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One for All, All for One: Coordinated Multi-Agent Diffusion Steering via Stochastic Optimal Control
Riccardo Barbano, Vincent Pauline, Runchang Li, George Webber, Alexander Denker, Željko Kereta, Stefan Bauer, Francisco Vargas, Esmeralda S. Whitammer
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Abstract
Abstract:Deep generative models often produce structured outputs composed of interacting components. Modelling these outputs with a single model requires learning both the component distributions and their interactions. We pursue a modular alternative: reuse independently trained component generators and learn only how to coordinate them to produce coherent structured outputs. Our framework, Coordinated Multi-Agent Diffusion Steering (CMDS), treats frozen pretrained diffusion models as reusable generative primitives and coordinates their reverse processes through a learned control. We formulate coordination as a stochastic optimal control problem, balancing an assembly-level reward that specifies the desired properties of the combined output against deviations from the pretrained dynamics. The learned control amortises this optimisation, allowing reuse across new task instances. Experiments show that CMDS can recover a known target distribution, satisfy different spatial constraints with the same trained control, and recover individual sources from degraded mixtures. Across multi-agent maze navigation, articulated robot planning, and text-conditioned human motion, CMDS turns frozen models into coordinated multi-agent generators.
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- Source: https://arxiv.org/abs/2610.08595v1
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One for All, All for One: Coordinated Multi-Agent Diffusion Steering via Stochastic Optimal Control Riccardo Barbano UCL Vincent Pauline TUM & MCML Runchang Li CUHK George Webber KCL Alexander Denker DESY Željko Kereta UCL Stefan Bauer TUM & MCML & CIFAR Francisco Vargas Proxima Bio Esmeralda S. Whitammer University of Edinburgh & CIFAR Abstract Deep generative models often produce structured outputs composed of interacting components. Modelling these outputs with a single model requires learning both the component distributions and their interactions. We pursue a modular alternative: reuse independently trained component generators and learn only how to coordinate them to produce coherent structured outputs. Our framework, Coordinated Multi-Agent Diffusion Steering (CMDS), treats frozen pretrained diffusion models as reusable generative primitives and coordinates their reverse processes through a learned control. We formulate coordination as a stochastic optimal control problem, balancing an assembly-level reward that specifies the desired properties of the combined output against deviations from the pretrained dynamics. The learned control amortises this optimisation, allowing reuse across new task instances. Experiments show that CMDS can recover a known target distribution, satisfy different spatial constraints with the same trained control, and recover individual sources from degraded mixtures. Across multi-agent maze navigation, articulated robot planning, and text-conditioned human motion, CMDS turns frozen models into coordinated multi-agent generators. 11footnotetext: Equal contribution. Correspondence to R.B. (r.barbano@ucl.ac.uk) and V.P. (vincent.paulinef@gmail.com). 1 Introduction Producing coherent structured outputs involves two distinct problems: modelling individual components and coordinating them when they are used together. End-to-end generative models absorb both into a single generative distribution. We instead build on a long-standing tradition of modular and compositional generative modelling (Grenander, 1978; Bienenstock et al., 1996; Williams, 2025; Du and Kaelbling, 2024). Component generators, pretrained without joint-system supervision, capture component-level variability, while coordination is learned separately at the level of the structured output. Diffusion models (Ho et al., 2020; Song et al., 2021; Lai et al., 2025) are well suited to this decomposition, as each pretrained reverse process provides a reusable component generative prior whose dynamics can be steered via a learned control while the underlying model remains fixed. Prior work on compositional diffusion generation typically resolves the required coordination at sampling time for each generated instance (Liu et al., 2022; Yang et al., 2023; Cao et al., 2025). This is sensible for one-off tasks, but becomes costly when related coordination problems recur across many instances. In settings where related coordination problems recur across many instances–for example, across relational design specifications, source-separation observations, or multi-agent start-goal configurations–the associated computation can instead be amortised into a reusable control. Building on recent work showing how controls or amortised samplers can steer pretrained diffusion processes towards task-specific target distributions (Domingo-Enrich et al., 2024; Domingo-Enrich et al., 2025; Venkatraman et al., 2024), we thus ask: How can we learn an amortised control that coordinates frozen component diffusion models using a reward on their assembled output, while remaining close to their pretrained dynamics? A framework for coordinated generation. The proposed framework begins with independently trained component generators, whose distributions form a reference product distribution over component configurations. An assembly map combines their outputs into a candidate, while an assembly-level reward evaluates the candidate. A learned control then coordinates the component generators towards high-reward joint outputs. To make this concrete, consider planning the movements of several agents through a shared environment. Each frozen copy of a single-agent trajectory model generates a path through the environment, using an agent-specific start and end as conditioning inputs. Stacking these paths produces a candidate multi-agent plan. Although each path may avoid walls in isolation, the assembled plan may still be invalid if agents collide. This illustrates the role of the assembly-level reward. It expresses requirements not captured by the independently trained component models and provides the signal used to coordinate their generation. Generally, the framework requires no training examples of valid structured outputs, only (i) component-level data or pretrained component generators; (i) an assembly map; and (i) an assembly-level reward. Continuous-time generative modelling. For each component i, let GiG^i denote the generator induced by the reverse-time sampling process of a pretrained diffusion model (Anderson, 1982; Haussmann and Pardoux, 1986; Song et al., 2021). We write X^si∈ℝd X_s^i ^d for component i’s state and X^s=(X^s1,…,X^sN)∈ℝD X_s=( X_s^1,…, X_s^N) ^D, D=NdD=Nd, for the joint state; hats denote reverse-time quantities. We use s∈[0,1]s∈[0,1] for reverse (generative) time with s=0s=0 corresponding to the noise state and s=1s=1 to the generated sample. The reverse process for component i starts from X^0i∼p^0i X^i_0 p^i_0, where p^0i p^i_0 is a standard Gaussian, and terminates at X^1i∼π~i X^i_1 π^i, where π~i π^i denotes the learned component distribution. Suppressing the index i, we write the reference reverse-time diffusion as, dX^s=f^s(X^s)ds+σ^sdW¯s,X^0∼p^0. X_s= f_s( X_s)\, s+ σ_s\, W_s, X_0 p_0. (1) Here, f^s f_s denotes the learned reverse drift, σ^s σ_s is the reverse-time diffusion scale, and W¯s W_s is a Brownian motion in reverse time. The terminal law of X^1 X_1 is the learned model distribution π~ π, which approximates the corresponding exact terminal law π. For N pretrained component samplers, their independent reference processes form the joint product reverse process X^s=(X^s1,…,X^sN) X_s=( X^1_s,…, X^N_s), with X^0i∼p^0i X^i_0 p^i_0 and X^1i∼π~i X^i_1 π^i. Its terminal marginal is therefore the product law ρ=π~1⊗⋯⊗π~N.ρ= π^1 ·s π^N. Controlled diffusion and stochastic optimal control (SOC). The connection between stochastic control, path-space relative entropy, and exponential tilting is well established (Chetrite and Touchette, 2015). Diffusion sampling admits a SOC interpretation in which a reward tilt is realised by controlling a reference diffusion under a quadratic path-action penalty (Nüsken and Richter, 2021; Berner et al., 2024; Uehara et al., 2024; Domingo-Enrich et al., 2025); see section 2.2. For a single diffusion model (N=1N=1), the problem can be written as, minu∈ _u ℙ^u[∫0112‖us(X^su)‖2ds−λr(X^1u)], _ P^u [ _0^1 12\|u_s( X^u_s)\|^2\, s-λ r( X^u_1) ], (2) s.t. .t. dX^us=[f^s(X^us)+σ^sus(X^us)]ds+σ^sdW¯s,X^u0∼p^0. X^u_s= [ f_s( X^u_s)+ σ_su_s( X^u_s) ] s+ σ_s W_s, X^u_0 p_0. Here, u∈u is an admissible control field, ℙ^u P^u is the path measure induced by the controlled process X^u X^u, λ>0λ>0 controls the reward strength, f^s f_s and σ^s σ_s denote the reverse-time drift and diffusion scale of the pretrained model, respectively, W¯s W_s is a Brownian motion, and p^0 p_0 is the initial noise distribution. Under the Girsanov conditions and a shared initial law, the expected accumulated quadratic control cost equals the relative entropy between the controlled and reference path measures (Nüsken and Richter, 2021; Berner et al., 2024). We apply this SOC formulation to the joint state X^s X_s of multiple frozen pretrained component diffusion models, using an assembly-level reward. Contributions. Our contributions can be summarised as follows: • Coordination as modular post-training. We formulate compositional generation as stochastic optimal control over the joint reverse dynamics of frozen pretrained component diffusion models. The independent component processes define a reference product law, while an assembly-level reward can induce cross-component dependencies, realised by a learned control that couples their reverse dynamics. • Comprehensive empirical validation. We demonstrate target-law recovery, amortised relational assembly, and source separation. In PointMaze, CMDS coordinates up to four agents and generalises to unseen start–goal regions. Using off-the-shelf KUKA and MDM checkpoints, it turns frozen single-agent models into coordinated multi-agent generators without joint training data. The learned KUKA control transfers to new geometries and additional arms without retraining. In four-person motion generation, CMDS achieves 90% joint success versus 75% for PCD++, with over 250×250× faster sampling. Further discussion on relevant background concepts can be found in Appendix A. 2 Coordinated Multi-Agent Diffusion Steering (CMDS) Figure 1: Overview of CMDS. Frozen component diffusion models are sampled concurrently, while a joint control conditions on the multi-agent state and applies component-wise controls throughout the reverse process, yielding a coordinated assembled output. We consider N pretrained diffusion agents, each generating one component of a structured output. Using the joint state X^s X_s defined above, the assembled state is Y^s=φ(X^s)∈ℝm Y_s= ( X_s) ^m, where φ:ℝD→ℝm :R^D ^m is a fixed assembly operator. Without coordination, the component processes evolve independently and need not produce a globally consistent assembly. We extend the single-process SOC formulation in eq. 2 to the joint state, with us:ℝD→ℝDu_s:R^D ^D, denoting the coordination control. Importantly, CMDS is non-autoregressive across components. All N reverse processes evolve concurrently along the same generative-time axis, and at each time s, the control usi(X^s)u_s^i( X_s) applied to agent i may depend on the current state of every agent. No component is therefore generated and fixed before the others. This avoids imposing an arbitrary ordering over components and enables corrections throughout generation, which is particularly important when the assembly-level criterion depends on their joint configuration. The component updates may also be evaluated in parallel, subject to the cost of the coordination modules. With this joint representation, coordination amounts to steering the combined reverse process so that its assembled terminal sample satisfies an assembly-level criterion while remaining close to the independent pretrained dynamics. Writing x=(x1,⋯,xN)∈ℝDx=(x^1,·s,x^N) ^D, we denote the joint reference reverse drift by f^s(x)=(f^s1(x1),…,f^sN(xN)). f_s(x)=( f_s^1(x^1),…, f_s^N(x^N)). The controlled terminal assembly is Y^1u=φ(X^1u)=φ(X^1u,1,…,X^1u,N), Y_1^u= ( X_1^u)= ( X_1^u,1,…, X_1^u,N), and we apply the SOC objective in eq. 2 to the reference product process, using the assembly-level reward r∘φr , minu∈ℙ^u[∫0112∑i=1N‖usi(X^su)‖2s−λr(φ(X^1u))], _u _ P^u [ _0^1 controlblue$ 12 _i=1^N \|u_s^i( X_s^u) \|^2$\,ds- rewardpink$ λ r\! ( ( X_1^u) )$ ], (3) where the reward of the assembled terminal output measures the global consistency, and the quadratic control cost penalises deviation from the pretrained agents’ distribution. The central hypothesis of CMDS is that locally valid components can be learned, while the dependencies required for global validity can be introduced through an assembly-level criterion. Further details on the forward noising process, its time reversal, the associated score-based parameterisation, and the controlled reverse-time SDE are provided in Sections B.1 and B.2. 2.1 Learning Coordinated Reverse Dynamics The exact optimal path distribution ℙ^u⋆ P^u is generally intractable. We therefore approximate the non-parametric optimal control us⋆(x)u _s(x) by learning a parametrised neural control field, Uθ(x,s)=(Uθ1(x,s),…,UθN(x,s)). controlleryellow$ U_θ(x,s)$= (U_θ^1(x,s),…,U_θ^N(x,s) ). The component controls may either share parameters across agents or use agent-specific parameters; in both cases, each UθiU_θ^i conditions on the full joint state. The resulting controlled reverse diffusion implicitly defines a stochastic generator Gθ=(Gθ1,…,GθN)G_θ=(G_θ^1,…,G_θ^N), with component dynamics, dX^sθ,i=[f^si(X^sθ,i)+σ^sUθi(X^sθ,s)]ds+σ^sdW¯si,i=1,…,N. X_s^θ,i= [ f^i_s( X_s^θ,i)+ σ_s controlleryellow$ U_θ^i( X_s^θ,s)$ ] s+ σ_s\, W_s^i, i=1,…,N. (4) We denote the corresponding assembled process by Y^sθ:=φ(X^sθ) Y_s^θ:= ( X_s^θ). Equivalently, Gθi:X^0↦X^1θ,iG_θ^i: X_0 X_1^θ,i denotes the stochastic solution map induced by the controlled reverse dynamics, with X^0i∼p^0i X_0^i p_0^i. The corresponding training objective balances control cost and global reward, (θ)=ℙ^θ[∫0112∑i=1N‖Uθi(X^sθ,s)‖2ds−λr(φ(X^1θ))], (θ)=E_ P^θ [ ^1_0 12 _i=1^N\|U_θ^i( X_s^θ,s)\|^2 s-λ r( ( X^θ_1)) ], (5) where ℙ^θ P^θ is the path distribution induced by the parametrised controlled process. Adjoint matching CMDS instantiation (Domingo-Enrich et al., 2025). To train the control efficiently, we use an adjoint matching (AM) version of the original objective. Along sampled controlled trajectories, an adjoint variable estimates the direction in which each component should move to improve the final global reward. The control is trained to match this optimal correction, ℒAM(θ)=12ℙ^θ¯[∫01‖Uθ(X^sθ¯,s)+σ^sasθ¯‖2ds],θ¯=stopgrad(θ) _ AM(θ)= 12E_ P θ [ _0^1 \| controlleryellow$ U_θ( X_s θ,s)$+ σ_sa_s θ \|^2\, s ], θ=stopgrad(θ) (6) where asθ¯a_s θ denotes the adjoint along the sampled controlled trajectory X^θ¯ X θ, obtained by solving the following terminal-value problem backwards from s=1s=1 to s=0s=0, ddsasθ¯ sa θ_s =−Dx(f^s(X^sθ¯)+σ^sUθ¯(X^sθ¯,s))⊤asθ¯−12∇x∑i=1N‖Uθ¯i(X^sθ¯,s)‖2, =-D_x ( f_s( X_s θ)+ σ_sU_ θ( X_s θ,s) ) a_s θ- 12 _x _i=1^N\|U_ θ^i( X θ_s,s)\|^2, a1θ¯ a θ_1 =−λDxφ(X^1θ¯)⊤∇yr(Y^1θ¯), =-λ D_x ( X θ_1) _yr( Y θ_1), where DxD_x denotes the Jacobian with respect to x. The full pathwise adjoint, its block-wise multi-agent form, and the corresponding adjoint matching surrogate are derived in Appendix C. Strategies for learning the coordination control. CMDS is defined by jointly controlling the reference product process under the SOC objective in eq. 3, whereas AM is one possible strategy for learning the coordination control UθU_θ, rather than part of the framework. When the reverse solver and assembly-level reward are differentiable, UθU_θ may instead be learned by directly optimising the discretised SOC objective; we refer to reverse-mode differentiation through the solver transitions as discrete-adjoint optimisation (Domingo-Enrich, 2024). Alternatively, path-space objectives such as relative trajectory balance (Venkatraman et al., 2024) could be adapted to the joint controlled reverse process to target the same reward-tilted law, provided that the corresponding controlled-to-reference trajectory-density ratio is tractable. We adopt AM since it propagates gradient information without backpropagating through the complete computational graph of the sampling rollout. 2.2 Path-Space Interpretation of Coordination The SOC objective in eq. 3 admits a path-space interpretation. Let ℙ P denote the reference path law induced by the independent pretrained reverse diffusion processes, and let ℙ^u P^u denote the controlled path law. By Girsanov’s theorem (Øksendal, 2003), under the usual change-of-measure conditions and a shared initial law, the quadratic control cost equals the path-space relative entropy, DKL(ℙ^u∥ℙ^)=12ℙ^u[∫01∥us(X^su)∥2ds].D_KL ( P^u\,\|\, P )= 12E_ P^u[ _0^1\|u_s( X_s^u)\|^2\, s]. The control problem admits the path-space form, u⋆∈argminu∈DKL(ℙ^u∥ℙ^)−λℙ^u[r(φ(X^1u))].u ∈ _u \ controlblue$ D_KL ( P^u\,\|\, P )$- rewardpink$ _ P^u [r\! ( ( X_1^u) ) ]$ \. (7) This formulation makes explicit that coordination balances the assembly-level reward of the assembled terminal object against deviation from the pretrained product process. By the standard variational identity for KL-regularised stochastic control (Chetrite and Touchette, 2015), the optimal controlled path law is an exponential tilt of the reference law. Conditionally on the initial noise state, dℙ^u⋆(⋅|X^0)dℙ^(⋅|X^0)(X^⋅)∝exp(λr(φ(X^1))). d P^u (\,·\,| X_0 )d P (\,·\,| X_0 ) ( X_· ) \! (λ r ( ( X_1) ) ). (8) Thus, optimal coordination re-weights joint trajectories by the assembly-level reward; see Section B.3 for the change of measure and Remark 2 for the induced dependencies. 3 Experimental Evaluation Figure 2: Two-agent OU problem. Terminal samples from the independent OU reference process and the rotation-equivariant CMDS control for a prescribed separation dT=4d_T=4. Yellow and blue points denote the two agents, and each line joins the terminal states belonging to the same sample. We evaluate CMDS along three axes: (i) a tractable stochastic-control problem with a known optimal solution, to assess recovery of the target distribution rather than constraint satisfaction alone; (i) source separation, to assess posterior inference from blurred mixtures when paired source–mixture training data are unavailable; and (i) multi-agent motion, to assess scalability to larger models and more agents using off-the-shelf pretrained checkpoints, including zero-shot transfer to changed geometry and additional agents without retraining the control. Additionally, we study constraint satisfaction and amortised coordination across relational specifications in Section D.2. These constrained visual-assembly experiments provide a controlled test-bed to isolate the effects of component factorisation, reward design, amortisation, and optimisation. 3.1 Two-agent OU coordination Setup. We test target-law recovery in a two-agent SOC problem with a computable value function, i.e., the minimum expected remaining cost, whose gradient determines the optimal control, providing a numerical oracle. We consider two independent Ornstein–Uhlenbeck (OU) reference processes Xt1,Xt2∈ℝ2X_t^1,X_t^2 ^2, with joint state Xt=(Xt1,Xt2)∈ℝ4X_t=(X_t^1,X_t^2) ^4. The agents are initialised and evolve according to X01,X02∼iid(0,I2),dXti=−Xtidt+dWti,t∈[0,1],i∈1,2,X_0^1,X_0^2 N(0,I_2), dX_t^i=-X_t^i\,dt+dW_t^i, t∈[0,1], i∈\1,2\, where W1W^1 and W2W^2 are independent two-dimensional Brownian motions. Consequently, the joint initial law is X0∼(0,I4)X_0 (0,I_4). The reference components evolve independently, while coordination is induced by the terminal reward rOU(x1,x2)=−(∥x2−x1∥2−dT)2r_OU(x^1,x^2)=-( x^2-x^1 _2-d_T)^2, with dT=4d_T=4, and realised through the resulting joint optimal control. Because the problem reduces to the relative coordinate Rt=Xt2−Xt1R_t=X_t^2-X_t^1, we can compute a value-function oracle for the optimal stochastic control (refer to Section D.1). We also include importance sampling (IS) (Robert and Casella, 2004), using samples from the (uncontrolled) OU reference process re-weighted according to the terminal reward. Constraint satisfaction. Figure 3: Recovery of the SOC target. The reference OU process yields a terminal separation of 1.3±0.71.3± 0.7, whereas the rotation-equivariant CMDS control achieves 4.0±0.14.0± 0.1, see fig. 2. The equivariant control is used only to respect a known symmetry of the oracle target; it is not a requirement of CMDS. Distributional recovery. We compare terminal samples against the oracle’s joint terminal law, i.e., the t=1t=1 marginal of the optimal path law characterised in eq. 8, using the 2-Wasserstein distance (W2W_2), maximum mean discrepancy (MMD), and Sinkhorn (Sink) transport cost over five seeds. As reported in fig. 3, CMDS achieves discrepancies comparable to oracle self-comparisons ** * Oracle self-comparisons provide a reference level for each reported discrepancy, using two independent oracle sample sets, each matching the evaluation sample size. and outperforms importance sampling (IS). These results show that CMDS does more than reach the prescribed separation as it also recovers the distributional structure of the reward-tilted target law. Section D.1 shows that an unconstrained control can satisfy the scalar distance constraint while failing to recover angular structure of the target distribution. 3.2 Source separation Compositional posterior formulation. We consider posterior inference in which N frozen pretrained diffusion samplers independently define component priors through their reverse processes, with reference terminal samples X^1i∼π~i X_1^i π^i. Let X=(X1,…,XN)X=(X^1,…,X^N) denote the unknown component state, whose non-injective additive assembly is Y=φ(X):=∑i=1NXiY= (X):= _i=1^NX^i, and suppose that we observe a blurred mixture b=Aτ(Y)+εb= A_τ(Y)+ , where Aτ A_τ is a known Gaussian blurring operator with standard deviation τ, and ε∼(0,σε2I) (0, _ ^2I). Since the additive assembly φ is non-injective, the observation does not uniquely identify its constituent sources; the independently pretrained component priors therefore regularise the resulting compositional posterior over valid decompositions. Assuming Gaussian observation noise, we define the assembly-level reward rb(y):=−12σε2‖Aτ(y)−b‖2r_b(y):=- 12 _ ^2\| A_τ(y)-b\|^2. The pretrained diffusion models define the reference product law ρ=π~1⊗⋯⊗π~Nρ= π^1 ·s π^N, while CMDS performs compositional posterior inference by coordinating their reverse dynamics, producing controlled terminal components X^1θ=(X^1θ,1,…,X^1θ,N) X_1^θ=( X_1^θ,1,…, X_1^θ,N) whose assembly Y^1θ=φ(X^1θ) Y_1^θ= ( X_1^θ) jointly explains the observation b, without requiring training samples from the joint source distribution. For comparison, we adapt diffusion posterior sampling (DPS) (Chung et al., 2023) to the product prior, paralleling the posterior-sampling construction of Luan et al. (2025). At each reverse step, we evaluate the observation likelihood on the Tweedie estimates (Efron, 2011) of both processes and apply the resulting component-wise gradients to their reverse updates; details are given in Section D.3. Figure 4: Source separation from a blurred mixture (τ=1τ=1). Left: circle–square reconstructions for two held-out mixtures; red and blue hatching behind the CMDS reconstructions shows same-seed matched reference sources. Right: comparison of the reference, DPS, and CMDS over 1,0241,024 held-out mixtures, measuring clean-mixture recovery (RMSE, ↓ ), labelled-source overlap (IoU, ↑ ), and shape distortion (↓ ). Full metric definitions are in Section D.3; aggregate results and representative reconstructions under milder blur (τ=0.5τ=0.5) are shown in figs. 15 and 16. Recovery under ambiguity. As shown in fig. 4, CMDS achieves the strongest overall recovery under the challenging τ=1τ=1 blur. It yields the lowest clean-mixture reconstruction error and the highest source intersection over union (IoU), while maintaining the low shape distortion induced by the frozen component priors. DPS substantially improves source localisation over independent reference sampling, but markedly distorts the components and yields a larger reconstruction error than CMDS. This localisation–distortion trade-off reflects the non-identifiability of the problem. The likelihood depends only on the assembled mixture X1+X2X^1+X^2, through Aτ(X1+X2) A_τ(X^1+X^2), and therefore contains no direct information about the differential mode X1−X2X^1-X^2 that distinguishes the individual sources. Consequently, the step-wise likelihood corrections used by DPS can improve the reward by redistributing or deforming mass in ways that are poorly aligned with the pretrained component distributions. By contrast, rather than applying step-wise likelihood guidance, CMDS learns joint feedback over the product of the pretrained reverse processes, while the quadratic path-action penalty favours small coordinated deviations from the frozen dynamics, helping preserve component structure. Quantitative results under milder blur are provided in Section D.3. 