map recovers, both read from the same data as Section 4.5. Figure 4: Warm- against cold-started runs on the partially coupled arena, joined per seed, with the engineered and silent references drawn as horizontal lines. The steep bundle is the discovery effect: seeds that sit near the silent floor when cold-started reach the engineered level when handed a competent receiver. Seeds that do not improve are visible rather than absorbed into a mean. Values are those of Section 4.3. Appendix E Does the failure survive a different learner? Every headline number comes from IPPO at one configuration, so “learned communication fails here” could be read as “this optimizer fails here”. We re-run the partially coupled arena under MAPPO, a centralized critic, which is a genuinely different credit-assignment scheme rather than a retuning of the same one, with its own engineered and silent references, so the contrast is internally paired. critic n learned engineered −- learned IPPO (main text) 25 0.562±0.1270.562± 0.127 +0.374±0.125+0.374± 0.125 MAPPO (centralized critic) 11 0.531±0.1910.531± 0.191 +0.447±0.206+0.447± 0.206 The centralized critic does not close the matched-rate gap (+0.447±0.206+0.447± 0.206, p=<0.001p=<0.001), and its learned condition lands inside the IPPO interval. Sharing a critic changes how credit is assigned, not whether the sender’s search finds a protocol, which is what the warm-start result already implied. This comparison is reported at the n it reached. Appendix F Arena specifications Every quantity below is read from the environment configuration that the runs use. All three arenas integrate at 55 ms with five physics substeps per control step, giving a 2525 ms control period (4040 Hz); actions are continuous and clipped to [−1,1][-1,1]. The hidden variable is one of four values in each arena, drawn uniformly at reset, and is visible to agent 00 (the sender) only. Observation widths include the 88-slot message input. The partner_blind flag zeroes the listed coordinates of the receiver’s observation rather than removing them, so widths and network shapes do not change when the flag is set. uncoupled partially coupled rigidly coupled (UAV–UGV) (bimanual, blind) (bi-humanoid, blind) horizon (control steps) 160 120 150 observation dim, sender 23 29 30 observation dim, receiver 19 25 26 zeroed when blind — 10 of 17 9 of 18 action dim, sender / receiver 3 / 2 3 / 3 4 / 4 hidden variable goal sector target target zone cardinality 4 4 4 target ring radius (m) 1.9 0.22 0.95 success tolerance (m) 0.35 0.07 0.30 lift requirement (m) — 0.16 0.14 above start success reward (once) 1.0 1.0 1.0 progress shaping (per m) 0.35 1.2 0.6 lift shaping (per m) — 0.6 0.9 step cost 0.004 0.004 0.004 drop penalty — 0.25 0.30 Success criteria. Uncoupled: the UGV body comes within 0.350.35 m of the goal, which sits at a jittered point (±0.45± 0.45 m) on one of four sector centres at radius 1.91.9 m; the UGV cannot sense the goal at all beyond 0.70.7 m, which is what makes the message the only information path. Partially coupled: the bar centre clears 0.160.16 m and comes within 0.070.07 m in the xyxy plane of the target. The hands begin already gripping the bar (a 0.020.02 m inward preload, settled for 4040 physics steps), so the task is coordinated transport rather than grasp discovery, though the grip can still be lost. Rigidly coupled: the box is carried 0.140.14 m above where it sits when control starts and comes within 0.300.30 m of the target zone; a box roll or pitch beyond 0.550.55 rad counts as a one-sided lift and ends the episode as a drop. Success is latched on first attainment and rewarded once. Episodes end on success, on a drop, or at the horizon.