3.3 From Single-Agent Priors to Coordinated Multi-Agent Motion Coordination across motion domains. Figure 5: PointMaze (Medium) coordination. Dashed ( ) and solid ( ) paths use the same initial noise. Reference (uncontrolled) process trajectories collide, whereas CMDS steers to avoid collisions while preserving their start–end pairs. We investigate multi-agent navigation problems where agents move through a shared environment, between prescribed start- and end-points, while satisfying task-specific constraints and avoiding collisions. We consider planar navigation (PointMaze) (Fu et al., 2020), articulated robot planning (KUKA) (Luo et al., 2024), and text-conditioned human motion (MDM) (Tevet et al., 2023). Pretrained single-agent priors generate component trajectories, the assembly operator combines them, and an assembly-level reward coordinates their movement. Throughout, the component priors remain frozen and only the coordination control is learned. PointMaze and KUKA condition the component models on prescribed start and end points. For MDM, component models are text-conditioned and end points are supplied to the coordination control. Success requires every agent to satisfy the task-specific goal, avoid collisions, and pass task-specific feasibility checks. Further details are in Sections D.4, D.5 and D.6. Medium Large N=2N=2 N=4N=4 N=2N=2 N=4N=4 Method Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Reference 0.174±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.195±0.008\, [rgb]0.5,0.5,0.5 ± 0.008 0.068±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.010±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.195±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.219±0.000\, [rgb]0.5,0.5,0.5 ± 0.000 0.039±0.000\, [rgb]0.5,0.5,0.5 ± 0.000 0.049±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 Best-of-256 0.216±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.263±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.076±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.044±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.232±0.009\, [rgb]0.5,0.5,0.5 ± 0.009 0.328±0.021\, [rgb]0.5,0.5,0.5 ± 0.021 0.076±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.070±0.000\, [rgb]0.5,0.5,0.5 ± 0.000 FK-Steering 0.797±0.028\, [rgb]0.5,0.5,0.5 ± 0.028 0.820±0.034\, [rgb]0.5,0.5,0.5 ± 0.034 0.398±0.036\, [rgb]0.5,0.5,0.5 ± 0.036 0.331±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.924±0.030\, [rgb]0.5,0.5,0.5 ± 0.030 0.953±0.008\, [rgb]0.5,0.5,0.5 ± 0.008 0.622±0.009\, [rgb]0.5,0.5,0.5 ± 0.009 0.729±0.023\, [rgb]0.5,0.5,0.5 ± 0.023 DPS 0.964±0.024\, [rgb]0.5,0.5,0.5 ± 0.024 0.956±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.958±0.025\, [rgb]0.5,0.5,0.5 ± 0.025 0.919±0.033\, [rgb]0.5,0.5,0.5 ± 0.033 0.984±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.977±0.000\, [rgb]0.5,0.5,0.5 ± 0.000 0.880±0.030\, [rgb]0.5,0.5,0.5 ± 0.030 0.927±0.009\, [rgb]0.5,0.5,0.5 ± 0.009 CMDS 0.990±0.012\, [rgb]0.5,0.5,0.5 ± 0.012 0.919±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.701±0.009\, [rgb]0.5,0.5,0.5 ± 0.009 0.669±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.911±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.831±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.880±0.012\, [rgb]0.5,0.5,0.5 ± 0.012 0.753±0.027\, [rgb]0.5,0.5,0.5 ± 0.027 CMDS + TRG 0.995±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.992±0.000\, [rgb]0.5,0.5,0.5 ± 0.000 0.974±0.012\, [rgb]0.5,0.5,0.5 ± 0.012 0.935±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.966±0.005\, [rgb]0.5,0.5,0.5 ± 0.005 0.898±0.016\, [rgb]0.5,0.5,0.5 ± 0.016 0.966±0.012\, [rgb]0.5,0.5,0.5 ± 0.012 0.977±0.000\, [rgb]0.5,0.5,0.5 ± 0.000 Table 1: PointMaze joint success. Mean ± standard deviation (std) across sampling seeds. Training-seed variability is reported separately in Section D.4.8. Each pool contains 128128 instances per layout and agent count. For CMDS and CMDS+TRG, we train five independent seeds and report the seed with median Pool 1 success for each setting, breaking ties using Pool 2. PointMaze: amortised multi-agent navigation. We pretrain one conditional single-agent trajectory prior for each of the “Medium” and “Large” layouts (Fu et al., 2020), then coordinate two or four copies. The reward penalises insufficient wall clearance, inter-agent collisions, path length, and trajectory acceleration. We evaluate on two sets of unseen start–end instances. Pool 1 places starts and goals in navigable grid squares used during control training, whereas Pool 2 uses squares excluded from control training (Section D.4.3). Tables 1 and 5 compare independent reference sampling, Best-of-256, FK-Steering (Singhal et al., 2025), DPS (Chung et al., 2022), and our learned control. We evaluate two parameterisations of the coordination control: CMDS uses an attention-based control conditioned on the joint multi-agent state, while CMDS+TRG additionally incorporates the Tweedie reward gradient (TRG), namely the reward gradient evaluated at the Tweedie estimate, inspired by prior work (Denker et al., 2024; Venkatraman et al., 2024). CMDS+TRG improves or matches the reported base-CMDS rates, achieving Pool 2 success between 89.8%89.8\% and 99.2%99.2\% across the reported settings; DPS is the best performing on “Large” with two agents. Across five independent training seeds per setting, CMDS+TRG achieves mean Pool 2 success between 90.6%90.6\% and 98.3%98.3\%, with training-seed standard deviations no larger than 1.51.5 percentage points (Section D.4.8). To stabilise adjoint-matching training against large residuals, we use a pseudo-Huber regression loss in place of the squared loss; details are in Section D.4.7. Figure 6: A dual-arm coordination example. Poses share normalised path progress s=0.135s=0.135; crosses mark arm–arm contact, and red intervals mark strict mesh collisions. PBDM uses ncand=6n_cand=6 with up to five repair attempts per candidate, versus ncand=1n_cand=1 for the other methods; its modified, native-accepted path fails strict checking between knots. This outcome-selected illustration is not a matched-budget comparison. Additional poses and clearance traces are provided in Section D.5. KUKA: multi-arm planning from single-arm priors. We coordinate two seven-joint robot arms in a shared environment containing obstacles. Each task specifies the obstacle layout and the start and goal configurations of both arms. A shared CMDS+TRG control coordinates frozen pretrained single-arm diffusion models to generate their paths jointly. Method Strict success (%, ↑ ) Planning time (s, ↓ ) PBDM 60.9±8.0\, [rgb]0.5,0.5,0.5 ± 8.0 0.427±0.003\, [rgb]0.5,0.5,0.5 ± 0.003 Dual KUKA + guidance 74.3±5.7\, [rgb]0.5,0.5,0.5 ± 5.7 0.356±0.002\, [rgb]0.5,0.5,0.5 ± 0.002 Dual KUKA prior 36.3±8.7\, [rgb]0.5,0.5,0.5 ± 8.7 0.274±0.033\, [rgb]0.5,0.5,0.5 ± 0.033 Single-arm priors + guidance 51.0±0.6\, [rgb]0.5,0.5,0.5 ± 0.6 0.355±0.001\, [rgb]0.5,0.5,0.5 ± 0.001 Reference single-arm priors 11.4±1.5\, [rgb]0.5,0.5,0.5 ± 1.5 0.193±0.002\, [rgb]0.5,0.5,0.5 ± 0.002 CMDS+TRG 77.3±0.7\, [rgb]0.5,0.5,0.5 ± 0.7 0.368±0.002\, [rgb]0.5,0.5,0.5 ± 0.002 Table 2: Dual KUKA planning with one initial joint candidate. Mean ± sample std. strict successes on 413 held-out test tasks. Planning time is the median value and includes scene setup, generation, verification and optional refinement. Publicly released checkpoints (Luo et al., 2024) predict noise as an energy gradient, ϵψ=∇xEψ _ψ= _xE_ψ. Adjoint propagation therefore requires Hessian–vector products through the energy, while the checkpoint parameters remain frozen. Strict success requires both arms to reach their goals, remain within the allowed angular range of every robot joint, and pass checks for inter-arm collisions, self-collisions, and collisions with the obstacles and worktable. We evaluate on the test layouts released by Luo et al. (2024), unseen by both the priors and the controller (Section D.5.1), and denote by ncandn_cand the number of initial candidate multi-arm plans generated per task. With one candidate plan, CMDS+TRG achieves 77.3%77.3\% strict success, compared with 11.4%11.4\% for the reference independent single-arm sampling and 51.0%51.0\% when adding collision guidance (see table 2). Using the separately pretrained two-arm diffusion prior, guided Dual KUKA achieves 74.3%74.3\%, while Potential Based Diffusion Motion Planning (PBDM; Luo et al., 2024) achieves 60.9%60.9\% using the same prior within its native candidate-selection and refinement procedure. We also apply the learned control without retraining to shifted or enlarged obstacles, wider arm-base spacing, and additional arms (table 3). On a separate set of tasks with five arms and four candidates per task (ncand=4n_cand=4), CMDS + TRG achieves 68.3%68.3\% strict success, compared with 48.3%48.3\% for guided single-arm priors. Additional information is provided in Section D.5. Method ncandn_cand Geometry transfer Number of arms Shifted obstacles Larger obstacles Wider bases 3 4 5 Single-arm KUKA + guidance 1 53.4±4.0\, ± 4.0 14.3±1.3\, ± 1.3 49.2±2.4\, ± 2.4 33.8±0.7\, ± 0.7 28.4±1.7\, ± 1.7 19.3±2.0\, ± 2.0 4 83.2±2.8\, ± 2.8 42.1±4.1\, ± 4.1 84.0±0.7\, ± 0.7 69.1±1.5\, ± 1.5 55.7±3.5\, ± 3.5 48.3±2.4\, ± 2.4 Dual KUKA + guidance 1 68.9±3.9\, ± 3.9 31.1±10.0\, ± 10.0 58.2±8.1\, ± 8.1 – – – 4 89.7±0.8\, ± 0.8 66.9±3.8\, ± 3.8 87.8±2.8\, ± 2.8 – – – 1 74.0±1.7\, ± 1.7 42.7±3.7\, ± 3.7 72.1±2.2\, ± 2.2 59.5±0.3\, ± 0.3 52.3±3.4\, ± 3.4 41.9±1.7\, ± 1.7 CMDS + TRG 4 91.6±0.5\, ± 0.5 73.0±1.0\, ± 1.0 91.8±0.8\, ± 0.8 86.8±1.0\, ± 1.0 76.2±2.6\, ± 2.6 68.3±1.2\, ± 1.2 Table 3: Zero-shot KUKA generalisation. Strict success (%), mean ± sample std. across three sampling seeds, for ncand=1,4n_cand=1,4. The left block reports geometry transfer; the right block reports transfer to larger numbers of arms. Method Success (%) ↑ Collision free (%) ↑ Goal err. (m) ↓ Time (s) ↓ Best-of-256 0.0±0.00.0\, [rgb]0.5,0.5,0.5 ± 0.0 16.7±5.916.7\, [rgb]0.5,0.5,0.5 ± 5.9 2.456±0.1462.456\, [rgb]0.5,0.5,0.5 ± 0.146 69.90±0.2969.90\, [rgb]0.5,0.5,0.5 ± 0.29 DPS 0.0±0.00.0\, [rgb]0.5,0.5,0.5 ± 0.0 61.7±10.861.7\, [rgb]0.5,0.5,0.5 ± 10.8 3.751±0.2843.751\, [rgb]0.5,0.5,0.5 ± 0.284 8.57±0.028.57\, [rgb]0.5,0.5,0.5 ± 0.02 PCD 4.2±2.94.2\, [rgb]0.5,0.5,0.5 ± 2.9 8.3±2.98.3\, [rgb]0.5,0.5,0.5 ± 2.9 0.500±0.0000.500\, [rgb]0.5,0.5,0.5 ± 0.000 9.99±0.069.99\, [rgb]0.5,0.5,0.5 ± 0.06 PCD++ 75.0±6.675.0\, [rgb]0.5,0.5,0.5 ± 6.6 98.3±2.398.3\, [rgb]0.5,0.5,0.5 ± 2.3 0.487±0.0010.487\, [rgb]0.5,0.5,0.5 ± 0.001 145.34±4.46145.34\, [rgb]0.5,0.5,0.5 ± 4.46 CMDS 90.0±6.390.0\, [rgb]0.5,0.5,0.5 ± 6.3 90.0±6.390.0\, [rgb]0.5,0.5,0.5 ± 6.3 0.158±0.0080.158\, [rgb]0.5,0.5,0.5 ± 0.008 0.54±0.000.54\, [rgb]0.5,0.5,0.5 ± 0.00 Table 4: Four-person motion coordination. Mean ± sample std. across five sampling seeds on 2424 scenes; CMDS uses one trained control. Failures count as zero successes. Full metrics appear in table 15. Figure 7: Four-person motion coordination. We compare reference MDM, PCD++, and CMDS on the four-person coordination task. Solid curves show left- and right-foot trajectories, with intermediate poses faded; the top-down insets summarise the trajectories and target completion. Here, the reference produces collisions and misses targets, whereas PCD++ and CMDS reach all targets without collisions. Figure 22 in Section D.6 compares all methods. MDM: from single-person motion to coordinated crowds. We use the 50-step MDM checkpoint (Tevet et al., 2023) to coordinate four independently text-conditioned people over six-second motions (see Section D.6). A generated scene is successful when all four people move from their starting positions to their targets, shown in the top-down insets of fig. 7, and their motions pass both the inter-person collision and kinematic-validity checks. Baselines include Best-of-256256, DPS, PCD (Luan et al., 2025), and our PCD++ extension, which jointly optimises noisy motions through the frozen denoiser for goal attainment and clearance (see Algorithm 2). Tables 4 and 7 report 90%90\% joint success for CMDS versus 75%75\% for PCD++. PCD++ has a higher collision-free rate, but lower joint success and longer sampling time. 4 Conclusion We introduced CMDS, a modular post-training framework for coordinating frozen pretrained diffusion models through stochastic optimal control. An assembly-level reward specifies joint requirements, while an amortised control couples the component reverse processes under a quadratic path-action penalty. Experiments demonstrate target-law recovery in a tractable setting, amortised relational assembly, and source recovery, while released KUKA and MDM priors extend coordination to multi-arm planning and multi-person motion. Transfer to additional robot arms further illustrates the reuse enabled by learning coordination separately from component generation. Together, these results show that frozen component models can support structured tasks beyond their individual training settings through learned interactions. Future work includes improving reward design, characterising how robust matching and control parameterisation affect target-law approximation, and extending coordination to larger, more heterogeneous generative models. AI use statement In this work, we used generative AI tools to assist with text editing, reformatting, and polishing; LaTeX formatting and styling; and software-code review, debugging, and editing. We also used generative AI to generate code for visualisation and presentation of results; all such code was reviewed by the authors and was not used to implement the methods or produce the experimental results reported in this paper. All AI-assisted text and code were reviewed and edited by the authors. AI-assisted code was manually inspected and tested before use. We did not use generative AI tools to generate synthetic datasets, help develop theoretical models or conceptual frameworks, formulate mathematical claims, provide critical ingredients for proving mathematical claims, assist in the writing of proofs, propose or refine hypotheses, clean and reformat datasets, support qualitative and thematic data analysis, or interpret results; assisting with translation is not applicable to this work. We take responsibility for the final content of this work, including text, claims, code, and artifacts produced with the aid of generative AI. Acknowledgments RB acknowledges support from the EPSRC (EP/V026259/1). GW acknowledges support from the EPSRC CDT in Smart Medical Imaging [EP/S022104/1] and by a GSK Studentship. AD acknowledges support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. VP acknowledges the financial support from the Munich Center for Machine Learning (MCML). The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V. (https://w.gauss-centre.eu) for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUPITER—JUWELS at Jülich Supercomputing Centre (JSC). Furthermore, the authors appreciate the computational resources provided by the National High Performance Computing Centre (https://w.nhr.kit.edu). Finally, the authors would also like to acknowledge the fruitful discussions with Lourdes de Agapito Vicente, Simon Arridge, Mirgahney Mohammed, Marcelo Pereyra, and Neill Campbell. References Anderson (1982) B. D. O. Anderson Reverse-time diffusion equation models. Stochastic Processes and their Applications 12 (3), p. 313–326. External Links: Document, Link Cited by: §B.1, §1. Bar-Tal et al. (2023) O. Bar-Tal, L. Yariv, Y. Lipman, and T. 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External Links: Document, Link Cited by: Appendix A. Øksendal (2003) B. Øksendal Stochastic differential equations: an introduction with applications. 6 edition, Springer. External Links: Document, Link Cited by: §B.3, §2.2. Appendix Overview Appendix A Related Work Modular and compositional generative modelling. Modular generative modelling has a long history in pattern theory and structured scene representations (Grenander, 1978; Bienenstock et al., 1996; Williams, 2025), and recent work advocates composing specialised generators rather than learning every combination within a single model (Du and Kaelbling, 2024). For diffusion models, this perspective has motivated compositional visual generation through combinations of conditional scores (Liu et al., 2022) and sampling methods for composed energy-based models (Du et al., 2023). Related approaches compose learned constraint factors (Yang et al., 2023) or diffusion policies across skills, domains, and input modalities (Wang et al., 2024; Cao et al., 2025). CMDS follows this modular perspective while explicitly distinguishing component states from their assembled output. Each frozen generator supplies a component, and an assembly-level reward specifies the interactions introduced through learned joint control. Joint generation with pretrained diffusion models. A related line of work coordinates multiple pretrained diffusion processes directly during generation. MultiDiffusion binds multiple denoising paths through shared image-generation constraints (Bar-Tal et al., 2023), while SyncDiffusion synchronises image patches using perceptual-similarity gradients (Lee et al., 2023). MMDisCo coordinates frozen single-modal diffusion models using a discriminator trained on paired audio–video examples (Hayakawa et al., 2025). In contrast to paired-data supervision, CMDS learns coordination from an assembly-level reward under a path-space control objective, without requiring training examples of valid joint outputs. Cost-based coordination of diffusion processes. Coordinated Diffusion (CoDi; Peters et al., 2026) and Projected Coupled Diffusion (PCD; Luan et al., 2025) are particularly close to our setting. CoDi constructs a KL-regularised cooperative objective relative to single-agent base policies and guides their diffusion processes simultaneously. Its sampling-based guidance estimator uses online cost evaluations without requiring cost gradients. PCD likewise introduces interactions between pretrained diffusion processes at test time, combining cost-based guidance with projections onto prescribed constraint sets. CMDS shares the use of separately pretrained generators and a joint criterion, but learns a reusable control across task instances under a quadratic path-action penalty. Coordination is therefore amortised into a joint-state control, rather than reconstructed solely through inference-time guidance. The approaches also differ in their constraint treatment: PCD explicitly projects onto supplied constraint sets, whereas the finite reward penalties used in our experiments do not guarantee hard constraint satisfaction. Our current adjoint-based implementation additionally assumes differentiable rewards, unlike CoDi’s cost estimator. Since CoDi and PCD act at inference time, they are complementary to amortised coordination: their guidance or projection mechanisms could in principle be applied on top of a learned CMDS control when additional instance-specific constraints must be imposed at sampling time. Diffusion-based multi-agent planning. Diffuser formulates planning as guided generation of trajectories (Janner et al., 2022). Multi-agent extensions such as MADiff learn coordinated behaviour from offline multi-agent trajectories (Zhu et al., 2024). Closer to our data assumptions, Multi-robot Multi-model planning Diffusion (MMD) combines single-agent diffusion priors with constraint-based search and replanning (Shaoul et al., 2025). Our motion experiments similarly begin from frozen component priors, but learn coordination separately from component generation. In PointMaze, we coordinate copies of a single-agent trajectory prior across start–goal instances; in KUKA, we coordinate copies of a released single-arm energy-based prior for multi-arm planning; and in the MDM experiment, we coordinate copies of a released text-conditioned single-person motion diffusion prior to generate collision-aware multi-person motion. Across these settings, the component models are not trained on the corresponding multi-agent task, while coordination is amortised into a reusable joint control rather than resolved solely through instance-specific search or guidance. Inference-time steering and compositional posterior inference. Diffusion posterior sampling (DPS) conditions pretrained diffusion models on observations through approximate likelihood guidance evaluated using denoised estimates (Chung et al., 2023). FK-Steering instead uses reward-dependent particle reweighting and resampling during generation (Singhal et al., 2025); its particles represent alternative complete candidates, rather than the components of one assembled output. Our source-separation experiment places the observation model after an additive assembly and performs inference over the constituent sources. We compare against DPS adapted to the same product prior, while learning observation-dependent joint feedback under a path-action penalty. Amortisation is compatible with additional test-time information: our Tweedie reward-gradient parameterisation retains explicit reward-gradient features within the learned control. The test-time coordination mechanisms of PCD (Luan et al., 2025) and CoDi (Peters et al., 2026) are closely related to the cooperative DPS (CDPS) baseline introduced in earlier work (Barbano et al., 2026). CDPS replaces a learned cooperative control with per-step guidance obtained from a joint cost evaluated on the agents’ Tweedie estimates. PCD explicitly derives an analogous posterior-sampling variant, in which a coupling cost evaluated on Tweedie-denoised component estimates is differentiated through each diffusion process. CoDi follows the same general principle of augmenting the product of single-agent diffusion scores with an online joint cost-guidance term, but estimates this correction gradient-free. These methods therefore provide non-amortised, inference-time alternatives to CMDS, whereas CMDS learns a reusable joint feedback control across task instances. Stochastic control and amortised diffusion steering. Optimal-control perspectives connect diffusion generation and reward-based fine-tuning to controlled dynamics and relative-entropy regularisation (Berner et al., 2024; Uehara et al., 2024). Stochastic optimal control matching and adjoint matching develop regression-based approaches for learning such controls (Domingo-Enrich et al., 2024; Domingo-Enrich et al., 2025). DEFT learns a generalised h-transform correction while keeping the pretrained diffusion model frozen (Denker et al., 2024), and relative trajectory balance amortises posterior sampling through a path-space balance objective (Venkatraman et al., 2024). Building on these foundations, CMDS coordinates a reference product process through an assembly-level reward and joint-state feedback, with the component generators remaining frozen. Adjoint matching is one learning strategy within this framework; our discrete implementation uses the full controlled-transition adjoint, rather than the lean-adjoint variant of Domingo-Enrich et al. (2025). Appendix B Diffusion and Path-Space Formulation of CMDS Key Takeaways: The pretrained models define an independent reference process. The N component diffusion models form a product reverse process, and because the reference density factorises across agents, its reverse drift acts agent-wise. Cross-agent dependence is introduced only by the coordination control (section B.1). The assembled process has well-defined marginal dynamics, but itself need not be Markov. CMDS controls the full component state X^s X_s, while the assembled state Y^s=φ(X^s) Y_s= ( X_s) is obtained through the assembly map. Its fixed-time law is the pushforward of the joint-component-state law, and its instantaneous marginals satisfy an exact projected Fokker-Planck equation, but Y Y need not be Markovian when φ discards information about dynamics (section B.2). Coordination is a minimum-KL change of controls in path measure. The quadratic control action equals the relative entropy from the independent pretrained product process eq. 34, so CMDS trades assembly reward against the smallest path-space deviation from the component priors (section B.3). The optimal law is reward tilting. The SOC optimum satisfies dℙ^⋆dℙ^(X^⋅)=exp(λr(φ(X^1)))Z(X^0), P P( X_·)= (λ r( ( X_1)))Z( X_0), where the X^0 X_0-dependent normaliser preserves the prescribed initial marginal. In the complete-noising limit, X^1 X_1 is independent of X^0 X_0, and this reduces to the terminal reward tilt in eq. 41. The reward-tilting formulation holds for general assembly-level rewards. In the particularly relevant and interesting case in which r∘φr is non-separable across components, the exponential tilt generally no longer factorises, so the resulting target law introduces intended dependencies between components that have no intended dependencies under the reference product law (section B.4). The change of measure can be transferred to the assembled process. If φ is injective, the assembled path-density ratio can be written directly through φ−1 ^-1. If φ is non-injective, the assembled density ratio is instead obtained by conditionally averaging the component-space likelihood ratio over all trajectories mapping to the same assembly (section B.5). We use the notation summarised in table 5. Particularly, s∈[0,1]s∈[0,1] denotes reverse, generative time; hatted variables denote reverse-time quantities; i∈1,…,Ni∈\1,…,N\ is an agent index; and X^s=(X^s1,…,X^sN)∈ℝD X_s=( X_s^1,…, X_s^N) ^D, with D=NdD=Nd. The fixed assembly map φ:ℝD→ℝm :R^D ^m differs from experiment to experiment, and Y^s=φ(X^s) Y_s= ( X_s). The system-level reward is r and λ>0λ>0 is its strength. We use reference to denote independent sampling from the component reference processes, without additional assembly-level steering. In discrete-time algorithms and implementation descriptions, XℓX_ denotes the reverse-rollout state with its hat and control superscript suppressed; X^1|ℓ X_1 denotes the clean-state estimate (in discrete time). Quantity Notation Variables and processes Abstract component & assembly Xi,X,Y=φ(X)X^i,\;X,\;Y= (X) (Uncontrolled) reference reverse process X^si,X^s,Y^s X_s^i,\; X_s,\; Y_s Generic controlled reverse process X^su,i,X^su,Y^su X_s^u,i,\; X_s^u,\; Y_s^u Parametrised controlled reverse process X^sθ,i,X^sθ,Y^sθ X_s^θ,i,\; X_s^θ,\; Y_s^θ Stop-gradient controlled rollout X^sθ¯,i,X^sθ¯,Y^sθ¯ X_s θ,i,\; X_s θ,\; Y_s θ Forward-time OU experiment Xti,Xtθ,iX_t^i,\;X_t^θ,i Densities and distributions Initial reverse-time noise density p^0 p_0 Initial forward-time data density p0p_0 Terminal density of the reference reverse process p^1 p_1 Terminal forward-time noise density p1p_1 Exact target law of a generic component π Learned terminal law of a generic component π~ π Joint terminal law of the independent pretrained samplers ρ=π~1⊗⋯⊗π~Nρ= π^1 ·s π^N Table 5: Notation convention. Hatted variables denote reverse-time quantities. The superscript u denotes a generic control, while θ denotes the process induced by the parametrised control UθU_θ. For the continuous-time process notation, quantities without a control superscript refer to the uncontrolled reference process, unless stated otherwise. In the discrete-time algorithms and implementation descriptions, XℓX_ denotes the rollout state, with the control superscript suppressed when the generating method is clear from the context. Standing assumptions. Throughout the Appendix, we work under the following convenient sufficient assumptions. For each i∈1,…,Ni∈\1,…,N\, the drift (t,xi)↦fti(xi)(t,x^i) f_t^i(x^i) is jointly measurable, locally Lipschitz in xix^i, uniformly on compact time intervals, and has at most linear growth. The diffusion scale σ:[0,1]→[0,∞)σ:[0,1]→[0,∞) is continuous and satisfies σt>0 _t>0 for t∈(0,1)t∈(0,1). The joint forward process admits strictly positive one-time densities ptp_t, and the map p:(0,1)×ℝD→(0,∞)p:(0,1)×R^D→(0,∞), defined by p(t,x):=pt(x)p(t,x):=p_t(x), belongs to C1,2((0,1)×ℝD)C^1,2((0,1)×R^D) and satisfies the integrability conditions required for the time-reversal identities below. B.1 Forward and Reverse Diffusion Processes For clarity, we present the derivation for N component processes, each taking values in ℝdR^d and driven by a common scalar diffusion schedule, so that the full component state is their concatenation, Xt=(Xt1,…,XtN)∈ℝD,D=Nd. X_t=(X_t^1,…,X_t^N) ^D, D=Nd. (9) Under the standing assumptions stated above, we define Xtt∈[0,1]\X_t\_t∈[0,1] by the time-forward SDE in a general form dXt=ft(Xt)dt+σtdWt,Wt is D-dimensional,Xt∈ℝD. X_t=f_t(X_t)\, t+ _t\, W_t, W_t is D-dimensional, X_t ^D. (10) XtX_t is the expanded process corresponding to the concatenation of the N agents, with Xti∈ℝdX_t^i ^d. Here Wt=((Wt1)⊤,…,(WtN)⊤)⊤,W_t=((W_t^1) ,…,(W_t^N) ) , where W1,…,WNW^1,…,W^N are independent d-dimensional Wiener processes, and the initial component states are sampled independently. We have D=N×dD=N× d and Xt=[Xt1XtN],ft(Xt)=[ft1(Xt1)ftN(XtN)]. X_t= bmatrixX^1_t\\ \\ X^N_t bmatrix, f_t(X_t)= bmatrixf^1_t(X^1_t)\\ \\ f_t^N(X^N_t) bmatrix. (11) Consequently, the reference process factorises across agents, and its one-time density satisfies pt(x)=∏i=1Npti(xi).p_t(x)= _i=1^Np_t^i(x^i). By the Fokker–Planck equation, the corresponding forward density ptp_t satisfies ∂tpt(x) _tp_t(x) =−∇x⋅(ft(x)pt(x))+12σt2Δxpt(x). =-\, _x\!·\! (f_t(x)\,p_t(x) )+ 12\, _t^2\, _xp_t(x). (12) Under the standard regularity assumptions ensuring time reversal (Nelson, 1967; Anderson, 1982; Haussmann and Pardoux, 1986), define X^s:=X1−s,p^s:=p1−s,σ^s:=σ1−s∈[0,1]. X_s:=X_1-s, p_s:=p_1-s, σ_s:= _1-s s∈[0,1]. (13) Then X^s X_s solves the reverse-time SDE dX^s X_s =f^s(X^s)ds+σ^sdW¯s,f^s(x):=−f1−s(x)+σ^s2∇xlogp1−s(x), = f_s( X_s)\, s+ σ_s\, W_s, f_s(x):=-\,f_1-s(x)+ σ_s^2\, _x p_1-s(x), (14) where W¯s W_s is a standard D-dimensional Brownian motion with respect to the reverse-time filtration. Since the reference density factorises across agents, the reverse drift also acts component-wise, f^s(x)=[f^s1(x1)f^sN(xN)],f^si(xi)=−f1−si(xi)+σ^s2∇xilogp1−si(xi). f_s(x)= bmatrix f_s^1(x^1)\\ \\ f_s^N(x^N) bmatrix, f_s^i(x^i)=-f_1-s^i(x^i)+ σ_s^2 _x^i p_1-s^i(x^i). Its density satisfies ∂sp^s(x) _s p_s(x) =−∇x⋅(f^s(x)p^s(x))+12σ^s2Δxp^s(x). =-\, _x\!·\! ( f_s(x)\, p_s(x) )+ 12\, σ_s^2\, _x p_s(x). (15) Let φ:ℝD→ℝm :R^D ^m be the fixed assembly map. We assume that φ∈C2(ℝD,ℝm) ∈ C^2(R^D;R^m) and that its first and second derivatives are bounded. This condition includes the linear masking, stacking, and additive assembly maps considered in our experiments. We define Y^s=φ(X^s)∈ℝm. Y_s= ( X_s) ^m. (16) Its marginal law is the push-forward measure μ^sY:=φ#(p^s(x)dx). μ_s^Y:= _\# ( p_s(x)\,dx ). (17) Whenever μ^sY μ_s^Y is absolutely continuous with respect to Lebesgue measure, we denote its density by p^sY p_s^Y. B.2 Controlled Reverse Diffusions and the Assembled Process We control the joint component process, and hence the assembled process, through a normalised drift perturbation. Let u:[0,1]×ℝD→ℝD,(s,x)↦us(x),u:[0,1]×R^D ^D, (s,x) u_s(x), be an admissible Markov control such that the controlled SDE below admits a weak solution and satisfies the integrability conditions. We take the controlled and reference processes to share the same initial law, X^0u∼ν0 X_0^u _0. Then the controlled reverse diffusion is dX^su=[f^s(X^su)+σ^sus(X^su)]ds+σ^sdW¯s, X_s^u= [ f_s( X_s^u)+ σ_su_s ( X_s^u ) ]\, s+ σ_s\, W_s, (18) where W¯ W is a standard D-dimensional Brownian motion under the controlled law ℙ^u P^u. When the law of X^su X_s^u admits a density, denoted p^su p_s^u, this density satisfies ∂sp^su(x)=−∇x⋅([f^s(x)+σ^sus(x)]p^su(x))+12σ^s2Δxp^su(x). _s p_s^u(x)=-\, _x\!·\! ( [ f_s(x)+ σ_su_s(x) ] p_s^u(x) )+ 12\, σ_s^2\, _x p_s^u(x). (19) In practice, the drift perturbation may be parameterised using assembled features, for example as Uθ(x,s)=U~θ(x,φ(x),s),U_θ(x,s)= U_θ(x, (x),s), while remaining a full-state feedback map from ℝDR^D to ℝDR^D. Applying Itô’s formula to Y^su:=φ(X^su) Y_s^u:= ( X_s^u) gives dY^su Y_s^u =[Dxφ(X^su)f^s(X^su)+12σ^s2Δxφ(X^su)+σ^sDxφ(X^su)us(X^su)]ds = [D_x ( X_s^u)\, f_s( X_s^u)+ 12\, σ_s^2\, _x ( X_s^u)+ σ_sD_x ( X_s^u)\,u_s( X_s^u) ]\, s (20) +σ^sDxφ(X^su)dW¯s. + σ_s\,D_x ( X_s^u)\, W_s. (21) Here, Δxφ:=(Δxφ1,…,Δxφm) _x :=( _x _1,…, _x _m) is understood component-wise. For the linear masking, stacking, and additive assembly maps used in our experiments, Δxφ=0 _x =0. Equivalently, introducing b^sφ(x):=Dxφ(x)f^s(x)+12σ^s2Δxφ(x), b_s (x):=D_x (x)\, f_s(x)+ 12\, σ_s^2\, _x (x), (22) we may write dY^su=[b^sφ(X^su)+σ^sDxφ(X^su)us(X^su)]ds+σ^sDxφ(X^su)dW¯s. Y_s^u= [ b_s ( X_s^u)+ σ_sD_x ( X_s^u)\,u_s( X_s^u) ] s+ σ_s\,D_x ( X_s^u)\, W_s. (23) We can also recover the evolution of marginals. For μ^su,Y μ_s^u,Y-almost every y, define the conditional projected coefficients, b¯su(y) b_s^u(y) :=ℙ^u[b^sφ(X^su)+σ^sDxφ(X^su)us(X^su)|Y^su=y], :=E_ P^u\! [ b_s ( X_s^u)+ σ_sD_x ( X_s^u)u_s( X_s^u)\, |\, Y_s^u=y ], (24) Q¯su(y) Q_s^u(y) :=ℙ^u[σ^s2Dxφ(X^su)Dxφ(X^su)⊤|Y^su=y]. :=E_ P^u\! [ σ_s^2D_x ( X_s^u)D_x ( X_s^u) \, |\, Y_s^u=y ]. (25) Then, for test functions ψ∈Cc2(ℝm)ψ∈ C_c^2(R^m), the one-time marginals μ^su,Y=Law(Y^su) μ_s^u,Y=Law( Y_s^u) satisfy dds∫ψ(y)μ^su,Y(dy)=∫[∇yψ(y)⋅b¯su(y)+12Tr(Q¯su(y)Dy2ψ(y))]μ^su,Y(dy), s ψ(y)\, μ_s^u,Y( y)= [ _yψ(y)· b_s^u(y)+ 12Tr\! ( Q_s^u(y)D_y^2ψ(y) ) ] μ_s^u,Y( y), (26) where Dy2ψD_y^2ψ denotes the Hessian of ψ. If μ^su,Y(dy)=p^su,Y(y)dy μ_s^u,Y( y)= p_s^u,Y(y) y, then p^su,Y p_s^u,Y satisfies, in the sense of distributions, ∂sp^su,Y(y)=−∇y⋅(b¯su(y)p^su,Y(y))+12∑j,k=1m∂yj∂yk((Q¯su(y))jkp^su,Y(y)). _s p_s^u,Y(y)=- _y· ( b_s^u(y) p_s^u,Y(y) )+ 12 _j,k=1^m _y_j _y_k (( Q_s^u(y))_jk p_s^u,Y(y) ). (27) Remark 1 (Projected marginal dynamics and the law of the assembly). The eq. 27 characterises only the one-time marginals of Y^u Y^u, and it should be distinguished from the standard Fokker-Planck equation of a Markov diffusion on the assembled space. Even though X^su X_s^u is Markovian, the assembled process Y^su=φ(X^su) Y_s^u= ( X^u_s) need not be Markovian if φ drops information about dynamics. The coefficients b¯su b_s^u and Q¯su Q_s^u defined above average over unresolved component states consistent with the same assembly state y. Under the conditions of Brunick and Shreve (2013, Corollary 3.7), there exists a weak solution of an SDE with drift b¯su b_s^u and covariance Q¯su Q_s^u whose one-time marginals coincide with those of Y^u Y^u. If, moreover, the controlled projected drift b^sφ(x)+σ^sDxφ(x)us(x) b_s (x)+ σ_sD_x (x)u_s(x) and the projected covariance σ^s2Dxφ(x)Dxφ(x)⊤ σ_s^2D_x (x)D_x (x) depend on x only through y=φ(x)y= (x), and the associated martingale problem on the assembled space is well posed, then Y^u Y^u is Markov (Kurtz, 1998, Corollary 3.5 and Remark 3.6). At each fixed s, independently of this Markov question, the law of the assembled state is simply the pushforward μ^su,Y=φ#(p^su(x)dx). μ^u,Y_s= _\#( p^u_s(x) x). (28) For m=Dm=D, if φ is injective and detDφ(x)≠0detD (x)≠ 0 for almost every x∈ℝDx ^D in Lebesgue measure, the law of Y^su Y^u_s admits a density given by p^su,Y(y)=p^su(φ−1(y))|detDφ(φ−1(y))|, p_s^u,Y(y)= p_s^u( ^-1(y))|detD ( ^-1(y))|, (29) for almost every y∈φ(ℝD)y∈ (R^D) with zero density outside φ(ℝD) (R^D). More generally, for m≤Dm≤ D, if the rank of DφD equals m for almost every x, an explicit expression for the density of Y Y can be given by a generalised coarea formula (Negro, 2024) under appropriate conditions but without injectivity. Without these assumptions, eq. 28 still holds, but an explicit expression of the density (with respect to Lebesgue measure) may not exist. B.3 Path-Space Change of Measure and KL Cost Let ℙ^:=ℙ^u=0 P:= P^u=0 and ℙ^u P^u denote the path-space laws on C([0,1],ℝD)C([0,1],R^D) induced by the reference and controlled reverse diffusions, respectively. Throughout this subsection, X X denotes the canonical coordinate process on path space. For completeness, we allow the controlled and reference processes to have potentially different initial marginals, although CMDS uses the common-initial-law setting. Let e0:C([0,1],ℝD) e_0:C([0,1],R^D) ⟶ℝD,e0(ω):=ω0, ^D, e_0(ω):= _0, (30) and define ν0:=(e0)#ℙ^,ν0u:=(e0)#ℙ^u. _0:=(e_0)_\# P, _0^u:=(e_0)_\# P^u. Assume that ν0u≪ν0 _0^u _0, that DKL(ν0u∥ν0)<∞D_KL( _0^u\| _0)<∞, and that Novikov’s condition holds under the reference law: ℙ^[exp(12∫01‖us(X^s)‖2ds)]<∞. _ P [ \! ( 12 _0^1 \|u_s( X_s) \|^2 s ) ]<∞. (31) Girsanov’s theorem (Øksendal, 2003) then gives dℙ^udℙ^(X^⋅) P^u P( X_·) =dν0udν0(X^0)exp(∫01us(X^s)⋅dW¯s−12∫01‖us(X^s)‖2ds), = _0^u _0( X_0) \! ( _0^1u_s( X_s)· W_s- 12 _0^1 \|u_s( X_s) \|^2 s ), (32) where W¯ W denotes the Brownian motion under the reference law ℙ P. In the common case ν0u=ν0 _0^u= _0, the initial-density factor disappears. Taking the expectation of the log-density ratio under the controlled law gives DKL(ℙ^u∥ℙ^) D_KL ( P^u\,\|\, P ) =DKL(ν0u∥ν0)+12ℙ^u[∫01‖us(X^su)‖2ds]. =D_KL ( _0^u\,\|\, _0 )+ 12E_ P^u [ _0^1 \|u_s( X_s^u) \|^2 s ]. (33) In particular, if ν0u=ν0 _0^u= _0, then DKL(ℙ^u∥ℙ^) D_KL ( P^u\,\|\, P ) =12ℙ^u[∫01‖us(X^su)‖2ds]. = 12E_ P^u [ _0^1 \|u_s( X_s^u) \|^2 s ]. (34) For a neural parameterisation using assembled features, the same identities apply with us(x)=Uθ(x,s)=U~θ(x,φ(x),s),u_s(x)=U_θ(x,s)= U_θ(x, (x),s), provided that the corresponding controlled process satisfies the admissibility and integrability conditions above. Finally, define the pathwise assembly map Φ:C([0,1],ℝD) :C([0,1],R^D) ⟶C([0,1],ℝm),(Φ(ω))s:=φ(ωs), C([0,1],R^m), ( (ω) )_s:= ( _s), (35) and the induced assembled path laws Π :=Φ#ℙ^,Π^u:=Φ#ℙ^u. := _\# P, ^u:= _\# P^u. (36) Since ℙ^u≪ℙ P^u P, the data-processing inequality gives DKL(Π^u∥Π^)≤DKL(ℙ^u∥ℙ^). D_KL ( ^u\,\|\, )≤ D_KL ( P^u\,\|\, P ). (37) Thus, the relative-entropy cost of controlling the assembled paths is bounded above by the corresponding cost on the full component path space. B.4 Optimal Coordinated Path and Terminal Laws The idea of the proof of the result in this section mainly follows (Domingo-Enrich et al., 2025, Section 4). Recall that ℙ P and ℙ^u⋆ P^u denote the uncontrolled reference path measure and the optimal controlled path measure, respectively. In addition, we define an evaluation map e1e_1 at time 11, e1:C([0,1],ℝD)→ℝD,e1(ω)=ω1. e_1:C([0,1],R^D) ^D, e_1(ω)= _1. (38) The marginal distributions at time 11 are then defined as ρ⋆:=(e1)#ℙ^u⋆ρ:=(e1)#ℙ^. ρ :=(e_1)_\# P^u ρ:=(e_1)_\# P. (39) Based on these, we can formulate the result mentioned in section 2.2 as the following result. Proposition: The optimal and reference distributions of the full path X X satisfy ℙ^u⋆(dX^)=ℙ^(dX^)exp(λr(φ(X^1)))Z(X^0), P^u ( X)= P( X) (λ r( ( X_1)) )Z( X_0), (40) where Z(X^0)Z( X_0) is a constant only depending on X^0 X_0. Furthermore, if X^1 X_1 is independent of X^0 X_0, then their marginal distributions at time 11 satisfy ρ⋆(dX^1)∝ρ(dX^1)exp(λr(φ(X^1))) ρ ( X_1) ρ( X_1) (λ r( ( X_1)) ) (41) Proof. By the tower property and definition of KL divergence, we can rewrite our SOC objective, DKL(ℙ^u∥ℙ^)−λℙ^u[r(φ(X^1))] D_KL( P^u\| P)- _ P^u [r( ( X_1)) ] (42) = = X^∼ℙ^u[logdℙ^udℙ^(X^)−λr(φ(X^1))] _ X P^u [ P^u P( X)-λ r( ( X_1)) ] (43) = = X^0∼ν0[X^∼ℙ^u|X^0(logdℙ^udℙ^(X^)−λr(φ(X^1)))|X^0] _ X_0 _0 [E_ X P^u|_ X_0 ( P^u P( X)-λ r( ( X_1)) ) | X_0 ] (44) = = X^0∼ν0[DKL(ℙ^u|X^0∥ℙ^|X^0)−λX^∼ℙ^u|X^0[r(φ(X^1))]], _ X_0 _0 [D_KL( P^u|_ X_0\| P|_ X_0)- _ X P^u|_ X_0[r( ( X_1))] ], (45) so the SOC objective can be rewritten as minuX^0∼ν0[DKL(ℙ^u|X^0∥ℙ^|X^0)−λX^∼ℙ^u|X^0[r(φ(X^1u))]] _u \E_ X_0 _0 [D_KL( P^u|_ X_0\| P|_ X_0)- _ X P^u|_ X_0[r( ( X^u_1))] ] \ (46) Let ℙ^|X^0 P|_ X_0 and ℙ^u|X^0 P^u|_ X_0 denote path measures of the uncontrolled process in eq. 14 and the controlled processes in eq. 18 starting from the same X^0 X_0 respectively. By the Girsanov theorem, we have dℙ^|X^0dℙ^u|X^0 P|_ X_0 P^u|_ X_0 =exp(−∫01u(X^su,s)dW¯s−12∫01∥u(X^su,s)∥2ds) = (- _0^1u( X^u_s,s) W_s- 12 ^1_0\|u( X^u_s,s)\|^2 s ) (47) ⇔log(dℙ^|X^0dℙ^u|X^0) ( P|_ X_0 P^u|_ X_0 ) =−∫01u(X^su,s)dW¯s−12∫01∥u(X^su,s)∥2ds, =- _0^1u( X^u_s,s) W_s- 12 ^1_0\|u( X^u_s,s)\|^2 s, (48) then we get DKL(ℙ^u|X^0∥ℙ^|X^0)=X^u∼ℙ^u|X^0[12∫01∥u(X^su,s)∥2ds|X^0] D_KL( P^u|_ X_0\| P|_ X_0)=E_ X^u P^u|_ X_0 [ 12 ^1_0\|u( X^u_s,s)\|^2ds | X_0 ] (49) Define the path measure ℙ^u⋆ P^u by dℙ^u⋆|X^0dℙ^|X^0:=exp(λr(φ(X^1)))Z(X^0),Z(X^0):=X^∼ℙ^|X^0[exp(λr(φ(X^1)))], P^u |_ X_0 P|_ X_0:= (λ r( ( X_1)))Z( X_0), Z( X_0):=E_ X P|_ X_0 [ (λ r( ( X_1))) ], (50) Notice that DKL(ℙ^u|X^0||ℙ^u⋆|X^0) D_KL( P^u|_ X_0|| P^u |_ X_0) =X^∼ℙ^u|X^0[logdℙ^u|X^0dℙ^u⋆|X^0] =E_ X P^u|_ X_0 [ P^u|_ X_0 P^u |_ X_0 ] (51) =X^∼ℙ^u|X^0[logdℙ^u|X^0dℙ^|X^0−λr(φ(X^1))+log(Z(X^0))] =E_ X P^u|_ X_0 [ P^u|_ X_0 P|_ X_0-λ r( ( X_1))+ (Z( X_0)) ] (52) =DKL(ℙ^u|X^0||ℙ^|X^0)−X^∼ℙ^u|X^0[λr(φ(X^1))]+log(Z(X^0)) =D_KL( P^u|_ X_0|| P|_ X_0)-E_ X P^u|_ X_0[λ r( ( X_1))]+ (Z( X_0)) (53) Rearranging terms, we get DKL(ℙ^u|X^0||ℙ^|X^0)−X^∼ℙ^u|X^0[λr(φ(X^1u))]=DKL(ℙ^u|X^0||ℙ^u⋆|X^0)−log(Z(X^0)). D_KL( P^u|_ X_0|| P|_ X_0)-E_ X P^u|_ X_0[λ r( ( X^u_1))]=D_KL( P^u|_ X_0|| P^u |_ X_0)- (Z( X_0)). (54) Since relative entropy is non-negative and vanishes if and only if its arguments coincide, the conditional measure-level objective is uniquely minimised when ℙ^u|X^0=ℙ^⋆|X^0. P^u_| X_0= P _| X_0. Hence, ℙ^u⋆|X^0 P^u _| X_0 is the unique minimiser of the conditional path-space objective. Equation 50 is equivalent to ℙ^u⋆|X^0(dX^)=exp(λr(φ(X^1)))Z(X^0)ℙ^|X^0(dX^) P^u |_ X_0( X)= (λ r( ( X_1)))Z( X_0) P|_ X_0( X) (55) Multiplying both sides of the equation by ν0(X^0) _0( X_0), we get ℙ^u⋆(dX^)=ℙ^u⋆|X^0(dX^)ν0(dX^0)=exp(λr(φ(X^1)))Z(X^0)ℙ^|X^0(dX^)ν0(dX^0)=exp(λr(φ(X^1)))Z(X^0)ℙ^(dX^), aligned P^u ( X)&= P^u |_ X_0( X)\, _0( X_0)\\ &= (λ r( ( X_1)))Z( X_0) P|_ X_0( X)\, _0( X_0)\\ &= (λ r( ( X_1)))Z( X_0) P( X), aligned (56) which finishes the proof of eq. 40. Applying the endpoint map e0,1:C([0,1],ℝD)→ℝD×ℝD,e0,1(ω):=(ω0,ω1),e_0,1:C([0,1],R^D) ^D×R^D, e_0,1(ω):=( _0, _1), to eq. 40 yields (e0,1)#ℙ^u⋆(dx0,dx1) (e_0,1)_\# P^u ( x_0, x_1) =exp(λr(φ(x1)))Z(x0)(e0,1)#ℙ^(dx0,dx1). = \! (λ r\! ( (x_1) ) )Z(x_0)(e_0,1)_\# P( x_0, x_1). (57) Under the assumed independence of X^0 X_0 and X^1 X_1 under the reference law, as in the idealised complete-noising limit, we can split ℙ^(dX^0,dX^1)=ν0(dX^0)ρ(dX^1) P( X_0, X_1)= _0( X_0)ρ( X_1), and then evaluate the expectation on X^0 X_0 to get ρ⋆(dX^1) ρ ( X_1) =X^0∼ν0[exp(λr(φ(X^1)))Z(X^0)ρ(dX^1)] =E_ X_0 _0 [ (λ r( ( X_1)))Z( X_0)ρ( X_1) ] (58) =ρ(dX^1)exp(λr(φ(X^1)))X^0∼ν0[1Z(X^0)] =ρ( X_1) (λ r( ( X_1)) )E_ X_0 _0 [ 1Z( X_0) ] (59) ∝ρ(dX^1)exp(λr(φ(X^1))), ρ( X_1) (λ r( ( X_1)) ), (60) which finishes the proof of eq. 41. Remark 2 (Reward-induced coordination). In the idealised complete-noising limit, where X^0 X_0 and X^1 X_1 are statistically independent, marginalising the optimal controlled path law yields the terminal distribution, ρ⋆(x)∝ρ(x)exp(λr(φ(x))),x∈ℝD,ρ (x) ρ(x)\, \! (λ r\! ( (x) ) ), x ^D, (61) where ρ=π~1⊗⋯⊗π~Nρ= π^1 ·s π^N is the terminal law of the independently sampled pretrained agents. When r∘φr is non-separable across components, this reward tilt breaks the product structure of ρ and introduces the task-specific dependencies required by the assembled system. The learned control provides a dynamic realisation of this change of measure. Rather than explicitly re-weighting complete pretrained trajectories, it steers the joint reverse dynamics towards the reward-tilted terminal law. Relation to probabilistic couplings. A coupling of π~1,…,π~N π^1,…, π^N is a joint law on the product space whose i-th marginal is π~i π^i. The reference product law π~1⊗⋯⊗π~N π^1 ·s π^N is their independent coupling. By contrast, the reward tilt above generally changes both the dependence structure and the component marginals, and the optimal terminal law is therefore not, in general, a coupling of the original pretrained laws. We instead use coupled dynamics to describe the system constructed through a joint control whose component-wise actions may depend on the full multi-agent state. B.5 Pushforward to the Assembled Process Starting from the optimal path-law characterisation established in section B.4, we now examine the corresponding change of measure after pushing the reference and controlled processes through the assembly map φ . This distinction is particularly relevant when φ is non-injective, since multiple component trajectories may correspond to the same assembled trajectory. By similar results discussed in Blessing et al. (2025), and Domingo-Enrich et al. (2024), we already have that, for the objective and terminal cost we considered (u) (u) :=ℙ^u[∫01C(X^su,s)+12‖us(X^su)‖2ds+g(X^1u)] :=E_ P^u [ _0^1C( X^u_s,s)+ 12\|u_s( X^u_s)\|^2\, s+g( X^u_1) ] (62) =ℙ^u[∫0112∥us(X^su)∥2ds−λr(φ(X^1u)),] =E_ P^u [ _0^1 12\|u_s( X^u_s)\|^2\, s-λ r\! ( ( X^u_1) ), ] (63) where C and g denote general running cost and terminal cost, respectively, and we consider zero running cost and take the terminal cost to be negative reward g=−λr∘φg=-λ r in this paper. The parametrised version of the objective expression is eq. 74. If the function g(x)g(x) is regular enough, the Radon–Nikodym derivative of the optimal path measure ℙ^u⋆ P^u is given as dℙ^u⋆dℙ^(X^)=exp(−(X^))Z(X^0), d P^u d P( X)= (-W( X) )Z( X_0), (64) where ℙ P is the path measure of the uncontrolled process X X, Z is a normalizing constant and (X^) ( X) :=∫01C(X^s,s)ds+g(X^1) := _0^1C( X_s,s)\, s+g( X_1) =−λr(φ(X^1)), =-λ r( ( X_1)), Equivalently, their densities give pX^u⋆(x(⋅))=exp(λr(φ(x1)))Z(x0)pX^u=0(x(⋅)),p_ X^u (x(·))= (λ r( (x_1)) )Z(x_0)p_ X^u=0(x(·)), where x(⋅)=x(s)s∈[0,1]x(·)=\x(s)\_s∈[0,1] is a whole trajectory rather than a single point. This means that we can use the above relation to generate samples of X^u⋆ X^u and X X, and then apply φ on those generated samples to get samples of Y^u⋆ Y^u and Y Y respectively. If φ is injective, then we further get d(φ#ℙ^u⋆)d(φ#ℙ^)(Y^) d( _\# P^u )d( _\# P)( Y) =1Z(φ−1(Y^0))exp(λr(Y^1))exp(−∫01C(φ−1(Y^s),s)ds) = 1Z( ^-1( Y_0)) (λ r( Y_1)) (- _0^1C( ^-1( Y_s),s)ds ) (65) =exp(λr(Y^1))Z(φ−1(Y^0)) = (λ r( Y_1))Z( ^-1( Y_0)) (66) where φ#ℙ^u⋆ _\# P^u and φ#ℙ _\# P denote path measures of optimal control Y^u⋆ Y^u and uncontrolled Y Y respectively. They are actually just push-forward measures of ℙ^u⋆ P^u and ℙ P under φ respectively. More generally, without requiring φ to be injective, we have d(φ#ℙ^u⋆)d(φ#ℙ^)(Y^) d( _\# P^u )d( _\# P)( Y) =ℙ^[exp(λr(φ(X^1)))Z(X^0)exp(−∫01C(X^s,s)ds)|φ(X^)=Y^] =E_ P [ (λ r( ( X_1)) )Z( X_0) (- _0^1C( X_s,s)ds ) | ( X)= Y ] (67) =ℙ^[exp(λr(Y^1))Z(X^0)|φ(X^)=Y^] =E_ P [ (λ r( Y_1) )Z( X_0)\, |\, ( X)= Y ] (68) =exp(λr(Y^1))ℙ^[1Z(X^0)|φ(X^)=Y^], = (λ r( Y_1) )E_ P [ 1Z( X_0) | ( X)= Y ], (69) where notations without subscripts denote the whole paths or processes. When φ is injective, the assembled path determines the component path through φ−1(Y^s) ^-1( Y_s). Consequently, the conditional expectation in eq. 69 reduces to 1/Z(φ−1(Y^0))1/Z( ^-1( Y_0)) and recovers eq. 66. Appendix C Adjoint Matching Optimisation Key Takeaways: The adjoint turns the global assembly objective into local learning signals. Starting from the terminal cost gradient, it propagates feedback backwards along the reverse trajectory, accounting for the remaining control cost and cross-agent dependencies introduced by the joint controller (section C.1). We retain the full adjoint, rather than the lean variant. Backward replay differentiates both the controlled transition and the quadratic control cost, retaining all state dependencies not removed by the task-specific detach conventions (section C.2). Discrete matching uses noise-normalised controls and next-state adjoints. At each sampling step, the controller-induced state displacement is normalised by the transition noise scale and matched to the negative next-state adjoint multiplied by that same scale. Rollout states and adjoint targets are held fixed during regression (Algorithm 1). We use reverse, generative time s∈[0,1]s∈[0,1]. In this subsection, f^s f_s denotes the reference reverse drift and σ^s σ_s denotes the corresponding reverse-time diffusion scale. Let X^sθ=(X^sθ,1,…,X^sθ,N),Y^sθ=φ(X^sθ,s), X_s^θ=( X_s^θ,1,…, X_s^θ,N), Y_s^θ= ( X_s^θ,s), (70) where φ:ℝD×[0,1]→ℝm :R^D×[0,1] ^m can be a time-dependent assembly function in general cases. The controlled CMDS process is dX^sθ=[f^s(X^sθ)+σ^sUθ(X^sθ,s)]⏟≕f^θ(X^sθ,s)ds+σ^sdW¯s. X_s^θ= [ f_s( X_s^θ)+ σ_sU_θ( X_s^θ,s) ]_ \, f_θ( X_s^θ,s)\, s+ σ_s\, W_s. (71) Here, Uθ(x,s)=(Uθ1(x,s),…,UθN(x,s))∈ℝD U_θ(x,s)= (U_θ^1(x,s),…,U_θ^N(x,s) ) ^D (72) is the actual full-state normalised control field. The block form of eq. 71 is dX^sθ,i=[f^i(X^sθ,s)+σ^sUθi(X^sθ,s)]ds+σ^sdW¯si,i=1,…,N. X_s^θ,i= [ f^i( X_s^θ,s)+ σ_sU_θ^i( X_s^θ,s) ] s+ σ_s\, W_s^i, i=1,…,N. (73) If the reference component diffusions are independent, then f^i(X^sθ,s)=f^i(X^sθ,i,s) f^i( X_s^θ,s)= f^i( X_s^θ,i,s); the derivation below does not require this simplification. The approximated objective in the neural network is (θ) (θ) =ℙ^θ[∫01C(X^sθ,s)+12‖Uθ(X^sθ,s)‖2ds+g(X^1θ)], =E_ P^θ [ _0^1C( X_s^θ,s)+ 12\|U_θ( X^θ_s,s)\|^2\, s+g( X_1^θ) ], (74) where C is an additional running cost and g=−λr∘φg=-λ r is the terminal cost. In this work, we consider the relatively simpler case φ(x,s)=φ(x) (x,s)= (x) which does not change with time, and treat φ(x,s) (x,s) as φ(x) (x) in the following derivations. C.1 Full Pathwise Adjoint Fix a sampled controlled path X^θ X^θ. The full adjoint corresponds to the gradient of the cost-to-go from X^s X_s as a_s :=∇X^s[∫s1C(X^τθ,τ)+12‖Uθ(X^τθ,τ)‖2dτ+g(X^1θ)]. := _ X_s [ _s^1C( X_τ^θ,τ)+ 12\|U_θ( X^θ_τ,τ)\|^2\, τ+g( X_1^θ) ]. (75) With column-vector adjoints and the Jacobian convention, we have: DxF(x)=(∂xjFi(x))i,j. D_xF(x)= ( _x_jF_i(x) )_i,j. (76) The full adjoint solves ddsas sa_s =−Dxf^θ(X^sθ,s)⊤as−∇x(C(X^sθ,s)+12‖Uθ(X^sθ,s)‖2), =-D_x f_θ( X_s^θ,s) a_s- _x (C( X_s^θ,s)+ 12\|U_θ( X^θ_s,s)\|^2 ), (77) a1 a_1 =∇xg(X^1θ)=−λDxφ(X^1θ)⊤∇yr(Y^1θ). = _xg( X_1^θ)=-λ D_x ( X_1^θ) _yr( Y_1^θ). (78) We can further develop the expression of the Jacobian and the gradient: Dxf^θ(x,s)=Dxf^s(x)+σ^sDxUθ(x,s). D_x f_θ(x,s)=D_x f_s(x)+ σ_sD_xU_θ(x,s). (79) Moreover, ∇x(C(x,s)+12‖Uθ(x,s)‖2)=DxUθ(x,s)⊤Uθ(x,s)+∇xC(x,s), _x (C(x,s)+ 12\|U_θ(x,s)\|^2 )=D_xU_θ(x,s) U_θ(x,s)+ _xC(x,s), (80) where DxUθ⊤UθD_xU_θ U_θ denotes the gradient of 12‖Uθ‖2 12\|U_θ\|^2. Write as=(as1,…,asN)a_s=(a_s^1,…,a_s^N) with asi∈ℝda_s^i ^d. The equivalent block-wise representation of eq. 77 is given as ddsasi sa_s^i =−∑j=1N[Dxif^sj(X^sθ)+σ^sDxiUθj(X^sθ,s)]⊤asj =- _j=1^N [D_x^i f^j_s( X_s^θ)+ σ_sD_x^iU_θ^j( X_s^θ,s) ] a_s^j (81) −∑j=1N[DxiUθj(X^sθ,s)]⊤Uθj(X^sθ,s) - _j=1^N [D_x^iU_θ^j( X_s^θ,s) ] U_θ^j( X_s^θ,s) (82) −∂xiC(X^sθ,s), - _x^iC( X^θ_s,s), (83) with terminal condition a1i=−λDxiφ(X^1θ)⊤∇yr(Y^1θ) a_1^i=-λ D_x^i ( X_1^θ) _yr( Y_1^θ) (84) for each i. C.2 Adjoint Matching Instantiation of CMDS Under the assumptions of Domingo-Enrich et al. (2025) (Proposition 2 and Appendices E.2 and E.3), the adjoint matching loss in eq. 6 and the original objective in eq. 74 share the same gradient with respect to θ. Moreover, in the corresponding control-space formulation, the unique critical point of the adjoint matching loss in eq. 6 is the optimal control for the original objective in eq. 74. Let θ¯≔stopgrad(θ) θ (θ), and sample paths from the frozen controlled process X^θ¯∼ℙ^θ¯. X θ P θ. (85) Let aθ¯a θ solve eq. 77–eq. 78 along X^θ¯ X θ, with all dependence on θ¯ θ stopped. The basic CMDS adjoint matching objective used in our work and shown in the main part of this paper is ℒCMDS-AM(θ)=12ℙ^θ¯[∫01‖Uθ(X^sθ¯,s)+σ^sasθ¯‖2ds]. _ CMDS-AM(θ)= 12E_ P θ [ _0^1 \|U_θ( X_s θ,s)+ σ_sa_s θ \|^2\, s ]. (6) Block-wise, ℒCMDS-AM(θ)=12ℙ^θ¯[∫01∑i=1N‖Uθi(X^sθ¯,s)+σ^sasθ¯,i‖2ds]. _ CMDS-AM(θ)= 12E_ P θ [ _0^1 _i=1^N \|U_θ^i( X_s θ,s)+ σ_sa_s θ,i \|^2\, s ]. (86) The gradients of J can be recovered using the adjoint (Domingo-Enrich et al., 2025). ∇θ(θ)=[∫01DθUθ(X^sθ,s)⊤(Uθ(X^sθ,s)+σ^sas)ds]. _θJ(θ)=E [ _0^1D_θU_θ( X_s^θ,s) (U_θ( X_s^θ,s)+ σ_sa_s )\, s ]. (87) C.2.1 Relation to Lean Adjoint Matching and Agent-Wise Structure The adjoint matching formulation above uses the full adjoint, so it keeps the same gradient with respect to the parameter θ as direct optimisation over the original stochastic control objective. Domingo-Enrich et al. (2025) also proposed a lean adjoint, obtained by removing terms depending on control from the adjoint dynamics, since their expectation equals 00 at the optimum. Consequently, the resulting matching loss has a different gradient from the original SOC objective in general away from the optimum, but the unique optimum is proven to be the same as the optimal control under their continuous control-affine assumptions. In CMDS, it is formulated as a control-affine SOC problem, and thus the same lean adjoint construction can be applied to the continuous time CMDS formulation. Specifically, the lean version of the original full adjoint dynamics (eq. 77) is da~sds=−[Dxf^s(X^sθ)]⊤a~s−∇xC(X^sθ,s), a_s s=- [D_x f_s( X_s^θ) ] a_s- _xC( X_s^θ,s), (88) while a~1 a_1 is defined to be the same as a1a_1 in eq. 78. The corresponding adjoint matching objective is ℒLeanAM(θ)=12ℙ^θ¯[∫01‖Uθ(X^sθ¯,s)+σ^sa~sθ¯‖2ds]. _ LeanAM(θ)= 12E_ P θ [ _0^1 \|U_θ( X_s θ,s)+ σ_s a_s θ \|^2\, s ]. (89) Agent-wise decomposition The simplification is natural for the reference product process. As pretrained agents evolve independently, the state Jacobian of f^s(x) f_s(x) is in the block diagonal form Dxf^s(x)=diag(Dx1f^s1(x1),Dx2f^s2(x2),…,DxNf^sN(xN)),D_x f_s(x)=diag(D_x^1 f^1_s(x^1),D_x^2 f^2_s(x^2),...,D_x^N f^N_s(x^N)), and thus the corresponding lean adjoint in agent-wise form satisfies da~sids=−[Dxif^si(X^sθ,i)]⊤a~si−∇xiC(X^sθ,s). a_s^i s=- [D_x^i f_s^\,i( X_s^θ,i) ] a^i_s- _x^iC( X_s^θ,s). (90) In the main CMDS setting considered in this work, C=0C=0, and thus the backward propagation of a~ a separates among agents after the terminal adjoint has been initialised. Importantly, this does not remove the coordination as a~1i=−λDxiφ(X^1θ)⊤∇yr(Y^1θ). a_1^i=-λ D_x^i ( X_1^θ) _yr( Y_1^θ). (91) Therefore, a precise interpretation is that the assembly reward performs joint credit assignment across agents at the terminal time, and the lean adjoint dynamics subsequently propagate the information assigned to each agent backward through each agent’s own frozen dynamics. Computational Interpretation For factorised reference dynamics, the lean-adjoint propagation only needs agent-wise vectors. If the cost of each component model is fixed, the computational cost of this lean-adjoint backward propagation grows linearly with respect to the number of agents, and can be parallelly computed. On the other hand, the full adjoint additionally differentiates through the joint control UθU_θ, and its block form contains cross-agent terms DxiUθjD_x^iU^j_θ representing dependencies introduced by the coordination network. The lean adjoint matching removes these interactions from the adjoint propagation through the reference dynamics. Note that this does not imply that the overall CMDS computation is necessarily linear in the number of agents N, as the evaluation of the assembly reward and joint control could potentially involve higher-order interactions among different agents. Relation to our implementation For the reasons discussed above, in this work we retain the full adjoint rather than the lean variant. In particular, the discrete DDIM recursion below differentiates the controlled transition and includes the quadratic control-cost derivative, thereby preserving the full cross-agent dependencies that are not detached by the task-specific implementation. A discrete lean CMDS variant could instead propagate adjoints through the frozen base transitions only, in analogy with the lean DDIM construction of Domingo-Enrich et al. (2025). C.2.2 Discrete-time adjoint matching for DDIM sampling We use the rollout convention of Algorithm 1: K is the number of DDIM transitions, ℓ=0,…,K−1 =0,…,K-1 indexes transitions, and ζ0>⋯>ζK−1 _0>·s> _K-1 are their diffusion time-steps. The rollout runs from the initial noise state X0X_0 to the terminal sample XKX_K. Let Xℓ=(Xℓ1,…,XℓN)∈ℝDX_ =(X_ ^1,…,X_ ^N) ^D denote the current realised joint state along the reverse DDIM trajectory, with the control superscript suppressed, and let Yℓ:=φ(Xℓ),Ξℓ=(Ξℓ1,…,ΞℓN)∼(0,ID),Y_ := (X_ ), _ =( _ ^1,…, _ ^N) (0,I_D), with Ξℓi∼(0,Id) _ ^i (0,I_d). For agent i, parameterise the fine-tuned denoiser as ϵθ,ℓi(x,y) _θ, ^i(x,y) :=ϵb,ℓi(xi)+δϵθ,ℓi(x,y),x=(x1,…,xN),y=φ(x), :=ε^i_ b, (x^i)+δ _θ, ^i(x,y), x=(x^1,…,x^N), y= (x), (92) where ϵb,ℓi(xi):=ϵbi(xi,ζℓ)ε^i_ b, (x^i):=ε^i_ b(x^i, _ ). The control parameters are shared across DDIM transitions; the native diffusion timestep ζℓ _ is supplied through the time input rather than indexing a separate parameter vector. In the two-agent implementation, δϵθ,ℓ1=Cθ1(x1,y,x2,ζℓ),δϵθ,ℓ2=Cθ2(x2,y,x1,ζℓ).δ _θ, ^1=C_ _1(x^1,y,x^2, _ ), δ _θ, ^2=C_ _2(x^2,y,x^1, _ ). For each DDIM transition, let α¯src,ℓ α_src, and α¯dst,ℓ α_dst, denote its source and destination cumulative alphas, respectively. Here α¯src,ℓ=α¯ζℓ α_src, = α_ _ , and α¯dst,ℓ=α¯ζℓ+1 α_dst, = α_ _ +1 for ℓ<K−1 <K-1; the terminal destination value is modified as specified below. For η>0η>0, define ςℓ _ :=η[1−α¯dst,ℓ1−α¯src,ℓ(1−α¯src,ℓα¯dst,ℓ)]1/2, :=η [ 1- α_dst, 1- α_src, (1- α_src, α_dst, ) ]^1/2, (93) γℓ _ :=[1−α¯dst,ℓ−ςℓ2]+1/2−(α¯dst,ℓα¯src,ℓ)1/2(1−α¯src,ℓ)1/2. := [1- α_dst, - _ ^2 ]_+^1/2- ( α_dst, α_src, )^1/2 (1- α_src, )^1/2. (94) The component DDIM map is Fℓ(z,ϵ,ξ) F_ (z,ε,ξ) :=(α¯dst,ℓα¯src,ℓ)1/2z+γℓϵ+ςℓξ. := ( α_dst, α_src, )^1/2z+ _ ε+ _ ξ. (95) At the terminal reverse transition ℓ=K−1 =K-1, we use α¯dst,ℓ=(1+α¯src,ℓ)/2 α_dst, =(1+ α_src, )/2, rather than α¯dst,ℓ=1 α_dst, =1. Hence ςℓ>0 _ >0 at that transition as well, and its control cost is included. The base and controlled component updates, evaluated from the same current state and using the same noise, are Xℓ+1b,i X_ +1 b,i :=Fℓ(Xℓi,ϵb,ℓi(Xℓi),Ξℓi), :=F_ \! (X_ ^i,ε^i_ b, (X_ ^i), _ ^i ), (96) Xℓ+1θ,i X_ +1^θ,i :=Fℓ(Xℓi,ϵθ,ℓi(Xℓ,Yℓ),Ξℓi). :=F_ \! (X_ ^i, _θ, ^i(X_ ,Y_ ), _ ^i ). (97) Using the same noise Ξℓi∼(0,Id) _ ^i (0,I_d) in the base and controlled DDIM steps, the one-step displacement induced by the epsilon residual is Xℓ+1θ,i−Xℓ+1b,i X_ +1^θ,i-X_ +1 b,i =γℓδϵθ,ℓi(Xℓ,Yℓ). = _ δ _θ, ^i(X_ ,Y_ ). (98) The normalised discrete control is therefore wθ,ℓi(Xℓ) w_θ, ^i(X_ ) :=1ςℓ(Xℓ+1θ,i−Xℓ+1b,i) := 1 _ (X_ +1^θ,i-X_ +1 b,i ) =γℓςℓδϵθ,ℓi(Xℓ,φ(Xℓ))≈ΔsℓUθi(Xℓ,sℓ), = _ _ δ _θ, ^i (X_ , (X_ ) )≈ s_ \,U_θ^i(X_ ,s_ ), (99) where ςℓ>0 _ >0 denotes the stochastic scale of DDIM transition ℓ . For the continuous-time interpretation, sℓs_ belongs to a generative-time grid 0=s0<⋯<sK=10=s_0<·s<s_K=1, with Δsℓ:=sℓ+1−sℓ>0 s_ :=s_ +1-s_ >0. The equality preceding the approximation is exact; the final relation is the continuous-time interpretation of the discrete control. Consequently, 12∑i=1N‖wθ,ℓi(Xℓ)‖2 12 _i=1^N \|w_θ, ^i(X_ ) \|^2 is the one-step discrete SOC control cost. Define the joint base and controlled DDIM maps by Fℓb(Xℓ,Ξℓ) F_ b(X_ , _ ) :=(Xℓ+1b,1,…,Xℓ+1b,N), := (X_ +1 b,1,…,X_ +1 b,N ), (100) Fℓθ(Xℓ,Ξℓ) F_ ^θ(X_ , _ ) :=(Xℓ+1θ,1,…,Xℓ+1θ,N)=Xℓ+1θ. := (X_ +1^θ,1,…,X_ +1^θ,N )=X_ +1^θ. (101) The base map is a one-step counterfactual used to define the discrete control; only the controlled map advances the sampled trajectory. As in the continuous-time formulation, let θ¯=stopgrad(θ) θ=stopgrad(θ), and let Xℓθ¯ℓ=0K\X_ θ\_ =0^K denote a trajectory generated by the frozen controlled DDIM maps Fℓθ¯F_ θ. Define the terminal cost g(x):=−λr(φ(x)).g(x):=-λ r\! ( (x) ). The full discrete adjoint used by the CMDS objective is initialised at K and propagated for ℓ=K−1,…,0 =K-1,…,0: aKθ¯ a_K θ =∇xg(XKθ¯) = _xg(X_K θ) =−λDxφ(XKθ¯)⊤∇yr(φ(XKθ¯)), =-λ D_x (X_K θ) _yr\! ( (X_K θ) ), (102) aℓθ¯ a_ θ =DxFℓθ¯(Xℓθ¯,Ξℓ)⊤aℓ+1θ¯+∇x[12∑i=1N‖wθ¯,ℓi(x)‖2]|x=Xℓθ¯. =D_xF_ θ (X_ θ, _ ) a_ +1 θ+ . _x [ 12 _i=1^N \|w_ θ, ^i(x) \|^2 ] |_x=X_ θ. (103) Here aℓ+1θ¯a_ +1 θ is the costate of the post-transition state Xℓ+1θ¯X_ +1 θ; it is paired with the action at transition ℓ before the recursion is propagated to aℓθ¯a_ θ. Equivalently, define Δℓθ¯(x,ξ) _ θ(x,ξ) :=Fℓθ¯(x,ξ)−x. :=F_ θ(x,ξ)-x. (104) Then the adjoint recursion can be written as aℓθ¯ a_ θ =aℓ+1θ¯+∇x[⟨aℓ+1θ¯,Δℓθ¯(x,Ξℓ)⟩+12∑i=1N‖wθ¯,ℓi(x)‖2]|x=Xℓθ¯. =a_ +1 θ+ . _x [ a_ +1 θ, _ θ(x, _ ) + 12 _i=1^N \|w_ θ, ^i(x) \|^2 ] |_x=X_ θ. (105) Because the DDIM stochastic term ςℓΞℓ _ _ is additive and independent of the current state, the state derivative in the adjoint recursion is unchanged if Ξℓ _ is set to zero during adjoint re-computation. The discrete CMDS adjoint matching loss is ℒCMDS(θ) _ CMDS(θ) =[∑ℓ∈12∑i=1N‖wθ,ℓi(stopgrad(Xℓθ¯))+ςℓstopgrad(aℓ+1θ¯,i)‖2], =E [ _ 12 _i=1^N \|w_θ, ^i (stopgrad(X_ θ) )+ _ stopgrad (a_ +1 θ,i ) \|^2 ], (106) where ⊆0,…,K−1K \0,…,K-1\ is the set of stochastic DDIM transitions selected for the matching loss. The adjoint recursion itself is evaluated over all DDIM transitions; when K is a strict subset, the matching terms are not importance re-weighted. The implementation evaluates eq. 103 through vector–Jacobian products of the controlled joint map Fℓθ¯F_ θ, while including 12∑i‖wθ¯,ℓi‖2 12 _i\|w_ θ, ^i\|^2 in the adjoint recursion. The complete discrete-time adjoint-matching procedure, including the controlled rollout, backward adjoint replay, and control regression, is summarised in Algorithm 1. It therefore implements the full discrete adjoint of the controlled DDIM map under the task-specific detach conventions, including all non-detached cross-agent derivatives induced by φ and the controls. Note that in our implementation, the matched-transition set K is resampled at each optimisation iteration, while the adjoint recursion is always evaluated over all reverse transitions. Algorithm 1 Adjoint matching for CMDS (discrete time) 1: frozen component denoisers ϵbi _b^i, i=1,…,Ni=1,…,N; parametrised component-wise residual δϵθδ _θ; assembly map φ ; task-dependent context builder κC_κ; terminal reward rκr_κ; reward strength λ; reverse-time grid ζ0>⋯>ζK−1 _0>·s> _K-1; joint controlled reverse maps FℓθF_ ^θ, signed epsilon-displacement coefficients γℓ _ , and stochastic scales ςℓ>0 _ >0; matched transitions ⊆0,…,K−1K \0,…,K-1\; batch size B 2: Sample task conditions κ1,…,κB _1,…, _B # e.g., target dΔxd_ x, observation b, or starts/goals (si,gi)i=1N\(s_i,g_i)\_i=1^N 3: Sample X0i,(b)∼(0,I)X_0^i,(b) (0,I) and independent rollout noise Ξℓi,(b)∼(0,I) _ ^i,(b) (0,I) 4: θ¯←stopgrad(θ) θ (θ) 5: 1. Controlled rollout without gradient tracking 6: for ℓ=0,…,K−1 =0,…,K-1 do 7: for i=1,…,Ni=1,…,N do 8: ϵb,ℓi←ϵbi(Xℓi,ζℓ) _b, ^i← _b^i(X_ ^i, _ ) 9: Compute X^1|ℓi X_1 ^i # Tweedie: X^1|ℓi=Xℓi−1−α¯ζℓϵb,ℓiα¯ζℓ X_1 ^i= X_ ^i- 1- α_ _ \, _b, ^i α_ _ 10: end for 11: X^1|ℓ←(X^1|ℓ1,…,X^1|ℓN) X_1 ← ( X_1 ^1,…, X_1 ^N ) 12: Yℓ←φ(Xℓ),Y^1|ℓ←φ(X^1|ℓ)Y_ ← (X_ ), Y_1 ← ( X_1 ) 13: ℐℓ←κ(Xℓ,X^1|ℓ,Yℓ,Y^1|ℓ,ζℓ)I_ _κ (X_ ,\, X_1 ,\,Y_ ,\, Y_1 ,\, _ ) # The control for agent i depends on the full multi-agent state 14: (δϵθ¯,ℓ1,…,δϵθ¯,ℓN)←δϵθ¯(ℐℓ) (δ _ θ, ^1,…,δ _ θ, ^N )←δ _ θ(I_ ) 15: for i=1,…,Ni=1,…,N do 16: wθ¯,ℓi←γℓςℓδϵθ¯,ℓiw_ θ, ^i← _ _ δ _ θ, ^i 17: Xℓ+1i←Fℓ(Xℓi,ϵb,ℓi+δϵθ¯,ℓi,Ξℓi)X_ +1^i← F_ (X_ ^i,\, _b, ^i+δ _ θ, ^i,\, _ ^i ) # Controlled DDIM rollout; see eqs. 92, 95 and 99 18: end for 19: Store XℓX_ 20: end for 21: 22: 2. Backward discrete-adjoint replay 23: aK←∇XK[−λ∑b=1Brκb(φ(XK(b)))]a_K← _X_K [-λ _b=1^Br_ _b ( (X_K^(b)) ) ] # Terminal costate 24: for ℓ=K−1,…,0 =K-1,…,0 do 25: a~ℓ+1←stopgrad(aℓ+1) a_ +1 (a_ +1) # Destination costate 26: Re-evaluate Fℓθ¯(x,0)F_ θ(x,0) and wθ¯,ℓ(x)w_ θ, (x) at x=stopgrad(Xℓ)x=stopgrad(X_ ), using the task-specific detach conventions # Differentiate all non-detached state routes; keep θ¯ θ fixed 27: aℓ←stopgrad[a~ℓ+1+∇x(⟨a~ℓ+1,Fℓθ¯(x,0)−x⟩+12∑b=1B∑i=1N‖wθ¯,ℓi,(b)(x)‖2)|x=Xℓ]. a_ [ a_ +1+ . _x ( a_ +1,F_ θ(x,0)-x + 12 _b=1^B _i=1^N \|w_ θ, ^i,(b)(x) \|^2 ) |_x=X_ ]. 28: end for 29: 3. Control regression 30: for ℓ∈ do 31: Build ℐℓ=κ(stopgrad(Xℓ),X^1|ℓ,Yℓ,Y^1|ℓ,ζℓ)I_ =C_κ (stopgrad(X_ ), X_1 ,Y_ ,\, Y_1 ,\, _ ) using domain-specific detach conventions 32: (δϵθ,ℓ1,…,δϵθ,ℓN)←δϵθ(ℐℓ) (δ _θ, ^1,…,δ _θ, ^N )←δ _θ(I_ ) 33: for i=1,…,Ni=1,…,N do 34: wθ,ℓi←γℓςℓδϵθ,ℓiw_θ, ^i← _ _ δ _θ, ^i 35: eθ,ℓi←wθ,ℓi+ςℓstopgrad(aℓ+1i)e_θ, ^i← w_θ, ^i+ _ stopgrad (a_ +1^i ) # Adjoint-matching regression residual 36: end for 37: end for 38: ℒAM(θ)←12B∑b=1B∑ℓ∈∑i=1N‖eθ,ℓi,(b)‖2 _AM(θ)← 12B _b=1^B _ _i=1^N \|e_θ, ^i,(b) \|^2 # PointMaze uses the self-calibrated pseudo-Huber variant; see Section D.4.7 39: θ←OptimizerStep(θ,∇θℒAM(θ))θ← OptimizerStep(θ,\, _θL_AM(θ)) Note that in Algorithm 1, we report a generic context builder κ:(Xℓ,X^1|ℓ,Yℓ,Y^1|ℓ,ζℓ)↦ℐℓC_κ:(X_ ,\, X_1 ,\,Y_ ,\, Y_1 ,\, _ ) _ , which is experiment-specific. Thus, we use the basic, full-adjoint form of adjoint matching, discretised directly through the DDIM transitions. Unlike the lean-adjoint variant of Domingo-Enrich et al. (2025), the backward replay differentiates the controlled transition and the quadratic control cost. Appendix D Experiments D.1 Two-Agent OU Coordination: A Stochastic-Control Sanity Check Key Takeaways: Least-action coordination recovers the target law. Across figs. 8 and 9, we compare the learned terminal distribution against the stochastic-control oracle using the 22-Wasserstein distance (W2W_2), maximum mean discrepancy (MMD), and Sinkhorn transport cost (Sink.). Together with the terminal-separation and angular diagnostics, these metrics show whether the control recovers the full reward-tilted distribution, rather than merely satisfying the prescribed distance constraint. Relational structure matters. A rotation-equivariant control preserves the symmetry of the target law, whereas an unconstrained MLP can satisfy the distance constraint while collapsing onto a preferred direction. Constraint satisfaction alone may not be sufficient. A control may achieve the correct terminal separation while remaining far from the oracle distribution. In particular, the angular histograms expose symmetry collapse that is not visible from the distance constraint alone, while W2W_2, MMD, and Sinkhorn transport cost quantify this distributional mismatch. The experiment isolates the control mechanism. The reference dynamics are known and an exact reduced value-function oracle is available, providing a controlled sanity check that the coordination objective learns the intended low-action path-space perturbation. Figure 8: Two-agent SOC toy: discrete adjoint. Two agents start from an OU reference and are rewarded for reaching terminal configurations with a prescribed separation. We compare the (uncontrolled) OU reference, a value-function oracle, an importance-sampled reference (IS), and the learned control. Figure 9: Two-agent SOC toy: adjoint matching. Two agents start from an OU reference and are rewarded for reaching terminal configurations with a prescribed separation. We compare the (uncontrolled) OU reference, a value-function oracle, an importance-sampled reference (IS), and the learned control. To isolate the stochastic-control mechanism underlying our coordination framework, we consider two agents, each evolving in ℝ2R^2 according to an Ornstein–Uhlenbeck (OU) reference process. This experiment is a low-dimensional sanity check in which the reference dynamics are known and a high-accuracy value-function oracle can be computed in a reduced coordinate. We use ordinary forward control time t∈[0,1]t∈[0,1] in this experiment, and therefore omit the reverse-time hats used for diffusion sampling elsewhere in the paper. Let Xt=(Xt1,Xt2)∈ℝ4X_t=(X^1_t,X^2_t) ^4 denote the joint state. The controlled dynamics are: dXti=[−Xti+Uθi(Xt,t)]dt+dWti,i∈1,2, X^i_t= [-X^i_t+U^i_θ(X_t,t) ] t+ W^i_t, i∈\1,2\, where W1W^1 and W2W^2 are independent two-dimensional Brownian motions and the uncontrolled drift −Xti-X^i_t pulls each agent toward the origin. The two agents are coupled only through the terminal relational cost, g(X1)=β(‖X12−X11‖−dT)2,g(X_1)=β (\|X^2_1-X^1_1\|-d_T )^2, which encourages their terminal separation to match a prescribed distance dTd_T. The corresponding stochastic optimal-control objective is J(θ)=ℙθ[g(X1)+α2∫01∑i=12‖Uθi(Xt,t)‖2dt],J(θ)=E_P^θ [g(X_1)+ α2 _0^1 _i=1^2 \|U^i_θ(X_t,t) \|^2 t ], where ℙθP^θ denotes the path law induced by the controlled dynamics, α>0α>0 controls the path-action penalty, and β>0β>0 controls the strength of the terminal constraint. In the uncontrolled process, both agents concentrate near the origin, and their terminal separation is typically substantially smaller than dTd_T. A successful control should therefore separate the agents sufficiently to satisfy the relational constraint while avoiding unnecessary translation of their centre of mass. Reduced value-function oracle. Since the two agents have the same component-wise OU dynamics, their common and relative modes can be decoupled. Since the terminal cost only depends on the relative displacement and their quadratic action is minimised with zero common mode control (i.e., qt=ut1+ut22q_t= u^1_t+u^2_t2), the control problem can be reduced to the relative coordinate Rt≔Xt2−Xt1∈ℝ2.R_t X^2_t-X^1_t ^2. For controls ut1,ut2u^1_t,u^2_t, define the relative control vt≔ut2−ut1.v_t u^2_t-u^1_t. For any fixed vtv_t, the minimum-action decomposition across the two agents is ut1=−12vt,ut2=12vt,u^1_t=- 12v_t, u^2_t= 12v_t, and hence α2(‖ut1‖2+‖ut2‖2)=α4‖vt‖2. α2 (\|u^1_t\|^2+\|u^2_t\|^2 )= α4\|v_t\|^2. The relative controlled process therefore satisfies dRt=(−Rt+vt)dt+2dBt, R_t= (-R_t+v_t ) t+ 2\, B_t, where Bt=(Wt2−Wt1)/2B_t=(W^2_t-W^1_t)/ 2 is a standard two-dimensional Brownian motion. Writing gR(r)≔β(‖r‖−dT)2,g_R(r) β (\|r\|-d_T )^2, the reduced value function is V(t,r)≔infv[gR(R1)+α4∫t1‖vs‖2ds|Rt=r].V(t,r) _vE [g_R(R_1)+ α4 _t^1\|v_s\|^2 s\, |\,R_t=r ]. Its Hamilton–Jacobi–Bellman equation is ∂tV−r⊤∇rV+ΔrV−1α‖∇rV‖2=0,V(1,r)=gR(r). _tV-r _rV+ _rV- 1α\| _rV\|^2=0, V(1,r)=g_R(r). Introducing the desirability function ψ(t,r)≔exp(−1αV(t,r)),ψ(t,r) \! (- 1αV(t,r) ), linearises the HJB equation. Equivalently, by the Feynman–Kac representation, ψ(t,r)=[exp(−1αgR(R1))|Rt=r]=[exp(−βα(‖R1‖−dT)2)|Rt=r],ψ(t,r)=E [ (- 1αg_R(R_1) )\, |\,R_t=r ]=E [ (- βα (\|R_1\|-d_T )^2 )\, |\,R_t=r ], where β/αβ/α matches the λ in the SOC formulation, and V(t,r)=−αlogψ(t,r).V(t,r)=-α ψ(t,r). Under the uncontrolled OU reference process, dRt=−Rtdt+2dBt, R_t=-R_t\, t+ 2\, B_t, so that R1|Rt=r∼(e−(1−t)r,(1−e−2(1−t))I2).R_1 R_t=r (e^-(1-t)r, (1-e^-2(1-t) )I_2 ). Since the terminal cost depends only on ‖R1‖\|R_1\|, the value function is radial, and the conditional distribution of ‖R1‖\|R_1\| is Rice distributed. Consequently, ψ(t,r)ψ(t,r) can be evaluated accurately by one-dimensional quadrature over the terminal radius. Minimising the reduced Hamiltonian gives v⋆(t,r)=−2α∇rV(t,r)=2∇rlogψ(t,r),v (t,r)=- 2α _rV(t,r)=2 _r ψ(t,r), with the corresponding minimum-action split ut1,⋆=−12v⋆(t,Rt),ut2,⋆=12v⋆(t,Rt).u^1, _t=- 12v (t,R_t), u^2, _t= 12v (t,R_t). This defines the radial stochastic-control oracle used in figs. 8 and 9. The HJB construction above directly introduces the optimal change of path measure through ψ(t,r)=exp(−V(t,r)/α)ψ(t,r)= (-V(t, r)/α), which solves the backward Kolmogorov equation of the uncontrolled relative OU process dRt=−Rtdt+2dBt, R_t=-R_t t+ 2 B_t, with terminal condition ψ(1,r)=exp(−gR(r)/α)ψ(1,r)= (-g_R(r)/α). Then, by Itô’s formula, under the uncontrolled law, dlogψ(t,Rt)=2∇rlogψ(t,Rt)⋅dBt−∥∇rlogψ(t,Rt)∥2dt, ψ(t,R_t)= 2 _r ψ(t,R_t)· B_t-\| _r ψ(t,R_t)\|^2 t, and thus ψ(t,Rt)ψ(0,R0)=exp(∫0tv⋆(s,Rs)2⋅dBs−12∫0t‖v⋆(s,Rs)2‖2ds). ψ(t,R_t)ψ(0,R_0)= ( _0^t v (s,R_s) 2· B_s- 12 _0^t \| v (s,R_s) 2 \|^2 s ). By Girsanov’s theorem, the RHS of the above equation is exactly the Radon-Nikodym derivative of the optimally controlled path measure of RtR_t with respect to its uncontrolled counterpart up to time t. Evaluating at t=1t=1, it gives dℙR⋆dℙR(R⋅)=exp(−gR(R1)/α)ψ(0,R0). _R _R(R_·)= (-g_R(R_1)/α)ψ(0,R_0). Since our above formulation has shown that controls are only applied to RtR_t and the common mode is uncontrolled and since g(X1)=gR(R1)g(X_1)=g_R(R_1) in this OU example, substituting these relations back to the two-agent process gives dℙ⋆dℙ(X⋅)=exp(−g(X1)/α)ψ(0,X02−X01). (X_·)= (- g(X_1)/α)ψ(0,X^2_0-X^1_0). The importance-sampling baseline approximates this same change of measure by drawing trajectories from the uncontrolled OU process and weighting each sample proportionally to exp(−g(X1)/α)ψ(0,R0) (-g(X_1)/α)ψ(0,R_0). For the oracle samples, we numerically evaluate the Feynman–Kac representation of the reduced HJB solution using radial quadrature, and sample directly from its implied joint terminal marginal. This avoids the time-discretisation error of an Euler rollout of the optimal control. Learning the control. We train the same feedback-control parameterisation using either direct discrete adjoint (a.k.a., differentiable optimisation of J(θ)J(θ) or differentiable simulations) or an adjoint-matching surrogate (Domingo-Enrich, 2024). For the latter, let θ¯=stopgrad(θ) θ=stopgrad(θ) and sample a trajectory Xθ¯∼ℙθ¯X θ θ without gradient tracking. For this experiment, we train CMDS using the full discrete adjoint of the controlled Euler–Maruyama dynamics. Let h=1/Kh=1/K, tℓ=ℓht_ = h, and Xℓ=Xtℓθ¯X_ =X_t_ θ. Define Fℓ(x)=x+h[−x+Uθ¯(x,tℓ)],cℓ(x)=α2‖Uθ¯(x,tℓ)‖2.F_ (x)=x+h [-x+U_ θ(x,t_ ) ], c_ (x)= α2 \|U_ θ(x,t_ ) \|^2. The joint adjoint satisfies aKθ¯=∇xg(XK),aℓθ¯=DxFℓ(Xℓ)⊤aℓ+1θ¯+h∇xcℓ(Xℓ),ℓ=K−1,…,0.a_K θ= _xg(X_K), a_ θ=D_xF_ (X_ ) a_ +1 θ+h\, _xc_ (X_ ), =K-1,…,0. The parameters θ¯ θ are held fixed during this backward recursion, while the state derivatives of the feedback control and its quadratic running cost are retained. The adjoint-matching update at step ℓ uses −aℓ+1θ¯/α-a_ +1 θ/α as its control target, with both the sampled states and the computed adjoints detached during the matching update. The resulting discrete adjoint-matching surrogate is ℒAMOU(θ,θ¯)=h[∑ℓ=0K−1∑i=12‖Uθi(Xℓ,tℓ)+1αaℓ+1i,θ¯‖2].L_AM^OU(θ; θ)=h\,E\! [ _ =0^K-1 _i=1^2 \|U_θ^i(X_ ,t_ )+ 1αa_ +1^i, θ \|^2 ]. The expectation is over frozen Euler–Maruyama rollouts. Both the sampled states and the computed adjoints are detached during the matching update. Control parameterisation. We consider two feedback-control parameterisations. The first is an unconstrained multilayer perceptron that receives the complete joint state and time, Uθi=Uθi(Xt,t).U^i_θ=U^i_θ(X_t,t). The second explicitly encodes the rotational symmetry of the problem, Uθi(Xt,t)=cθi(‖Xtj−Xti‖2,t)(Xtj−Xti),j≠i.U_θ^i(X_t,t)=c_ _i\! (\|X_t^j-X_t^i\|^2,t )(X_t^j-X_t^i), j≠ i. The two scalar coefficient networks have separate parameters. This equivariant parameterisation is a natural inductive bias because both the reference dynamics and the terminal cost are invariant to a common rotation of the two-agent system. The optimal correction therefore acts along the relative displacement between the agents, pushing them apart when their predicted terminal separation is too small and pulling them together when it is too large. The comparisons in figs. 8 and 9 highlight the importance of this inductive bias. The uncontrolled OU process concentrates both agents near the origin and therefore produces terminal separations substantially below the target. The value-function oracle achieves the prescribed separation while preserving rotational symmetry. The importance-sampling baseline approximates the same terminal tilt but suffers from finite-sample variance. The unconstrained MLP control can substantially reduce the terminal-distance error, but it may select a preferred direction in the plane and thereby collapse the terminal-angle distribution. In contrast, the rotation-equivariant control achieves a terminal separation close to the target while maintaining an approximately uniform angular distribution, substantially closer to the oracle. Thus, terminal-distance accuracy alone is insufficient: a control may satisfy the relational constraint while distorting symmetries and diversity of the target law. This toy problem provides a controlled SOC sanity check in which the component dynamics are known, the global constraint is purely relational, and a reduced value-function oracle is available. The experiment tests whether a learned control can balance satisfaction of a terminal system-level constraint against quadratic path action. It therefore provides a low-dimensional analogue of the least-action principle underlying CMDS: instead of arbitrarily moving samples toward high reward, the control seeks the smallest path-space perturbation required to make independently evolving components satisfy a consistency criterion. A remark on the IS baseline. Despite using 2202^20 proposals per evaluation seed, conditionally normalised IS achieves a pre-resampling effective sample size (ESS) of only 99.6±15.399.6± 15.3, approximately 0.0095%0.0095\% of the proposal budget. The 4,0964,096 terminal pairs obtained by multinomial resampling contain only 343.4±20.6343.4± 20.6 distinct proposals, indicating severe weight concentration. Metrics are evaluated on 1,0241,024 equally weighted terminal pairs per distribution, subsampled without replacement from the endpoint banks, retaining repeated IS endpoints. Against the numerical SOC terminal oracle, IS yields 0.917±0.0350.917± 0.035, 0.0656±0.00750.0656± 0.0075, and 1.103±0.0301.103± 0.030 for W2W_2, MMD, and Sinkhorn transport cost, respectively, versus 0.497±0.0220.497± 0.022, 0.0180±0.00820.0180± 0.0082, and 0.888±0.0110.888± 0.011 for CMDS. Reported uncertainties are sample standard deviations across five independent evaluation banks, with the trained controller held fixed. Thus, at the reported proposal budget, conditionally normalised IS exhibits severe weight concentration, while CMDS provides a close approximation to the SOC target terminal law, with empirical joint-terminal discrepancies comparable to oracle self-comparisons at the reported sample size. D.2 Relational Visual Assembly Key Takeaways: Compositionality helps. When the sampler is factorised according to the object structure, coordination is substantially easier than forcing a single process to learn the full constraint through control alone due to positive inductive bias of the decomposition. Rewards are fragile. Reward-based coordination is only as reliable as the system-level signal used to define it. Poorly specified rewards can be over-optimised or hacked. Least-work coordination. The SOC objective favours assemblies that satisfy the global design while requiring the smallest deviation from the independent pretrained dynamics. Thus CMDS is not only searching for valid component combinations; it is learning a low-action path to valid assemblies. CMDS is sampling high-reward assemblies while paying a quadratic control/action cost that penalises deviation from the pretrained reverse diffusion dynamics. This experiment is not reported in the main paper, but serves as a controlled testbed in which the coordination network can be trained cheaply, allowing us to study the method in detail from both theoretical and computational perspectives. In particular, it lets us isolate the effects of component factorisation, optimisation strategy, and reward design, and examine how different rewards trade off constraint satisfaction, diversity, and the amount of control required to steer the pretrained processes. Setup. Method Optim. scheme ELBO (↑)( ) logZIS Z_IS Valid (↑)( ) N=1N=1 (1-square) Adj. Match. -1303.1 -7.2 0.0% Disc. Adj. -1309.5 -7.2 0.0% N=1N=1 (2-square) Adj. Match. -289.1 -0.7 18.9% Disc. Adj. -173.4 -0.7 25.4% CMDS Adj. Match. -74.3 -1.6 99.2% Disc. Adj. -61.2 -1.6 98.5% Table 6: Classifier-only steering diagnostics. We report logZIS Z_IS as an importance-sampling estimate of the log normalising constant of the reward-tilted target, providing a calibration diagnostic for the strength of the induced tilt. We test two claims central to CMDS: whether a single amortised control can satisfy a family of relational constraints, and whether explicitly factorising the sampler by component simplifies relational steering. We consider one-channel 10×1010× 10 canvases with a white background and dark 2×22× 2 squares. The target distribution consists of valid two-square canvases satisfying the prescribed constraints. We pretrain a single diffusion model G on canvases containing one square, and instantiate two independent frozen copies at sampling time. No samples from the valid two-square distribution are provided to CMDS during either pretraining or coordination. Their controlled terminal states are combined through the fixed left/right assembly Y^1θ=φ(X^1θ,1,X^1θ,2)=ML⊙X^1θ,1+MR⊙X^1θ,2, Y_1^θ= ( X_1^θ,1, X_1^θ,2)=M_L X_1^θ,1+M_R X_1^θ,2, where ML,MR∈0,110×10M_L,M_R∈\0,1\^10× 10 mask the left and right halves and satisfy ML+MR=1M_L+M_R=1. Each component contributes only within its assigned half; its contribution outside that region is masked out. The masks therefore define the components’ regions of influence, but do not ensure that a complete square appears in each visible half or that the two squares satisfy the requested relative separation. Independent sampling can consequently produce plausible single-square components but invalid assemblies. Left-Right Steering. We first evaluate whether the agents can learn an additional drift on top of the pretrained dynamics that guides each square toward its prescribed region of interest. For this initial experiment, we train a simple multilayer perceptron to classify whether a square lies in the target half of the domain. The classifier-only assembly reward is rclf(y)=logσ(fψ(ML⊙y))+logσ(fψ(MR⊙y)),r_clf(y)= σ(f_ψ(M_L y))+ σ(f_ψ(M_R y)), where fψf_ψ denotes the trained classifier, σ(z)=(1+e−z)−1σ(z)=(1+e^-z)^-1 is the logistic sigmoid, and each term scores square presence in its corresponding masked half. Figure 10: Left-right steering overview. Two copies of a pretrained single-square diffusion generator are coordinated through a controlled reverse process and assembled by the fixed masking operator Y^1θ=φ(X^1θ,1,X^1θ,2). Y^θ_1= ( X^θ,1_1, X^θ,2_1). Black filled squares show the controlled terminal assembly Y^1θ Y^θ_1. Red and blue dashed outlines show the paired counterfactual uncontrolled terminal proposals X^11 X^1_1 and X^12 X^2_1, obtained from the same initial noise draw but with the learned control removed. Thus, each dashed outline indicates where the corresponding generator copy would have ended up under the reference product law ℙ P, before reward-based steering of the assembled canvas. The training curves report the terminal reward r(Y^1θ)r( Y^θ_1) and the path-space objective ELBO(θ)=ℙ^θ[λr(Y^1θ)−Aθ],ELBO(θ)=E_ P^θ[λ r( Y^θ_1)-A_θ], where AθA_θ denotes the accumulated control/action cost. The learned controls steer the two locally valid component generators into their assigned left/right regions while remaining close to the pretrained dynamics. The path-action cost remains low because the uncontrolled generator already places substantial mass near valid square configurations, so the control only needs a moderate path-space perturbation to steer samples into high-reward assemblies. Figure 11: Left-right steering (Samples). Samples from the aggregated canvas after training. Black filled squares denote the CMDS assembled output. Red and blue dashed outlines denote the paired uncontrolled terminal proposals from the two generator copies, obtained from the same initial/noise draw but with the learned control removed. Thus, for each sample, the outlines show where that instance would have landed without reward-based steering of the assembled canvas. The learned controls steer each square into its assigned half of the canvas, yielding valid stitched assemblies under the fixed left/right masking operator. In fig. 11, we show samples of the aggregated canvas, illustrating that fine-tuning produces samples that satisfy the prescribed constraints. In both cases, each component model generates a square in its assigned region, so that the final stitched assembly is valid. The output of the first model is masked on the right half of the canvas, while the output of the second model is masked on the left half. We optimise the controls using both the discrete adjoint method (Disc. Adj.) and adjoint matching (Adj. Match.), as described in section 2.1. We compare our approach against two single-process baselines: (i) tilting the same one-square base model used as the component generator in CMDS; and (i) tilting a generator trained directly to sample two 2×22× 2 squares within the 10×1010× 10 canvas. One-square baseline. The failure of the one-square baseline is expected as its pretrained support is misaligned with the target object class. Since the base model represents canvases with a single 2×22× 2 square, producing a valid two-square configuration requires the control to create (or hallucinate) an additional square through tilting alone, rather than merely steering existing component structure. Two-square baseline. The two-square baseline has the correct object count, but not the right factorisation. It can generate two-square canvases, yet the control must still solve the assignment problem of which square should satisfy which part of the left/right and distance reward (below). This creates an ambiguous credit-assignment problem. The reward gradients act on the whole image rather than on separately addressable components. In contrast, CMDS builds the assignment into the sampler by using one component process per square and applying the reward after assembly, so the same global signal induces more targeted, lower-action coordination. Metrics. In tables 6, 7 and 8, we report the path-space evidence lower bound (ELBO) ELBO(u)=ℙ^u[λr(φ(X^1u))−Au],ELBO(u)=E_ P^u\! [λ r\! ( ( X^u_1) )-A_u ], where ℙ^u P^u is the controlled path measure and AuA_u is the accumulated control path-action. The normalizing constant of the reward-tilted target, Z=ℙ^[exp(λr(φ(X^1)))],Z=E_ P\! [ \! (λ r\! ( ( X_1) ) ) ], is estimated independently from uncontrolled samples X^1(j) X_1^(j) which follow the terminal time marginal of ℙ P via logZIS=log(1M∑j=1Mexp(λr(φ(X^1(j))))). Z_IS= ( 1M _j=1^M \! (λ r\! ( ( X_1^(j)) ) ) ). Thus, logZ−ELBO(u) Z-ELBO(u) is the path-space KL divergence from the controlled sampler to the reference path distribution re-weighted by the exponential of the scaled terminal reward and normalised by Z. This unconditional reward tilt differs from the fixed-initial-law SOC optimum unless the conditional normaliser Z(X^0)Z( X_0) is constant. In the numerical diagnostic, logZ Z is replaced by its estimate logZIS Z_IS. Here AθA_θ denotes the quadratic path-action, i.e., the accumulated control cost along a parametrised controlled trajectory. Following Algorithm 1, we write XℓX_ for the controlled rollout state, suppressing its parameter superscript, and ζℓ _ for the native diffusion time-step at transition ℓ . Aθ=12∑ℓ=0K−1∑i=1N‖wθ,ℓi(Xℓ)‖22,A_θ= 12 _ =0^K-1 _i=1^N \|w_θ, ^i(X_ ) \|_2^2, where wθ,ℓiw_θ, ^i is the normalised discrete control associated with component i at DDIM transition ℓ . For multi-component samplers, the norm includes the sum over all controlled processes and pixels. This is the discrete Girsanov control cost appearing in the path-space change of measure from the uncontrolled process ℙ P to the controlled process ℙ^u P^u. Coupled Steering. We impose a constraint that can only be satisfied through coordination. Method Optim. scheme ELBO (↑)( ) logZIS Z_IS Valid (↑)( ) N=1N=1 (1-square) Adj. Match. -1706.8 -22.8 0.0% Disc. Adj. -1565.0 -22.8 0.0% N=1N=1 (2-square) Adj. Match. -1561.7 -4.5 8.3% Disc. Adj. -381.4 -4.5 9.6% CMDS Adj. Match. -156.7 -5.4 97.9% Disc. Adj. -190.9 -5.4 96.8% Table 7: Coupled steering diagnostics. We report logZIS Z_IS as an importance-sampling estimate of the log normalising constant of the reward-tilted target, providing a diagnostic for the strength of the induced tilt. Single-process baselines struggle to satisfy the coupled validity constraint, whereas CMDS coordinates two pretrained one-square generators through the assembled canvas and achieves near-perfect validity with improved ELBO. The assembly-level reward combines square presence with the requested horizontal separation: rdΔx(y)=rclf(y)−α|Δx(y)−dΔx|,r_d_ x(y)=r_clf(y)-α _x(y)-d_ x , where dΔxd_ x is the target separation and α is the distance penalty weight. The achieved separation is Δx(y)=μx(MR⊙y)−μx(ML⊙y), _x(y)= _x(M_R y)- _x(M_L y), with μx _x denoting the horizontal foreground barycentre. This introduces coordination through an analytic distance term computed between the barycentres of the two squares. This term makes each agent’s reward depend on the relative configuration of both squares, encouraging the learned drifts to coordinate rather than solve two independent localisation tasks. Training scheme ELBOdΔx=5_d_ x=5 (↑)( ) logZIS|dΔx=5 . Z_IS |_d_ x=5 Valid (↑)( ) dΔx=3d_ x=3 dΔx=4d_ x=4 dΔx=5d_ x=5 dΔx=6d_ x=6 Avg. Adj. Match. -172.2 -5.4 95.8% 95.4% 97.3% 97.1% 96.4% Disc. Adj. -158.5 -5.4 95.2% 95.7% 97.2% 95.5% 95.9% Table 8: Amortized steering diagnostics. ELBO and logZIS Z_IS are evaluated at dΔx=5d_ x=5. Validity is evaluated at every supported target using 4096 samples per target with matched random seeds across target conditions. Avg. is the macro-average over dΔx∈3,4,5,6d_ x∈\3,4,5,6\. In this setting, we also compare against the two single-process baselines. The failure for the two-square baseline becomes more pronounced for the distance constraint, where validity depends on the relative barycentres of the two squares. A unique control can improve the reward by moving mass in reward-favourable ways, but it does not inherit the component-level separation that makes the desired relation easy to express and control. This failure is therefore an inductive-bias failure, not just a capacity failure. The two-square generator represents valid objects, but its representation does not expose the component assignments needed for least-action steering. Figure 12: Fixed-target coupled steering (dΔx=5d_ x=5). Black squares show the controlled assembly, red and blue hatched squares the matched reference terminal proposals, green arrows the induced corrections. Yellow arrows indicate the achieved horizontal separation Δx x. Figure 13: Amortised coupled steering. Samples from a single pair of agent-specific controls trained jointly over dΔx∈3,4,5,6d_ x∈\3,4,5,6\. Black squares show the controlled assembly, red and blue hatched squares the matched reference terminal proposals, and green arrows the induced corrections. Yellow arrows indicate the achieved horizontal separation Δx x. Amortised coordination across design specifications. To test amortisation, we jointly train two agent-specific controllers with separate parameters, one acting on each frozen copy of the one-square reverse process, over dΔx∈3,4,5,6d_ x∈\3,4,5,6\. The requested separation is represented by a four-way one-hot vector, broadcast over the spatial grid and concatenated to each controller input at every DDIM transition. Each controller outputs an additive residual to its corresponding frozen denoiser’s noise prediction, thereby steering the two component processes jointly under the shared assembly-level reward. As shown in fig. 14, the amortised control steers the matched reference terminal proposals into valid assemblies, while the achieved separations closely match the requested targets. The discrete adjoint-matching formulation is provided in Section C.2.2, and the control implementation is described below. Figure 14: Amortised assembly from copies of a single-square prior. Two frozen copies of a generator pretrained only on one-square canvases are coordinated by a control and combined through a fixed left/right mask. Left: for each requested separation dΔx∈3,4,5,6d_ x∈\3,4,5,6\, red and blue dashed outlines show paired uncontrolled terminal proposals obtained from the same noise, black squares show the controlled assembly, and green arrows show the induced corrections. Dots indicate feasible barycentre locations within the two masked halves. Right: the separation closely matches the requested target dΔxd_ x, with points and error bars reporting the sample mean and standard deviation. Yellow arrows indicate the achieved horizontal separation Δx x. Control implementation. For the squares experiments, we attach two agent-specific convolutional control heads to two frozen copies of the same pretrained one-square denoiser. At each DDIM transition ℓ , the frozen denoiser provides Tweedie estimates X^1|ℓi X_1 ^\,i of the terminal component states, which are assembled as Y^1|ℓ=φ(X^1|ℓ 1,X^1|ℓ 2). Y_1 = ( X_1 ^\,1, X_1 ^\,2). The control head for component i receives the full two-agent configuration, catc[Xℓi,X^1|ℓi,Yℓ,Y^1|ℓ,Xℓj,X^1|ℓj],Yℓ=φ(Xℓ1,Xℓ2),j≠i,cat_c\! [X_ ^i,\, X_1 ^\,i,\,Y_ ,\, Y_1 ,\,X_ ^j,\, X_1 ^\,j ], Y_ = (X_ ^1,X_ ^2), j≠ i, together with the embedding of the native diffusion timestep ζℓ _ . Coordination therefore enters through joint conditioning: although the two heads have separate parameters and produce component-specific corrections, each control depends on both reverse processes and on their current and predicted assembly. Each head outputs an unbounded additive epsilon residual δϵθ,ℓiδ _θ, ^\,i, which is added to the frozen denoiser prediction and induces the normalised DDIM control wθ,ℓiw_θ, ^i. For the amortised variant, both heads are additionally conditioned on the requested separation dΔxd_ x. The Tweedie features are recomputed at every transition and remain differentiable with respect to the component states, while the denoiser parameters remain frozen. D.3 Source Separation Key Takeaways: Joint control improves reconstruction while preserving component structure. In the evaluated circle–square source-separation task, CMDS achieves lower clean-mixture reconstruction error and more accurate labelled source placement than DPS, while retaining the structural coherence of the frozen component priors. DPS has no equivalent control-cost regularisation and tends to enforce the measurement reward more aggressively, often at the expense of the individual components. The control resolves component-specific corrections. For additive mixtures, the measurement-reward gradient is identical with respect to either clean source. Providing this Tweedie-evaluated gradient to the joint controller is important: by combining it with both source states and their denoised estimates, the controller can turn shared measurement feedback into component-specific corrections. Under ambiguity, plausible recovery need not equal ground-truth recovery. For sufficiently ill-posed problems, CMDS can recover a plausible decomposition rather than necessarily the ground-truth one. Under stronger blur, components may preserve valid shapes while swapping their labelled locations and still explain the observation well; DPS tends to exhibit such swaps together with substantially greater component deformation. Source separation seeks to recover two or more latent signals from a combined observation. In many cases, datasets of individual source types and datasets of mixtures may each be available, while paired examples containing a mixture together with its constituent sources are unavailable or difficult to obtain. We investigate whether CMDS can coordinate samplers trained independently on the individual source distributions to explain an observed mixture without requiring paired source–mixture training data. The objective is to recover components whose assembly matches the observation while each component remains consistent with its respective pretrained distribution. Task formalization. We consider N=2N=2 frozen pretrained samplers, G1G^1 and G2G^2, defining circle and square priors π~1 π^1 and π~2 π^2, respectively. The ground-truth component images are single-channel 16×1616× 16 images, Xgt1,Xgt2∈−1,+11×16×16X_gt^1,X_gt^2∈\-1,+1\^1× 16× 16, with background value +1+1 and foreground value −1-1. Their clean additive assembly is Ygt=φ(Xgt):=Xgt1+Xgt2.Y_gt= (X_gt):=X_gt^1+X_gt^2. Observation model. We observe b=Aτ(Ygt),b=A_τ(Y_gt), where Aτ A_τ denotes Gaussian convolution with a 5×55× 5 kernel and blur standard deviation τ∈0.5,1.0.τ∈\0.5,1.0\. No additional observation noise is applied (ε=0 =0). Although the ground-truth sources are binary-valued, the blurred observations and generated sources are continuous-valued. Representative reconstructions under the two blur operators are shown in figs. 16 and 17. Component priors. The circle and square priors are trained independently using 40964096 sampled examples per component and epoch, defining the reference product law ρ=π~1⊗π~2.ρ= π^1 π^2. Each noise-prediction network projects a 3232-dimensional sinusoidal time-step embedding and concatenates it spatially with the noisy image. The resulting tensor is processed by two 3×33× 3, 6464-channel convolutional layers with SiLU activations, a residual four-head spatial self-attention layer, and two further 3×33× 3 convolutional layers producing a single-channel epsilon prediction. Both samplers use a linear 100100-step diffusion schedule, β1=10−4,β100=0.08. _1=10^-4, _100=0.08. They are trained for 10001000 epochs using Adam, batch size 128128, and a learning rate cosine-annealed from 10−310^-3 to 10−510^-5, and are frozen. Jointly controlled source processes. The reference process samples the circle and square independently. CMDS coordinates their reverse processes using a joint feedback control, producing X^1θ=(X^1θ,1,X^1θ,2),Y^1θ=φ(X^1θ)=X^1θ,1+X^1θ,2. X_1^θ= ( X_1^θ,1, X_1^θ,2 ), Y_1^θ= ( X_1^θ)= X_1^θ,1+ X_1^θ,2. At each reverse step, the control outputs a separate additive epsilon-space control for each component process. CMDS training. Following Algorithm 1, we use ℓ for rollout indices and ζℓ _ for the corresponding native diffusion timesteps. At reverse transition ℓ , the control receives the current noisy states Xℓ1,Xℓ2X_ ^1,X_ ^2, their Tweedie estimates X^1|ℓ 1,X^1|ℓ 2 X_1 ^\,1, X_1 ^\,2, the observation b, and the predicted observation b^ℓ=Aτ(X^1|ℓ 1+X^1|ℓ 2). b_ =A_τ\! ( X_1 ^\,1+ X_1 ^\,2 ). It additionally receives the residual b−b^ℓb- b_ , the component-wise reward gradients, i.e., ∇X^1|ℓ 1rb _ X_1 ^\,1r_b and ∇X^1|ℓ 2rb _ X_1 ^\,2r_b, and a sinusoidal embedding of ζℓ _ . It outputs two additive epsilon-space controls, one for each reverse process. The control uses joint multi-scale processing with hidden width 3232 and one output head per component. We train the control by adjoint matching for 10,00010,000 optimisation steps. Each iteration samples 512512 circle–square pairs, initial diffusion states, and rollout noise. Rollouts use 2020 DDIM transitions with η=1η=1, and adjoint matching uses 1515 selected matching transitions. We use Adam with cosine learning-rate decay from 10−410^-4 to 10−610^-6, gradient clipping at norm 1.01.0, and rejection of non-finite updates. An exponential moving average of the control parameters is maintained with decay 0.9950.995. Reward and control cost. The system-level reward is rb(y)=−1d‖Aτ(y)−b‖22,d=162.r_b(y)=- 1d \| A_τ(y)-b \|_2^2, d=16^2. Training uses reward multiplier 10001000 and the uniform quadratic control cost from the CMDS objective. The component-wise reward-gradient inputs are computed exactly through the known observation operator Aτ A_τ. Source-separation control implementation. For source separation, we use a centralised joint control with a shared representation and two source-specific output heads, one for each frozen component diffusion model. At each DDIM transition ℓ , the shared control receives the current noisy states Xℓ1,Xℓ2X_ ^1,X_ ^2, their Tweedie estimates X^1|ℓ1,X^1|ℓ2 X_1 ^1, X_1 ^2, the observation b, and the predicted observation b^ℓ=Aτ(X^1|ℓ1+X^1|ℓ2). b_ = A_τ ( X_1 ^1+ X_1 ^2 ). It is additionally conditioned on the measurement residual b−b^ℓb- b_ , the component-wise reward gradients, and the diffusion-time embedding ζℓ _ . Coordination arises through the shared representation as both output heads have access to both reverse processes, their predicted clean sources, and their joint measurement mismatch before producing source-specific corrections. The two heads output additive epsilon residuals δϵθ,ℓ1δ _θ, ^1 and δϵθ,ℓ2δ _θ, ^2, which modify their respective frozen denoisers and induce the normalised DDIM controls wθ,ℓ1w_θ, ^1 and wθ,ℓ2w_θ, ^2. The reward-gradient features are detached, while the Tweedie features remain differentiable. Baselines. The reference baseline samples the frozen component priors independently, X^11∼π~1,X^12∼π~2, X_1^1 π^1, X_1^2 π^2, and returns their assembly Y^1=X^11+X^12 Y_1= X_1^1+ X_1^2 without conditioning on b. We additionally adapt diffusion posterior sampling (DPS) (Chung et al., 2023) to the product of the two frozen component priors. At each reverse step, DPS evaluates the observation residual through the Tweedie estimates of both sources and applies the resulting component-wise data-fidelity gradients to their reverse updates. Writing X^1|ℓi X_1 ^\,i for the Tweedie estimate of component i at rollout index ℓ , its guidance is computed from ∇Xℓi‖Aτ(X^1|ℓ 1+X^1|ℓ 2)−b‖2,i∈1,2. _X_ ^i \| A_τ\! ( X_1 ^\,1+ X_1 ^\,2 )-b \|_2, i∈\1,2\. In this sense, product-DPS provides a non-amortised sampling-time analogue of coordination. The observation-dependent correction is recomputed for every mixture, whereas CMDS amortises this dependence into a learned joint feedback control. Evaluation protocol. All methods are evaluated on the same 10241024 procedurally generated circle–square instances with batch size 128128. All methods are evaluated against the same ground-truth components Xgt1,Xgt2X_gt^1,X_gt^2 and use matched initial diffusion states; observation-conditioned methods use the same b. The CMDS and matched reference rollouts share identical per-step noise, while DPS uses the same rollout-noise seed under its selected DDIM discretisation. Evaluation metrics. We report three complementary properties of the recovered components. In the following, XKiX_K^i denotes the recovered terminal component i, with the method superscript suppressed, while XgtiX_gt^i denotes the corresponding ground-truth component. We write YK=φ(XK)=XK1+XK2,Ygt=Xgt1+Xgt2.Y_K= (X_K)=X_K^1+X_K^2, Y_gt=X_gt^1+X_gt^2. Clean-mixture reconstruction RMSE measures whether the recovered assembly matches the unblurred ground-truth mixture: RMSErecon=[1d‖YK−Ygt‖22]1/2.RMSE_recon= [ 1d \|Y_K-Y_gt \|_2^2 ]^1/2. This metric evaluates recovery of the assembled mixture but does not determine whether each recovered structure has been assigned to the correct source. Source IoU compares the foreground mask of each recovered component with that of its corresponding labelled ground-truth component: IoUsource=12∑i=12|M(XKi)∩M(Xgti)||M(XKi)∪M(Xgti)|.IoU_source= 12 _i=1^2 |M(X_K^i)∩ M(X_gt^i) | |M(X_K^i)∪ M(X_gt^i) |. The darkness representation and foreground mask are D(x)=1−clip(x,−1,1)2,M(x)=p:D(x)p≥0.5.D(x)= 1-clip(x,-1,1)2, M(x)= \p:D(x)_p≥ 0.5 \. Equivalently, a pixel is classified as foreground when its clipped model value is non-positive. Source IoU equals 11 when both recovered components occupy exactly the correct pixels, and 00 when neither overlaps its corresponding ground-truth component. Shape distortion measures the difference between each recovered source and the nearest valid instance of its labelled prior shape, independently of location. Let 1P_1 contain all 100100 generator-valid circle placements and 2P_2 all 169169 generator-valid square placements. For each source, we define δi=minT∈i‖D(XKi)−T‖22‖T‖22,Distshape=12∑i=12δi. _i= _T _i \|D(X_K^i)-T \|_2^2 \|T \|_2^2, _shape= 12 _i=1^2 _i. where T∈iT _i is a binary foreground template in the darkness representation. Shape distortion is zero when each recovered component is a valid instance of its labelled prior shape at any generator-valid location. It increases when a component is blurred, deformed, missing, or has the wrong shape. Together, reconstruction RMSE, source IoU, and shape distortion measure assembly recovery, labelled source placement, and source-shape validity, respectively. Additional evaluations. The aggregate comparisons over 10241024 instances at τ=0.5τ=0.5 (low blur) and τ=1.0τ=1.0 (high blur) are shown in fig. 15 and fig. 4, respectively. CMDS achieves the lowest reconstruction RMSE and highest source IoU while retaining the low shape distortion of the frozen component priors. Representative reconstructions for the low-blur setting are shown in fig. 16; corresponding results under the stronger blur τ=1.0τ=1.0 are shown in fig. 17. Under stronger blur, the observation contains less information about the locations of the individual components, and the recovered circle and square can therefore exchange their ground-truth locations, as illustrated in the fourth row of fig. 17. Such decompositions preserve plausible component shapes and can produce a similar blurred assembly. The quadratic cost can consequently favour a nearby prior-consistent decomposition over the larger intervention required to recover the exact ground-truth placement. Figure 15: Source separation from a blurred mixture (τ=0.5τ=0.5). Each task contains a ground-truth circle Xgt1X_gt^1 and square Xgt2X_gt^2. We observe the blurred mixture b=A0.5(Ygt)=A0.5(Xgt1+Xgt2)b= A_0.5(Y_gt)= A_0.5(X_gt^1+X_gt^2) and recover the labelled sources. Left: two representative evaluation mixtures. The red and blue hatched shapes behind the CMDS reconstructions are the matched reference terminal sources. Right: results over 10241024 evaluation mixtures for the reference process, DPS, and CMDS. Reconstruction RMSE measures whether the two recovered sources add up to the correct clean mixture. Source IoU measures whether each recovered object occupies the same pixels as its corresponding ground-truth object (↑ ): it is 11 for an exact match and 00 when neither recovered source overlaps its corresponding ground-truth source. The bars show the evaluation-set mean with 95%95\% bootstrap confidence intervals computed from 10,00010,000 resamples. Shape distortion measures whether source 11 still looks like a circle and source 22 like a square, irrespective of their location (↓ ). It is zero for an exact valid shape and increases when an object is blurred, deformed, missing, or has the wrong shape. Figure 16: Representative samples (τ=0.5τ=0.5). Each row represents a different instance of the problem. The columns are: (i) the ground-truth mixture; (i) the observation b; (i) the DPS mixture reconstruction; (iv) the CMDS mixture reconstruction, with the matched reference circle and square shown using red and blue hatching, respectively; (v) the blurred CMDS reconstruction overlaid with the observation b; and (vi–vii) the CMDS estimates of the circle and square sources, respectively. In the measurement overlay, green indicates agreement between the predicted and observed mixtures, red indicates structure present only in the observation, and blue indicates structure present only in the predicted mixture. Figure 17: Representative samples (τ=1.0τ=1.0). Each row represents a different instance of the problem. The columns are: (i) the ground-truth mixture; (i) the observation b; (i) the DPS mixture reconstruction; (iv) the CMDS mixture reconstruction, with the matched reference circle and square shown using red and blue hatching, respectively; (v) the blurred CMDS reconstruction overlaid with the observation b; and (vi–vii) the CMDS estimates of the circle and square sources, respectively. In the measurement overlay, green indicates agreement between the predicted and observed mixtures, red indicates structure present only in the observation, and blue indicates structure present only in the predicted mixture. D.4 PointMaze Collision Coordination Key Takeaways: Multi-agent coordination can be amortised. A single control coordinates up to four frozen single-agent trajectory diffusion models, achieving high collision-free joint success and generalising to unseen start–goal regions. Cross-agent communication must scale. Factoring attention into temporal and cross-agent passes preserves the timing of predicted conflicts while reducing complexity from O(N2H2)O(N^2H^2) to O(NH(N+H))O(NH(N+H)). Tweedie reward gradients improve control. Adding a learned time-dependent scaling of the reward gradient at the Tweedie estimate improves performance, with Pool 2 joint success ranging from 89.8%89.8\% to 99.2%99.2\%. Robust regression stabilises adjoint matching. Squared adjoint matching can be seed-sensitive in harder settings. The self-calibrated pseudo-Huber loss improves mean joint success in six of eight comparisons and reduces variance in the most unstable cases, but does not inherit the gradient identity or critical-point guarantee of squared adjoint matching. This section provides detail on the protocol for the maze collision coordination experiment. D.4.1 Task, State, and Reward We first recall the setup of the task. Agent i generates a length-H trajectory τi=(pi(0),…,pi(H−1))∈ℝH×2 _i=(p_i(0),…,p_i(H-1)) ^H× 2 using the same pretrained single-agent diffusion model G, instantiated independently for each agent. We follow the discrete rollout convention in Algorithm 1. Here, K is the number of reverse diffusion transitions, ℓ∈0,…,K ∈\0,…,K\ indexes rollout states, and transitions ℓ=0,…,K−1 =0,…,K-1 use native diffusion time-steps ζℓ _ . Thus X0X_0 is the initial noise and XKX_K is the terminal sample; the physical waypoint index h∈0,…,H−1h∈\0,…,H-1\ is distinct from ℓ . The assembly operator stacks the N component states, Yℓ=φ(Xℓ)=(Xℓ1,…,XℓN), Y_ = (X_ )=(X_ ^1,…,X_ ^N), (107) where φ is the identity on the concatenated state. We write X^1|ℓ X_1 for the frozen prior’s terminal-state estimate and Y^1|ℓ=φ(X^1|ℓ) Y_1 = ( X_1 ) for its assembly. Start and goal positions condition the reconstructed trajectory, which is learned as a residual modification to a straight-line base path interpolating between start and goal (section D.4.4). Let κ=(si,gi)i=1Nκ=\(s_i,g_i)\_i=1^N denote these fixed conditions and ℛκR_κ the residual-to-world reconstruction, so that Zℓ=ℛκ(Yℓ),Z^1|ℓ=ℛκ(Y^1|ℓ).Z_ =R_κ(Y_ ), Z_1 =R_κ( Y_1 ). Coordination is introduced by a shared control conditioned on these reconstructed joint trajectories. Goal reach is enforced structurally by the reconstruction and is not part of the reward. D.4.2 Reward For an assembled state Y, let pi(h,Y)≔[ℛκ(Y)]i,h∈ℝ2p_i(h;Y) [R_κ(Y) ]_i,h ^2 denote agent i’s reconstructed world-space position at waypoint h. We suppress the dependence on Y below. The reward combines wall avoidance, inter-agent collision avoidance, and penalties on speed, acceleration, and path length. The terminal reward is evaluated at YKY_K. The path penalties use the waypoint grid =0,…,H−1T=\0,…,H-1\. Collision penalties use a denser linearly interpolated grid denseT_dense, with |dense|=5(H−1)|T_dense|=5(H-1) evaluation points. Positions at fractional waypoint indices are obtained by linear interpolation. We write [a]+=maxa,0[a]_+= \a,0\, so collision penalties vanish once the prescribed clearance is satisfied. Wall penalty. Wall avoidance is enforced through the signed distance to the maze geometry. We refer to world space as the two-dimensional maze coordinate system in which the start and goal positions, maze geometry, and reconstructed agent trajectories are expressed. For each agent, we penalise only positions whose clearance from the nearest wall is smaller than the agent radius: rwall(i)(Y)=−λwall∑h∈dense[ragent−sdf(pi(h))]+2. r_wall^(i)(Y)=- _wall _h _dense [r_agent-sdf\! (p_i(h) ) ]_+^2. (108) For a world-space position (p∈ℝ2p ^2), let (sdf(p)sdf(p)) denote its signed Euclidean distance to the boundary of the collision-free maze region, taken to be positive in free space, zero on the boundary, and negative inside an obstacle or outside the maze boundary. The hinge therefore vanishes whenever the trajectory maintains at least one agent radius of wall clearance. Inter-agent collision penalty. Collisions between agents are handled analogously by penalising insufficient pairwise separation along the densified trajectory. We define rpair(Y)=−λpair(N2)∑i<j∑h∈dense[2ragent+m−∥pi(h)−pj(h)∥]+2, r_pair(Y)=- _pair N2 _i<j _h _dense [2r_agent+m-\|p_i(h)-p_j(h)\| ]_+^2, (109) where m is an additional safety margin. The term is zero once two agents are separated by at least 2ragent+m2r_agent+m, and the factor (N2)−1 N2^-1 averages the penalty over agent pairs. Path regularisation. The collision terms constrain feasibility but do not by themselves favour short or smooth trajectories. We therefore include three per-agent regularisers on the raw waypoint grid T: a one-sided penalty on excessive waypoint displacement, a discrete acceleration penalty, and total path length. These are rspeed(i)(Y) r_speed^(i)(Y) =−λspeed∑h=0H−2[∥pi(h+1)−pi(h)∥−vmax]+2, =- _speed _h=0^H-2 [\|p_i(h+1)-p_i(h)\|-v_ ]_+^2, (110) raccel(i)(Y) r_accel^(i)(Y) =−λaccel∑h=1H−2‖pi(h+1)−2pi(h)+pi(h−1)‖2, =- _accel _h=1^H-2 \|p_i(h+1)-2p_i(h)+p_i(h-1) \|^2, (111) rlength(i)(Y) r_length^(i)(Y) =−λlength∑h=0H−2∥pi(h+1)−pi(h)∥. =- _length _h=0^H-2\|p_i(h+1)-p_i(h)\|. (112) The speed term is active only above vmaxv_ , whereas acceleration and path length are penalised throughout the trajectory. Total reward. Each agent receives its own wall and path penalties together with the shared inter-agent penalty: r(i)(Y) r^(i)(Y) =rwall(i)(Y)+rspeed(i)(Y)+raccel(i)(Y) =r_wall^(i)(Y)+r_speed^(i)(Y)+r_accel^(i)(Y) +rlength(i)(Y)+rpair(Y). +r_length^(i)(Y)+r_pair(Y). (113) For a world-space trajectory Z, let rκworld(Z)r_κ^world(Z) denote this reward evaluated directly at positions pi(h)=Zi,hp_i(h)=Z_i,h, so that rκ(Y)=rκworld(ℛκ(Y))r_κ(Y)=r_κ^world(R_κ(Y)). The subscripted penalty weights are distinct from the outer SOC reward strength λ. We use (λwall,λpair,λspeed,λaccel,λlength)=(200,400,0,1,1), ( _wall, _pair, _speed, _accel, _length )=(200,400,0,1,1), with m=0.05m=0.05 and vmax=0.15v_ =0.15. The Tweedie reward-gradient extension (section D.4.5) uses the same reward definition, except that λwall=300 _wall=300. D.4.3 Data splits. We use the medium and large PointMaze layouts (Fu et al., 2020). Each maze is mapped by a binary grid with 0 and 1 entries corresponding respectively to corridors and walls. The continuous PointMaze environment is obtained by fixing each grid entry to a physical size of 1.0×1.01.0× 1.0. Cells are partitioned once, in advance, into disjoint training and test pools: Layout Grid Navigable cells Pool 1 cells Pool 2 cells Medium 8×88× 8 26 18 8 Large 9×129× 12 46 32 14 The Pool 1 and Pool 2 cells are disjoint, spatially interspersed across each maze and fixed across runs. The exact partition of cells is presented in fig. 18. (a) Medium (8×88× 8): 26 navigable cells, comprising 18 training and 8 held-out cells. (b) Large (9×129× 12): 46 navigable cells, comprising 32 training and 14 held-out cells. Figure 18: Fixed PointMaze cell partitions. Coordinates use zero-based (row,column)(row,column) matrix indexing. The colours show the exact partition of cells between Pool 1 and Pool 2. A training sample is obtained by sampling every start and goal cell independently and uniformly from the selected pool, adding continuous within-cell jitter, and retaining the draw only if: 1. Every agent’s uncontrolled path has a minimum length of 4. 2. The independently planned paths contain collisions. 3. Conflict-Based Search (CBS) (Sharon et al., 2015) finds a joint solution within a specified compute budget. We report two held-out splits. Pool 1 contains unseen instances sampled from the training-cell pool; Pool 2 contains instances sampled from the disjoint test-cell pool. Every method and hyperparameter setting is evaluated on the same fixed set of 128128 instances per pool. D.4.4 Frozen Per-Agent Prior Each layout uses a single-agent trajectory diffusion model, pretrained once and frozen throughout coordination training. We adopt the Diffusion Veteran configuration (Lu et al., 2025): a DiT-style denoiser with six adaLN-zero blocks, hidden width 256256, four attention heads, and ≈7.3≈ 7.3M parameters. The model is trained as a 100100-step discrete-time DDPM with a linear schedule and mean-squared ϵε-prediction loss. This configuration reaches near-perfect (≈0.99≈ 0.99) wall feasibility on both layouts. D.4.5 Control Architecture Base control. The pretrained diffusion model represents each trajectory as a residual correction to the straight-line path between the start and goal. Before passing the trajectory to the control network, we reconstruct the corresponding absolute (x,y)(x,y) positions in the maze. Cross-agent attention therefore operates on the agents’ predicted physical positions rather than on the residual representation used by the diffusion model. At reverse transition ℓ , it receives the joint trajectory ZℓZ_ , the reconstructed Tweedie estimate Z^1|ℓ Z_1 , the start–goal conditions κ, and the native diffusion time-step ζℓ _ . The trajectory and Tweedie estimate are concatenated along the channel axis and projected to width 344344, with sinusoidal embeddings of diffusion time and physical waypoint position. An initial temporal self-attention block processes each agent’s trajectory separately. Two axial blocks then alternate temporal self-attention within each agent, cross-agent attention at each shared waypoint, and a feed-forward layer. Both blocks retain the full waypoint resolution and use four attention heads, a feed-forward multiplier of 44, residual connections, and layer normalisation. Cross-agent attention includes a learned geometric bias. For each agent pair, an eight-dimensional feature collects relative position (22), distance (11), forward-differenced relative velocity (22), closing speed (11), and relative goal offset (22). An MLP with dimensions →48\!→\!32\!→\!4 maps these features to an additive bias for each attention head, applied through a custom scaled-dot-product implementation. Separate output heads read the local and communication representations to produce the epsilon residual δϵθ,ℓδ _θ, . Both heads and the final layer of the geometric-bias MLP are zero-initialised, so the base control initially leaves the pretrained sampler unchanged. The network has ∼4.3 4.3M parameters. Rather than applying attention jointly over all N×HN× H agent–waypoint tokens, we factor it along the two natural axes of the trajectory tensor (Ho et al., 2019): temporal attention operates over waypoints within each agent, while cross-agent attention operates over agents at each shared waypoint. This reduces the per-block attention complexity from O(N2H2)O(N^2H^2) to O(NH(N+H))O(NH(N+H)). For H=50H=50 and N=4N=4, this gives approximately 3.73.7 times fewer attention-pair evaluations. Temporal attention is batched over agents, and cross-agent attention is batched over waypoints. Shared weights and attention along the agent axis make the control equivariant to a consistent permutation of agents and their conditions (Lee et al., 2019). Tweedie reward gradient extension. Inspired by Denker et al. (2024); Venkatraman et al. (2024), CMDS+TRG augments the base axial control with the joint reward gradient g^ℓraw=λ⋆∇Zrκworld(Z)|Z=Z^1|ℓ, g_ ^raw=λ . _Zr_κ^world(Z) |_Z= Z_1 , evaluated at the reconstructed Tweedie estimate Z^1|ℓ Z_1 where λ⋆>0λ >0 is a fixed scale, distinct from the outer SOC reward strength λ. This is normalised to unit root-mean-square magnitude and clamped element-wise to [−3,3][-3,3] to give g^ℓ g_ (λ⋆λ cancels under this normalisation, so its value does not affect g^ℓ g_ ). This is combined additively with the base network: δϵθ,ℓ=Nθ(Zℓ,Z^1|ℓ,g^ℓ,κ,ζℓ)−sθ(ζℓ)g^ℓ, δ _θ, =N_θ\! (Z_ , Z_1 , g_ ,κ, _ )-s_θ( _ )\, g_ , (114) where δϵθ,ℓδ _θ, is the additive epsilon residual, inducing the normalised control wθ,ℓ=(γℓ/ςℓ)δϵθ,ℓw_θ, =( _ / _ )δ _θ, as in eq. 99. Here NθN_θ is the same control described above, applied to a widened, three-channel input – the reconstructed noisy state, the reconstructed Tweedie estimate, and g^ℓ g_ , concatenated along the channel axis (so it can condition on the reward gradient rather than only react to its additive contribution) – and sθ(ζℓ)=1+tanh(MLPθ(ζℓ))s_θ( _ )=1+ (MLP_θ( _ )) is a single learned, time-dependent, agent-shared scalar gate (a function of the diffusion time-step only, not of Z^1|ℓ Z_1 or g^ℓ g_ ). The gate MLP is zero-initialised so that sθ(ζℓ)=1s_θ( _ )=1 at the start of training. This adds only ≈1,000≈ 1,000 parameters over the base control (approximately 4.34.3M total). We use this full configuration (both the network conditioning and the gated additive term) throughout; restricting to either component alone, or removing the network’s access to Z^1|ℓ Z_1 or g^ℓ g_ , trades off differently across cells but does not change the qualitative picture and is left to future ablation work. D.4.6 Training Objectives We apply the adjoint matching coordination objective introduced in section 2.1 to the stacked N-agent state. Adjoint matching rolls out the current controlled trajectory along the reverse grid with no gradient tracking. Starting from the terminal cost gradient aK=∇XK[−λrκ(YK)]a_K= _X_K[-λ r_κ(Y_K)], it propagates an adjoint variable backward one DDIM step at a time. This backward pass replays individual controlled transitions rather than retaining the computational graph of the full multi-step sampler. The control is then trained to match the correction implied by the next-state adjoint aℓ+1a_ +1. All headline results use the robust regression variant described in section D.4.7. At low signal-to-noise ratios, the DiT prior can also produce excessively large X^1|ℓ X_1 estimates; following Janner et al. (2022), we clip X^1|ℓ X_1 before converting it back to the equivalent noise prediction used in the control-penalty decomposition. The adjoint matching configuration is summarised below. Setting Value Optimiser steps 15,00015,000 Initial learning rate 10−410^-4 (Medium N=2N=2, Medium N=4N=4, Large N=2N=2); 3×10−43× 10^-4 (Large N=4N=4) Learning-rate schedule Cosine decay to 10−510^-5 EMA decay 0.99950.9995 Clipping Gradient norm 0.50.5, elementwise X^1|ℓ X_1 magnitude 44 Batch size 256256 for N=2N=2, 128128 for N=4N=4 Training instances 40964096 joint instances for N=2N=2; 20482048 for N=4N=4 The table above is CMDS’s recipe. CMDS+TRG converges in fewer steps and uses a correspondingly slower schedule: 7,0007,000 optimiser steps, initial learning rate 10−510^-5 with cosine decay to 10−610^-6, and reward wall-weight λwall=300 _wall=300. Batch size, training instances, EMA decay, and gradient/X^1|ℓ X_1 clipping are shared between both variants. D.4.7 Robust Adjoint-Matching: pseudo-Huber versus squared ℓ2 _2 Motivation. Adjoint matching is an on-policy stochastic regression problem whose inputs and targets depend on the current control. Because adjoints are propagated through Jacobians evaluated along stochastic, control-dependent trajectories, finite-batch gradient estimates can have high variance despite being unbiased in expectation (Domingo-Enrich et al., 2025). The squared matching loss can amplify this variability because its gradient grows with the adjoint residual. For collision rewards, early trajectories often contain persistent agent collisions or deep wall penetrations, which can dominate minibatch adjoints and make optimisation unstable and strongly seed-dependent. This motivates a modification of adjoint matching’s regression objective. Objective. Using the rollout convention of Algorithm 1, with the frozen-rollout superscript suppressed, we write the joint matching residual as eθ,ℓ≔coli=1N[wθ,ℓi(stopgrad(Xℓ))+ςℓstopgrad(aℓ+1i)]∈ℝD.e_θ, _i=1^N [w_θ, ^i\! (stopgrad(X_ ) )+ _ stopgrad\! (a_ +1^i ) ] ^D. Here D is the dimension of the stacked residual after vectorisation (D=2NHD=2NH for the full H×2H× 2 representation). This is the residual appearing in eq. 106. The normalised control is defined in eq. 99, and the adjoint target is obtained from the terminal condition eq. 102 and recursion eq. 103. To reduce the influence of rare high-norm residuals induced by these recursively propagated targets, we use a pseudo-Huber, or Charbonnier loss (Huber, 1992; Charbonnier et al., 1994; Barron, 2019). For a residual vector e∈ℝDe ^D, let rms(e)=‖e‖/Drms(e)=\|e\|/ D be its root-mean-square magnitude and define ρδℓ(e)≔Dδℓ2(1+‖e‖2Dδℓ2−1). _ _ (e) D _ ^2 ( 1+ \|e\|^2D _ ^2-1 ). (115) The resulting robust matching objective is ℒCMDS-PH(θ)≔[∑ℓ∈ρδℓ(eθ,ℓ)]. _ CMDS-PH(θ) \! [ _ _ _ (e_θ, ) ]. (116) For ‖e‖≪Dδℓ\|e\| D _ , ρδℓ(e)=12‖e‖2+O(‖e‖4Dδℓ2), _ _ (e)= 12\|e\|^2+O\! ( \|e\|^4D _ ^2 ), so ∇e2ρδℓ(0)=I _e^2 _ _ (0)=I and the loss retains the local curvature of squared matching. For ‖e‖/(Dδℓ)→∞\|e\|/( D _ )→∞, it instead satisfies ρδℓ(e)=Dδℓ‖e‖+O(1) _ _ (e)= D _ \|e\|+O(1), and ∇eρδℓ(e) _e _ _ (e) =e1+‖e‖2/(Dδℓ2),‖∇eρδℓ(e)‖≤Dδℓ. = e 1+\|e\|^2/(D _ ^2), \| _e _ _ (e)\|≤ D\, _ . (117) Because both the rollout and adjoint target are stopped in eq. 106, ∇θρδℓ(eθ,ℓ)=Dθwθ,ℓ⊤∇eρδℓ(eθ,ℓ). _θ _ _ (e_θ, )=D_θw_θ, _e _ _ (e_θ, ). Thus, pseudo-Huber bounds the influence of an individual target in residual space, although the parameter gradient can still be amplified by the control Jacobian Dθwθ,ℓD_θw_θ, . Self-calibration of the pseudo-Huber loss. Adjoint residual scales differ substantially across DDIM transitions, so a single global threshold would constrain some steps too aggressively and others too weakly. We first run a short calibration phase with the squared DDIM matching objective eq. 106 and record the per-instance RMS residual rms(eθ,ℓ(b))rms(e_θ, ^(b)) at each transition ℓ . We then set δℓ _ =maxδmin,Qq(rms(eθ,ℓ(b))b∈cal). = \! \ _ ,Q_q\! (\rms(e_θ, ^(b))\_b _cal ) \. (118) Here, QqQ_q is a fixed empirical quantile over the calibration set calD_cal; we freeze δℓ _ for the remainder of training. The calibration hyperparameters are fixed before the multi-seed comparison and shared across mazes and agent counts. The underlying SOC objective remains eq. 74; only the squared regression surrogate eq. 106 is replaced. The gradient identity and critical-point guarantee for the basic squared objective eq. 6, established by Domingo-Enrich et al. (2025), do not automatically extend to eq. 116. This modified regression objective is studied empirically and need not preserve the minimiser of squared adjoint matching; its connection to the original stochastic-control objective remains an open question. Table 9 reports the full squared-ℓ2 _2-versus-pseudo-Huber regression comparison, held-out joint success, mean ± standard deviation across 1010 independent training seeds per (maze, N) cell. All runs use identical hyperparameters tuned for ℓ2 _2-AM, identical control architecture (Section D.4.5), and evaluation protocol. The pseudo-Huber matching loss (115) generally improves mean success and substantially reduces variance in challenging settings, such as the Large maze with 4 agents. Medium (N=2N=2) Medium (N=4N=4) Large (N=2N=2) Large (N=4N=4) Matching loss Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Squared ℓ2 _2 0.940±0.0200.940\, [rgb]0.5,0.5,0.5 ± 0.020 0.884±0.0480.884\, [rgb]0.5,0.5,0.5 ± 0.048 0.806±0.1550.806\, [rgb]0.5,0.5,0.5 ± 0.155 0.667±0.1510.667\, [rgb]0.5,0.5,0.5 ± 0.151 0.833±0.0410.833\, [rgb]0.5,0.5,0.5 ± 0.041 0.792±0.0420.792\, [rgb]0.5,0.5,0.5 ± 0.042 0.723±0.1570.723\, [rgb]0.5,0.5,0.5 ± 0.157 0.606±0.1520.606\, [rgb]0.5,0.5,0.5 ± 0.152 Pseudo-Huber 0.988±0.0120.988\, [rgb]0.5,0.5,0.5 ± 0.012 0.933±0.0320.933\, [rgb]0.5,0.5,0.5 ± 0.032 0.805±0.0980.805\, [rgb]0.5,0.5,0.5 ± 0.098 0.665±0.1690.665\, [rgb]0.5,0.5,0.5 ± 0.169 0.907±0.0370.907\, [rgb]0.5,0.5,0.5 ± 0.037 0.833±0.0450.833\, [rgb]0.5,0.5,0.5 ± 0.045 0.873±0.0270.873\, [rgb]0.5,0.5,0.5 ± 0.027 0.744±0.0360.744\, ± 0.036 Table 9: Adjoint matching training-seed variance and its mitigation. Held-out joint success, mean ± population standard deviation across 1010 independent training seeds per (maze, N) cell. D.4.8 Training Variance We report the final CMDS+TRG training variance. We train CMDS+TRG with 5 different seeds for each setting and report the mean and standard deviation of joint success on each pool. Medium (N=2N=2) Medium (N=4N=4) Large (N=2N=2) Large (N=4N=4) Method Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ Pool 1 ↑ Pool 2 ↑ CMDS+TRG 0.998±0.0030.998\, [rgb]0.5,0.5,0.5 ± 0.003 0.983±0.0150.983\, [rgb]0.5,0.5,0.5 ± 0.015 0.984±0.0050.984\, [rgb]0.5,0.5,0.5 ± 0.005 0.947±0.0060.947\, [rgb]0.5,0.5,0.5 ± 0.006 0.973±0.0090.973\, [rgb]0.5,0.5,0.5 ± 0.009 0.906±0.0150.906\, [rgb]0.5,0.5,0.5 ± 0.015 0.975±0.0060.975\, [rgb]0.5,0.5,0.5 ± 0.006 0.969±0.0050.969\, [rgb]0.5,0.5,0.5 ± 0.005 Table 10: CMDS+TRG training variance. Held-out joint success rate (Pool 1: train-cell instances, Pool 2: instances from entirely held-out maze cells), mean ± population standard deviation across 5 independent training seeds, evaluated on 128 held-out instances per split. D.4.9 Baseline Suite and Protocol Matching We compare against independent reference sampling, Best-of-256256 reward selection, FK-Steering, and DPS test-time reward guidance. All baselines use the same frozen single-agent prior, trajectory representation, reward, agent radius ragent=0.12r_agent=0.12, X^1|ℓ X_1 -clip of 44, and fixed evaluation sets of 128128 instances per split. DPS and FK-Steering are tuned for each task. D.4.10 Qualitative Examples Figures 19I, 19I, 19I and 19IV give 66 held-out examples per (maze, N) cell, with the highest variation of the trajectory before and after introducing the control. I Medium PointMaze (N=2N=2). I Medium PointMaze (N=4N=4). I Large PointMaze (N=2N=2). IV Large PointMaze (N=4N=4). Figure 19: Qualitative maze-coordination examples. For each maze and agent count, the top row is the reference process trajectory (dashed) and the bottom row is the trajectory produced with the trained CMDS control (solid), for the same 66 held-out instances and the same initial noise. D.5 Multi-Arm KUKA Coordination from Frozen Single-Arm Priors Key Takeaways: Frozen energy-based priors can be coordinated. CMDS learns a shared control over independent seven-joint KUKA priors while keeping their pretrained weights fixed. Adjoint training differentiates the reverse dynamics through the learned energy using input Hessian–vector products, evaluated one transition at a time. A two-arm controller generalises without retraining. Trained to coordinate two frozen single-arm priors, CMDS provides zero-shot transfer to shifted or enlarged obstacles, wider robot base spacing, and three to five arms. It achieves higher mean strict success than the evaluated guidance baselines in all these transfer settings. D.5.1 Task, representation, and data The KUKA task follows Luo et al. (2024) and consists of two robotic arms whose single-arm trajectory priors are independently pretrained for obstacle-avoiding motion planning. Arm i generates joint angles qi=(q1i,…,qHi)∈ℝH×7q^i=(q^i_1,…,q^i_H) ^H× 7, with H=56H=56. All arms share a path-progress coordinate, and q1i,qHiq^i_1,q^i_H are fixed to the requested start and goal. The planner generates joint-angle paths without assigning physical times to their waypoints or enforcing joint-velocity limits. Evaluation checks geometric constraints, including joint limits and collisions along trajectories. With the saved prior-normalisation bounds qlo,qhiq_lo,q_hi, the normalised full path is x=2(q−qlo)/(qhi−qlo)−1x=2(q-q_lo)/(q_hi-q_lo)-1, with component-wise operations and an affine inverse. Only the H−2H-2 interior knots are sampled; endpoints are reinserted whenever the full trajectory is reconstructed. We denote the joint native state by X=(X1,…,XN)X=(X^1,…,X^N), where Xi∈ℝ(H−2)×7X^i ^(H-2)× 7 contains arm i’s normalised interior coordinates. For fixed task conditions κ, comprising the endpoints, arm bases, and obstacles, the reconstructed assembly is Y=φ(X)=q=(q1,…,qN)Y= (X)=q=(q^1,…,q^N); the dependence of φ on κ is suppressed. The two-arm state has 14(H−2)≈75014(H-2)≈ 750 free coordinates. For N arms, D=7N(H−2)D=7N(H-2); norms on the native state and control are Euclidean after vectorisation, without averaging over arms or coordinates. The arm bases are positioned at (−0.5,0,0)(-0.5,0,0) and (0.5,0,0)(0.5,0,0) metres, with both bases’ local coordinate axes aligned with the world axes. The scene contains a shared table beneath the arms and five cubic obstacles, each with side length 0.44m0.44\,m and edges aligned with the world axes. Single-arm priors receive obstacle centres expressed in each arm’s local base coordinates; the controller and collision reward use world-frame (i.e. shared coordinate system) forward kinematics and the actual obstacle extents. The pretrained conditioning interface remains fixed, while the joint control observes the transformed scene. Priors. We instantiate the priors from the pretrained checkpoints of Luo et al. (2024). The Dual-arm checkpoint is used only for evaluation and comparison with CMDS† † https://github.com/devinluo27/potential-motion-plan-release. Controller-training and evaluation populations. A layout specifies the obstacle arrangement, while a planning task specifies start and goal configurations for both arms within that layout. The released dataset provides separate training and test layouts. We divide its 2,500 training layouts into 2,250 layouts for learning the controller and 250 layouts reserved for validation. Validation is used to monitor training and select controller settings and guidance strengths. Test layouts are used to report performance. These three groups contain no shared layouts. For controller training, we extract 32 start–goal pairs from each of the 2,250 training layouts, giving 72,000 tasks. Removing tasks whose start or goal configurations violate joint limits or contain collisions leaves 50 thousand training tasks. For validation, we select two tasks from each of 100 of the 250 reserved layouts. For testing, we select two tasks from each of 300 released test layouts. Applying the same endpoint checks leaves 134 validation tasks and 413 test tasks, with the latter spanning 264 layouts. All methods use the same retained tasks, selected before planner evaluation. Collision-free start and goal configurations do not guarantee that a collision-free path exists between them. Zero-shot transfer. Each transformation below is applied to all test configurations. Obstacle perturbations are deterministic per layout and shared across methods. All controller and prior weights and guidance strengths are held fixed. The resulting number of tasks after feasibility filtering is detailed in table 11. Population Construction Evaluation tasks In-domain Released dual-arm scene and endpoints 413 Shifted obstacles Horizontal Gaussian shifts, standard deviation 0.10 m 191 Larger obstacles Each box half-extent multiplied by 1.2 121 Wider bases Base positions moved to x=±0.6x=± 0.6 m 208 Three arms Third base at (0,0.8,0)(0,0.8,0) m, identity rotation 172 Table 11: KUKA evaluation populations. For wider bases, joint-space endpoints and world-frame obstacle positions remain fixed, so world-space end-effector endpoints change. Transfer from two to five arms. CMDS is architecturally compatible with an arbitrary number of arms without retraining. For the arm-count comparison in the right-hand block of table 3, we select twenty start–goal tasks from each of the 300 test layouts, giving 6,000 tasks. Selection is fixed before running any planner. We keep the original two arms and obstacles unchanged, then add arms at (0,0.8,0)(0,0.8,0), (0,−0.8,0)(0,-0.8,0), and (0.8,0.8,0)(0.8,0.8,0) metres, all with the same base orientation as the original arms. Within each layout, we pair consecutive tasks in the fixed selection order. We assign endpoints to the added arms using other two-arm tasks from the same obstacle layout. We pair the twenty selected tasks in their fixed order: the first with the second, the third with the fourth, and so on. For each task, arm three receives the start and goal joint angles of the paired task’s first arm; arm four receives those of its second arm. Arm five receives the first arm’s start and goal joint angles from a third task. We select this third task from all released tasks in the same layout, sorted by task identifier, taking the first entry other than the current task, its paired task, or either of the two tasks used in the original two-arm benchmark. Only endpoint joint angles are reused; trajectories are generated anew for each arm count. Checking joint limits and endpoint collisions leaves 4 thousands 1,5 thousand, 500, and 200 tasks for two, three, four, and five arms, respectively. We compare all arm counts on the same 209 tasks whose endpoints pass these checks in every setting. These results are conditional on endpoint feasibility for all four counts; they do not describe the full retained population at each count or certify that a feasible connecting path exists. D.5.2 Energy-based priors and discrete adjoint matching Energy parameterisation. The released network fψf_ψ defines the energy parameterisation of Du et al. (2023); Luo et al. (2024). For a normalised full path x, diffusion index ζ, and scene conditioning c, the noise predictor is Eψ(x,ζ,c)=12‖fψ(x,ζ,c)‖22,ϵψ(x,ζ,c)=∇xEψ(x,ζ,c). E_ψ(x,ζ,c)= 12\|f_ψ(x,ζ,c)\|_2^2, _ψ(x,ζ,c)= _xE_ψ(x,ζ,c). (119) The prior applies classifier-free guidance with weight ωcfg=2 _cfg=2, which gives ϵψcfg=ϵψ∅+ωcfg(ϵψc−ϵψ∅) _ψ^cfg= _ψ + _cfg( _ψ^c- _ψ ). These are learned trajectory energies, distinct from the geometric collision reward defined below. Reverse transition and control. We use the state and transition-index convention of Algorithm 1, but retain K=10K=10 for the number of DDIM transitions in this subsection, since ncandn_cand denotes the number of initial joint trajectory candidates per task in the KUKA experiments. The rollout runs from the initial noise state X0X_0 to the terminal sample XKX_K, with transitions indexed by ℓ=0,…,K−1 =0,…,K-1 and native diffusion time-steps ζℓ=90−10ℓ _ =90-10 . Let α¯ζ α_ζ denote the checkpoint’s cumulative noise-schedule coefficient at native time-step ζ, and define α¯src,ℓ=α¯ζℓ,α¯dst,ℓ=α¯ζℓ+1,0≤ℓ<K−1,1,ℓ=K−1. α_src, = α_ _ , α_dst, = cases α_ _ +1,&0≤ <K-1,\\ 1,& =K-1. cases The frozen noise prediction is evaluated on each arm’s full normalised path with fixed endpoints, then restricted to its free interior. We write the concatenated prediction as ϵb,ℓ(Xℓ)=(ϵb,ℓi(Xℓi))i=1N _b, (X_ )=( _b, ^i(X_ ^i))_i=1^N, with evaluation at ζℓ _ , scene conditioning, and classifier-free guidance included. The reference transition first clips its clean estimate and recomputes the corresponding epsilon: X^1|ℓclip X_1 ^clip =clip[−1,1](Xℓ−1−α¯src,ℓϵb,ℓ(Xℓ)α¯src,ℓ), =clip_[-1,1] ( X_ - 1- α_src, \, _b, (X_ ) α_src, ), (120) ϵ~b,ℓ(Xℓ) ε_b, (X_ ) =Xℓ−α¯src,ℓX^1|ℓclip1−α¯src,ℓ, = X_ - α_src, \, X_1 ^clip 1- α_src, , ςℓ2 _ ^2 =η21−α¯dst,ℓ1−α¯src,ℓ(1−α¯src,ℓα¯dst,ℓ), =η^2 1- α_dst, 1- α_src, (1- α_src, α_dst, ), μb,ℓ(Xℓ) _b, (X_ ) =α¯dst,ℓX^1|ℓclip+1−α¯dst,ℓ−ςℓ2ϵ~b,ℓ(Xℓ). = α_dst, \, X_1 ^clip+ 1- α_dst, - _ ^2\, ε_b, (X_ ). For the learned-control sampler, η=1η=1, so ςℓ>0 _ >0 for 0≤ℓ<K−10≤ <K-1, whereas ςK−1=0 _K-1=0. For an epsilon correction δϵθ,ℓ(Xℓ)δ _θ, (X_ ), define on the stochastic transitions 0≤ℓ<K−10≤ <K-1, γℓ _ =1−α¯dst,ℓ−ςℓ2−α¯dst,ℓα¯src,ℓ1−α¯src,ℓ, = 1- α_dst, - _ ^2- α_dst, α_src, 1- α_src, , (121) wθ,ℓ(Xℓ) w_θ, (X_ ) =γℓςℓδϵθ,ℓ(Xℓ), = _ _ δ _θ, (X_ ), μℓθ(Xℓ) _ ^θ(X_ ) =μb,ℓ(Xℓ)+ςℓwθ,ℓ(Xℓ), = _b, (X_ )+ _ w_θ, (X_ ), Xℓ+1 X_ +1 =μℓθ(Xℓ)+ςℓΞℓ,Ξℓ∼(0,ID). = _ ^θ(X_ )+ _ _ , _ (0,I_D). At the deterministic final transition, no learned correction is applied: XK=μb,K−1(XK−1),wθ,K−1≡0.X_K= _b,K-1(X_K-1), w_θ,K-1≡ 0. The final control is set to zero by convention, not computed through division by ςK−1 _K-1. Unlike the modified endpoint in section C.2.2, KUKA retains the native deterministic endpoint; it contributes neither control energy nor a matching term. Adjoint matching with energy-based priors. We use the discrete adjoint recursion and matching loss of Section C.2.2. The control cost is averaged over arms, and the collision reward has weight β=40β=40. Adjoints propagate through all ten denoising transitions, while control costs and matching terms involve only the nine stochastic transitions. Since ϵψ=∇xEψ _ψ= _xE_ψ, the required vector–Jacobian products through the noise predictor are Hessian–vector products through the energy. We compute these by locally recomputing each transition with input differentiation enabled and checkpoint parameters frozen. The clean predictions, reward-gradient directions, and auxiliary geometry supplied to the controller are detached during this differentiation. The calibrated pseudo-Huber matching loss is specified in Section D.5.3. D.5.3 Collision reward, controller, and training Differentiable collision reward. We divide each trajectory segment into four subsegments by interpolating the joint angles, then use forward kinematics to compute the positions and orientations of the robot links at each sampled configuration. We approximate each link with four spheres to measure distances between arms and to obstacles and the table‡ ‡ Distances are estimated using a sampled-axis convex-hull approximation with a 2m2\,m offset. . Intended contact between the fixed mounting link and the table is excluded. For clearances d, safety margin m, and activation width δ=0.01mδ=0.01\,m, we define p(d,m)=[m+δ−dδ]+2,P(d,m)=meanp(d,m)+maxp(d,m),p(d;m)= [ m+δ-dδ ]_+^2, P(d;m)=meanp(d;m)+ p(d;m), (122) where [x]+=max(x,0)[x]_+= (x,0). The mean penalises violations across the sampled points, while the maximum emphasises the worst violation for each arm or arm pair. For an assembled trajectory Y=φ(X)=qY= (X)=q, the reward combines inter-arm, obstacle, table, and self-collision penalties: rκ(Y)=−[1(N2)∑i<jP(darmij,m)+1N∑i(2P(dobsi,0)+12P(dtablei,0)+2P(dselfi,0))].r_κ(Y)=- [ 1 N2 _i<jP(d^ij_arm;m)+ 1N _i (2P(d^i_obs;0)+ 12P(d^i_table;0)+2P(d^i_self;0) ) ]. The clearances are evaluated on Y in scene κ; these arguments are suppressed on the right-hand side. Penalties activate within 1 cm of contact; the additional inter-arm margin m is zero in these experiments. Start and goal configurations are fixed exactly. We use no endpoint, path-length, or curvature penalty. The reward and control cost do not guarantee collision-free, smooth, or dynamically feasible motion. Shared control. The controller follows the same implementation detailed in section D.4.5. Inputs include the noisy state, detached clean prediction, normalised reward gradient, endpoint and base conditions, and local kinematic features. During cross-arm attention, the controller uses the relative positions of each pair of arms, their clearance, goals, base positions, and motion to adjust how much attention they give each other. Consequently, the same controller can process an additional arm without changing parameter dimensions. The controller is parameterised using the Tweedie reward gradient, CMDS+TRG; see eq. 114, and uses the pseudo-Huber loss in section D.4.7. Detailed parameters are provided in table 12. Setting Final configuration Trajectory / reverse steps 56 knots / 10 DDIM steps Stochasticity / prior conditioning η=1η=1 / ωcfg=2 _cfg=2 Controller Width 344, 4 heads, 2 axial blocks Feed-forward / pair-bias hidden width 4×4× controller width / 32 Optimiser Adam, (β1,β2)=(0.9,0.999)( _1, _2)=(0.9,0.999), ϵ=10−8ε=10^-8 Updates / batch / microbatch 20,000 / 256 joint queries / 128 Learning rate 400-step warm-up to 10−410^-4, cosine decay to 10−510^-5 EMA / gradient-norm clipping 0.99 / 0.5 Reward coefficient / active matching steps β=40β=40 / all 9 stochastic transitions Robust calibration First 500 updates; 95th percentile; floor 10−310^-3 Primary / supplementary training seed 0 / 1; final-update EMA weights Table 12: KUKA controller-training configuration. Values are taken from the final saved training configuration; the priors remain frozen. D.5.4 Baselines and PBDM refinement Independent and joint priors. Single-arm KUKA samples each arm independently using the same frozen prior as CMDS; Dual KUKA uses the released joint prior. Single-arm sampling uses η=1η=1; Dual KUKA retains its native η=0η=0 sampler. We also evaluated single-arm sampling with η=0η=0, but it did not improve performance. Except for PBDM, methods return the first candidate that passes the common collision and endpoint checks. Direct collision guidance. Guided single-arm and joint priors use the same collision reward as CMDS, with gradient scale 0.6 on the first nine DDIM transitions. The reward gradient is evaluated at the predicted clean path, without backpropagating through the prior. The guidance strength was selected from 0.1,0.3,0.6,1.0,1.7\0.1,0.3,0.6,1.0,1.7\ by validation success across ncand∈1,2,3,4n_cand∈\1,2,3,4\ initial noises, then fixed for all tests. PBDM selection and refinement. PBDM uses the released joint prior and planning procedure (Luo et al., 2024). The release defaults to 20 initial candidates with refinement disabled; our main PBDM results enable its optional refinement with the stated candidate budget ncandn_cand equal to 1 or 4, matching the number of initial noise samples drawn by the other methods. 1. Generate ncandn_cand candidates and return the first accepted by PBDM’s own checker. 2. If none passes, try the first min(ncand,6) (n_cand,6) candidates in order, allowing up to R=5R=5 repair attempts each. Repair partially re-noises a path and applies three DDIM steps. Return the first accepted repaired path, or report failure. 3. Evaluate the returned path with our common strict checker. Success requires passing both checks; strict rejection does not trigger another candidate or repair. The budget ncandn_cand excludes repair attempts, but planning time includes all repair and checking. Single-arm priors with adapted PBDM. This baseline uses independent single-arm priors. During repair, the released partial denoiser runs separately for each arm. Joint segments are accepted using PBDM’s rules, with three repair attempts and three denoising steps per attempt. The adaptation is applied to three to five arms. D.5.5 Strict evaluation, timing, and uncertainty Strict success. An independent PyBullet mesh checker tests inter-arm, obstacle, table, and non-adjacent self collisions. Each trajectory segment has at least eight subdivisions, with joint increments no larger than 0.04 radians. Success requires no detected collisions and finite joint angles within joint limits, with no additional inter-arm clearance margin. Planning time. Each task generates its ncandn_cand candidates as one batch on a GH200 GPU using warm CUDA graphs. Timing includes scene setup, generation, transfer, checking, and any repair through the final decision. It excludes model and asset loading, graph preparation, diagnostics, logging, and clean-up. Figure 20: Synchronized KUKA configurations. Reference (two independent single-arm priors), Dual KUKA, PBDM and CMDS+TRG on the same task at common path progress s∈0,0.135,0.3,1s∈\0,0.135,0.3,1\: shared start, arm-arm collision, obstacle collision, shared goal. Squares mark starts, circles goals, red crosses collisions. Only CMDS+TRG remains collision-free. Figure 21: KUKA poses and inter-arm clearance. Reference (two independent single-arm priors), Dual KUKA, PBDM and CMDS+TRG on the same held-out task at normalized path progress s=0.135s=0.135 (dotted line), with the inter-arm clearance along each path below. Wrist paths fade from start to goal; squares mark starts, circles goals, red crosses collisions at the shown pose. Shaded negative clearance indicates collision; only CMDS+TRG stays positive over the full path. D.6 Goal-Directed Multi-Person Motion Coordination Key Takeaways: Frozen single-person priors can be coordinated into multi-person motion. A single shared control coordinates four independently text-conditioned MDM processes to reach prescribed destinations, avoid inter-person collisions, and maintain kinematic validity, without paired interaction data. Amortisation makes coordination efficient. CMDS achieves 90.0%90.0\% joint success compared with 75.0%75.0\% for PCD++, while requiring 0.540.54 s per assembly versus 145.34145.34 s for PCD++. We coordinate four text-conditioned human motions using a shared control and a frozen single-person prior. People cross a common region towards opposite destinations, requiring collision avoidance and consistent articulated motion. D.6.1 Task, representation, and frozen MDM prior Each person i∈1,…,Ni∈\1,…,N\, with N=4N=4, receives a caption cic_i, a planar start si∈ℝ2s_i ^2, an initial yaw ψi _i, and a destination gi=−sig_i=-s_i. Starts are arranged on a randomly rotated ring with nominal radius 2.147m2.147\,m, radial jitter ±0.15m± 0.15\,m, and angular jitter ±10∘± 10 . Yaw points towards the destination with an additional ±10∘± 10 jitter. Scene conditions remain fixed during generation, and waiting and lateral detours are allowed. A motion contains F=120F=120 frames at 20Hz20\,Hz, represented by 263263 normalised HumanML features per frame. Notation and assembly. We retain the discrete state and indexing convention of Algorithm 1: the joint component state is Xℓ=(Xℓ1,…,XℓN)∈ℝN×F×263X_ =(X_ ^1,…,X_ ^N) ^N× F× 263, the reverse-time grid is ζℓ=K−1−ℓ _ =K-1- , and the generated motion is XKX_K, with K=50K=50. Thus X0X_0 denotes Gaussian noise, rather than a clean motion, and ℓ is distinct from the physical frame index f∈0,…,F−1f∈\0,…,F-1\. We use τ for possibly fractional frame indices on the densified collision-checking grid. Each reverse transition updates the entire motion clip; the terminal reward is evaluated after denoising and can inspect all physical frames. The task condition is κ=(ci,si,gi,ψi)i=1Nκ=\(c_i,s_i,g_i, _i)\_i=1^N, together with the active-person and frame masks. A deterministic geometric decoder Di(⋅,si,ψi)D_i(·;s_i, _i) denormalises person i’s features, reconstructs 2222 joints, and applies the scene transform, enforcing pi(0)=sip_i(0)=s_i by construction. For a fixed task, the assembly retains both the native features and the decoded world geometry: φ(X)≔(X,Jκ(X)),Jκ(X)≔(Di(Xi,si,ψi))i=1N. (X) (X,J_κ(X) ), J_κ(X) (D_i(X^i;s_i, _i) )_i=1^N. (123) We suppress the fixed scene dependence of φ in the notation. Retaining native features is necessary for the rotation, displacement, and contact-consistency costs, which cannot be evaluated from world-space joint positions alone. State variables, noise, and optimisation are restricted to active people and frames; padded coordinates remain zero throughout. For motion-feature tensors, ∥⋅∥2\|·\|_2 denotes the Euclidean norm after vectorisation, without averaging over people, frames, or features. Frozen denoising and reverse transitions. We use the released HumanML MDM checkpoint trained for 5050 diffusion steps, together with its frozen CLIP ViT-B/32 text encoder. Each person’s frozen denoiser is conditioned on its own caption, not on the other people or the scene’s starts and goals, with classifier-free guidance scale 2.52.5. This is a separately trained native 5050-step model, not a respaced 1,0001,000-step checkpoint. Sampling retains its cosine schedule, fixed posterior variance, and unclipped clean predictions. Write αt=1−βt _t=1- _t, α¯t=∏j=0tαj α_t= _j=0^t _j, and α¯−1=1 α_-1=1 for the saved native schedule. For a candidate joint state Z=(Z1,…,ZN)Z=(Z^1,…,Z^N), define ℓi(Zi) _ ^i(Z^i) ≔Zi−1−α¯ζℓϵbi(Zi,ζℓ)α¯ζℓ, Z^i- 1- α_ _ \, _b^i(Z^i, _ ) α_ _ , 0≤ℓ<K, 0≤ <K, (124) ℓ(Z) _ (Z) ≔(ℓ1(Z1),…,ℓN(ZN)), (D_ ^1(Z^1),…,D_ ^N(Z^N) ), K(Z) _K(Z) ≔Z. Z. Thus X^1|ℓ=ℓ(Xℓ) X_1 =D_ (X_ ) is the clean-motion estimate, and Y^1|ℓ=φ(X^1|ℓ) Y_1 = ( X_1 ) is its assembly. MDM predicts the clean state directly; the expression above is its equivalent epsilon parameterisation, with classifier-free guidance included in ϵbi _b^i. The denoising map ℓD_ is distinct from the geometric decoder DiD_i. Although Algorithm 1 uses DDIM, the reverse maps here are MDM’s native DDPM transitions: Fℓb(Xℓ,Ξℓ)≔μb,ℓ(Xℓ)+ςℓΞℓ,ςℓ>0,0≤ℓ<K−1,ςK−1=0.F_ ^b(X_ , _ ) _b, (X_ )+ _ _ , cases _ >0,&0≤ <K-1,\\ _K-1=0.& cases (125) Here μb,ℓ _b, is the frozen classifier-free-guided DDPM posterior mean, acting component-wise. With ϵb(Z,t)=(ϵbi(Zi,t))i=1N _b(Z,t)=( _b^i(Z^i,t))_i=1^N, the native mean and effective variance are μb,ℓ(Z) _b, (Z) =1αζℓ[Z−βζℓ1−α¯ζℓϵb(Z,ζℓ)], = 1 _ _ [Z- _ _ 1- α_ _ _b(Z, _ ) ], (126) ςℓ2 _ ^2 =βζℓ(1−α¯ζℓ−1)1−α¯ζℓ. = _ _ (1- α_ _ -1)1- α_ _ . The final transition has ζK−1=0 _K-1=0 and is deterministic. No denoiser evaluation or additional grid point ζK _K is needed for K=IdD_K=Id. Controlled transition. We use wθ,ℓw_θ, for the normalised discrete control, consistently with Algorithm 1. For the stochastic transitions, CMDS uses Xℓ+1 X_ +1 =Fℓθ(Xℓ,Ξℓ) =F_ ^θ(X_ , _ ) (127) ≔μb,ℓ(Xℓ)+ςℓwθ,ℓ(Xℓ)+ςℓΞℓ,Ξℓ∼(0,I),0≤ℓ<K−1. _b, (X_ )+ _ w_θ, (X_ )+ _ _ , _ (0,I), 0≤ <K-1. Unlike the epsilon-residual parameterisation in Algorithm 1, the motion controller outputs wθ,ℓw_θ, directly, supplying the mean correction ςℓwθ,ℓ _ w_θ, . For ςℓ>0 _ >0, this is the same normalisation wθ,ℓ=(Fℓθ−Fℓb)/ςℓw_θ, =(F_ ^θ-F_ ^b)/ _ , with both maps evaluated at the same state and noise. The final update is XK=μb,K−1(XK−1)=K−1(XK−1)X_K= _b,K-1(X_K-1)=D_K-1(X_K-1), with no learned correction, control-energy term, or matching loss at that transition. In particular, the positive-noise terminal modification used for DDIM in Algorithm 1 is not used here. DDPM–DDIM control correspondence. Both samplers use the same control normalisation, but different network outputs. In Algorithm 1, an epsilon residual δϵθ,ℓδ _θ, induces the DDIM mean shift γℓδϵθ,ℓ _ δ _θ, , giving wθ,ℓ=(γℓ/ςℓ)δϵθ,ℓw_θ, =( _ / _ )δ _θ, for that sampler. For the native DDPM kernel, the corresponding signed epsilon coefficient and equivalent residual are γℓDDPM _ ^DDPM ≔−βζℓαζℓ1−α¯ζℓ, - _ _ _ _ 1- α_ _ , (128) δϵθ,ℓequiv δ _θ, ^equiv ≔ςℓγℓDDPMwθ,ℓ,0≤ℓ<K−1. _ _ ^DDPMw_θ, , 0≤ <K-1. The sign is retained: γℓDDPMδϵθ,ℓequiv=ςℓwθ,ℓ _ ^DDPMδ _θ, ^equiv= _ w_θ, . The motion implementation requires no such conversion because the controller outputs wθ,ℓw_θ, directly. For adjacent native timesteps, the same denoiser and schedule, and no clipping, stochastic DDIM with η=1η=1 reproduces the DDPM mean and posterior variance on stochastic transitions. This does not identify the complete rollouts: Algorithm 1 retains noise at its modified DDIM endpoint, whereas MDM uses its original control-free deterministic endpoint. Thus the adaptation changes the sampling kernel, output parameterisation, and endpoint treatment, rather than the underlying stochastic-control formulation. D.6.2 Cooperative control architecture The controller forms one token per person and physical frame. Its motion input concatenates the current noisy features, the prior’s clean-motion estimate, and the decoded world-space joints: J^ℓ,i,f J_ ,i,f ≔Di(X^1|ℓi,si,ψi)f, D_i( X_1 ^\,i;s_i, _i)_f, (129) zℓ,i,f z_ ,i,f ≔[(Xℓi)f,(X^1|ℓi)f,vec(J^ℓ,i,f)]∈ℝ263+263+66. [(X_ ^i)_f,\,( X_1 ^\,i)_f,\,vec( J_ ,i,f) ] ^263+263+66. The 6666 skeleton coordinates represent the 2222 joints in metres. We project zℓ,i,fz_ ,i,f to width 344344 and add separate projections of the 512512-dimensional CLIP embedding, the static condition [si,gi,sinψi,cosψi,F/196]∈ℝ7[s_i,g_i, _i, _i,F/196] ^7, and 3232-dimensional sinusoidal embeddings of the physical frame f and native diffusion index ζℓ _ , then apply SiLU. A local temporal encoder attends to each person’s entire clip. Two subsequent blocks alternate temporal attention within a person, attention across people at the same frame, and a feed-forward network. Communication is aligned in physical time, with whole-motion context supplied by the temporal passes. Weights are shared across people, making the controller equivariant to a consistent permutation of people and their conditions. Geometric communication. Person attention receives an additive bias computed from predicted planar root geometry. At a fixed reverse transition and physical frame, suppress these indices and write Δpij=p^j−p^i p_ij= p_j- p_i, dijroot=‖Δpij‖2d_ij^root=\| p_ij\|_2, and Δvij=v^j−v^i v_ij= v_j- v_i, where velocities use 20Hz20\,Hz backward differences. The bias is bij=MLP8→32→4([Δpij,dijroot,Δvij,−Δpij⊤Δvijmax(dijroot,10−4),gj−gi]).b_ij=MLP_8→ 32→ 4\! ( [ p_ij,\,d_ij^root,\, v_ij,\, - p_ij v_ij (d_ij^root,10^-4),\,g_j-g_i ] ). (130) Its four outputs are added to the corresponding attention-head logits qi⊤kj/86q_i k_j/ 86. The same bias is used in both interaction blocks. The root distance dijrootd_ij^root is distinct from the signed body-capsule clearance defined below. Control output. Separate linear heads read the local encoder output HlocalH_local and the second interaction block output HjointH_joint: wθ,ℓ= w_θ, = WlocalHlocal+blocal W_localH_local+b_local (131) +WjointHjoint+bjoint∈ℝN×F×263. +W_jointH_joint+b_joint ^N× F× 263. This 4,863,0424,863,042-parameter control supplies joint awareness to the separate denoising processes. Both output heads and the final layer of the geometric-bias MLP are zero-initialised, so the initial controller leaves the frozen sampling process unchanged. D.6.3 Assembly reward For a completed joint motion X, write Y=φ(X)Y= (X). The total cost combines collision avoidance, goal attainment, directional progress, and physical consistency: Cκ(X)= C_κ(X)= λcolEcol+λgoalEgoal+λprogEprog _colE_col+ _goalE_goal+ _progE_prog (132) +λboneEbone+λFKEFK+λdispEdisp + _boneE_bone+ _FKE_FK+ _dispE_disp +λcontactEcontact+λfloorEfloor+λspeedEspeed. + _contactE_contact+ _floorE_floor+ _speedE_speed. All terms are evaluated using X and φ(X) (X), with dependence on the fixed task suppressed on the right-hand side. We distinguish CκtrainC_κ^train, used to train CMDS, from CκbaseC_κ^base, used for baseline selection and guidance. They use identical terms and normalisations, but different goal weights: λgoal=50 _goal=50 for control training and λgoal=1 _goal=1 for the baselines. The corresponding assembly-level rewards are rκtrain(φ(X))≔−Cκtrain(X),rκbase(φ(X))≔−Cκbase(X).r_κ^train( (X)) -C_κ^train(X), r_κ^base( (X)) -C_κ^base(X). (133) The subscripted cost weights are distinct from the outer SOC reward multiplier λ, specified in section D.6.4. Their recorded values are reported in table 14. All geometric distances below are in metres. Body collision. Each person is represented by 1515 fixed-radius capsule primitives, including a spherical head primitive. For capsule centreline segment Sia(τ,Y)S_ia(τ;Y) and radius ρa _a, the signed surface clearance is dijab(τ,Y)≔dist(Sia(τ,Y),Sjb(τ,Y))−ρa−ρb.d_ij^ab(τ;Y) (S_ia(τ;Y),S_jb(τ;Y) )- _a- _b. (134) We divide each native frame interval into five linear subintervals and retain the final endpoint, giving the synchronised grid =n/5:n=0,…,5(F−1)T=\n/5:n=0,…,5(F-1)\, with ||=5(F−1)+1=596|T|=5(F-1)+1=596. The collision penalty is Ecol=1(N2)||∑i<j∑τ∈∑a,b=115([0.03−dijab(τ,Y)]+0.05)2,[z]+≔max(z,0).E_col= 1 N2|T| _i<j _τ _a,b=1^15 ( [0.03-d_ij^ab(τ;Y)]_+0.05 )^2, [z]_+ (z,0). (135) For N=4N=4, the denominator is 6||6|T|, as in the implemented cost. The penalty averages over person pairs and dense times, sums all 225225 capsule pairs, and becomes zero at 3cm3\,cm clearance. This soft-cost margin is distinct from both PCD++’s nonnegative-clearance constraint and evaluation’s 5m5\,m penetration tolerance. Table 13 specifies the primitive endpoints and radii. Table 13: Body-capsule geometry for collision costs and checks. Endpoints are zero-based HumanML joint indices; the repeated head endpoint defines a sphere. Primitive Joint endpoints Radius (m) Pelvis (1,2)(1,2) 0.120 Torso (0,9)(0,9) 0.150 Shoulder span (16,17)(16,17) 0.100 Upper torso–head (9,15)(9,15) 0.100 Head (15,15)(15,15) 0.110 Upper arms (16,18),(17,19)(16,18),\ (17,19) 0.055 Forearms (18,20),(19,21)(18,20),\ (19,21) 0.045 Thighs (1,4),(2,5)(1,4),\ (2,5) 0.085 Lower legs (4,7),(5,8)(4,7),\ (5,8) 0.065 Feet (7,10),(8,11)(7,10),\ (8,11) 0.055 Destination and progress. Let pi(f,Y)p_i(f;Y) be person i’s planar root position, and set the goal tolerance to εg=0.5m _g=0.5\,m. On the final five frames, i==F−5,…,F−1=115,…,119W_i=W=\F-5,…,F-1\=\115,…,119\, we use Egoal E_goal =1N∑i=1Nminf∈i‖pi(f,Y)−gi‖22εg2, = 1N _i=1^N _f _i\|p_i(f;Y)-g_i\|_2^2 _g^2, (136) q¯i q_i =1|i|∑f∈i(pi(f,Y)−si)⊤(gi−si)‖gi−si‖22,Eprog=1N∑i=1N[0.9−q¯i]+2. = 1|W_i| _f _i (p_i(f;Y)-s_i) (g_i-s_i)\|g_i-s_i\|_2^2, E_prog= 1N _i=1^N[0.9- q_i]_+^2. (137) People may arrive at different frames within the window. Progress is the window-mean displacement along the start-to-goal direction; its threshold shapes training but is not an extra success condition. Physical consistency. Let Qifj∈ℝ3Q_ifj ^3 be canonical decoded joints and JifjJ_ifj their world-space counterparts. Write LifbL_ifb for the 2121 bone lengths, Lb⋆L_b for fixed audit references, and QFKQ^FK for joints reconstructed from the native 6D rotations using those reference lengths. For denormalised displacement channels VifjV_ifj, root-local rotation AifA_if, and ΔQifj=Qi,f+1,j−Qifj Q_ifj=Q_i,f+1,j-Q_ifj, the costs are Ebone E_bone =⟨(L/L⋆−1)2⟩, = (L/L -1)^2 , EFK E_FK =⟨‖Q−QFK‖22⟩3(0.05)2, = \|Q-Q^FK\|_2^2 3(0.05)^2, (138) Edisp E_disp =⟨‖V−AΔQ‖22⟩3(0.05)2, = \|V-A Q\|_2^2 3(0.05)^2, Econtact E_contact =⟨(H−H⋆)2⟩, = (H-H )^2 , (139) Efloor E_floor =⟨[−Jy]+2⟩(0.05)2, = [-J_y]_+^2 (0.05)^2, Espeed E_speed =⟨[‖20ΔQ‖2/vmax−1]+2⟩. = [\|20 Q\|_2/v_ -1]_+^2 . (140) Angle brackets average over people, valid native frames or intervals, and the relevant bones or joints. The factors of three make vector errors coordinate-wise means. Encoded displacement is compared per frame, without multiplying it by the frame rate. Here vmaxv_ is the speed threshold in the soft cost. For denormalised contact channels H, corresponding to joints 7,10,8,117,10,8,11, the soft reference is H⋆=σ((0.08−h)/0.02)σ((0.2−v)/0.05)H =σ((0.08-h)/0.02)\,σ((0.2-v)/0.05), using foot height h, speed v, and logistic sigmoid σ. Root speed and jerk are additional hard evaluation checks. D.6.4 Training by adjoint matching Only controller parameters are updated. Training scenes are generated online, with captions drawn from five families: walking, jogging, high-knee marching, forward jumping, and crouched walking. Each family has three phrasings. A quarter of scenes use one shared action family; the remainder sample family assignments conditioned on having at least two distinct families. Phrasings are sampled per person. Training uses generated trajectories, without paired interaction data. Control objective and replay. The motion experiment uses reward strength λ=100λ=100 and includes all stochastic transitions in the matched set =0,…,K−2K=\0,…,K-2\. Its discrete stochastic-control objective is (θ)=[λCκtrain(XK)+12∑ℓ=0K−2‖wθ,ℓ(Xℓ)‖22].J(θ)=E\! [λ C_κ^train(X_K)+ 12 _ =0^K-2\|w_θ, (X_ )\|_2^2 ]. (141) The expectation covers task conditions, initial noise, and rollout noise. The norm sums over people, physical frames, and all 263263 motion features, without averaging; λ scales only the terminal cost. For a fixed task and state, the controlled and reference stochastic transitions have common covariance ςℓ2I _ ^2I and mean difference ςℓwθ,ℓ _ w_θ, , giving conditional KL divergence 12‖wθ,ℓ‖22 12\|w_θ, \|_2^2. With the same initial Gaussian distribution, the expected sum of these energies equals the discrete path-space KL relative to the frozen sampler at classifier-free guidance scale 2.52.5; the shared deterministic final map adds no conditional KL. This Gaussian mean-shift interpretation does not cover arbitrary nonzero corrections at zero-noise transitions, including deterministic DDIM with η=0η=0. As in Algorithm 1, set θ¯=stopgrad(θ) θ=stopgrad(θ), generate a controlled rollout without gradient tracking, and replay individual transitions backwards. Suppressing the rollout superscript, the state adjoints satisfy aK a_K =λ∇XKCκtrain(XK), =λ _X_KC_κ^train(X_K), (142) aK−1 a_K-1 =[DXμb,K−1(XK−1)]⊤aK, = [D_X _b,K-1(X_K-1) ] a_K, aℓ a_ =[DXμb,ℓ(Xℓ)+ςℓDXwθ¯,ℓ(Xℓ)]⊤aℓ+1 = [D_X _b, (X_ )+ _ D_Xw_ θ, (X_ ) ] a_ +1 +[DXwθ¯,ℓ(Xℓ)]⊤wθ¯,ℓ(Xℓ),ℓ=K−2,…,0. + [D_Xw_ θ, (X_ ) ] w_ θ, (X_ ), =K-2,…,0. The deterministic final transition remains in adjoint propagation, but contributes neither control energy nor a matching term. Each transition is differentiated using a fresh state variable at the stored value XℓX_ , with θ¯ θ fixed and the next-state adjoint detached. The controller’s clean-motion, world-joint, text, and geometry-derived observations are detached and held fixed during this state differentiation, while its explicit noisy-state input and the frozen-prior mean remain differentiable. The recursion follows these observation-detach conventions, rather than differentiating through the recomputation of every controller observation; graphs are released between transitions. Calibrated robust matching. Training regresses against detached next-state adjoints using eθ,ℓ≔wθ,ℓ(stopgrad(Xℓ))+ςℓstopgrad(aℓ+1),ℓ∈.e_θ, w_θ, (stopgrad(X_ ) )+ _ stopgrad(a_ +1), . (143) The control is evaluated with the same task and detached observations as in replay. With D=NF⋅263=4⋅120⋅263D=NF· 263=4· 120· 263 motion-feature coordinates, the robust matching loss is ℒPH(θ)=[∑ℓ=0K−2Dδℓ2(1+‖eθ,ℓ‖22Dδℓ2−1)].L_PH(θ)=E\! [ _ =0^K-2D _ ^2 ( 1+ \|e_θ, \|_2^2D _ ^2-1 ) ]. (144) The first 500500 successful updates use squared matching and collect per-instance RMS residuals ‖eθ,ℓ‖2/D\|e_θ, \|_2/ D. Each δℓ _ is then fixed to their 9595th percentile, floored at 10−310^-3. With squared matching, current-policy rollouts, and all stochastic transitions included, the matching gradient agrees with direct differentiation of eq. 141 under the same observation-detach conventions. Pseudo-Huber matching downweights large residuals and generally changes that gradient; it leaves the terminal reward, adjoint recursion, and quadratic control energy unchanged. Table 14: Reward weights and controller training settings for the reported N=4N=4 experiment. Baselines share the cost normalisations but use a smaller goal weight. Setting Recorded value (λcol,λgoal,λprog)( _col, _goal, _prog), training (100,50,1)(100,50,1) (λcol,λgoal,λprog)( _col, _goal, _prog), baselines (100,1,1)(100,1,1) (λbone,λFK,λdisp)( _bone, _FK, _disp) (500,5,5)(500,5,5) (λcontact,λfloor,λspeed)( _contact, _floor, _speed) (1,200,10)(1,200,10) Outer SOC reward multiplier λ 100100, control training only Optimiser / updates / learning rate Adam / 15,00015,000 / constant 10−410^-4 Effective batch size 88 scenes: microbatch 22, accumulation 44 Gradient-norm clipping / EMA decay 0.50.5 / 0.99950.9995 Raw and EMA checkpoints were validated during training. Selection ranked joint success, then terminal cost plus control energy, preferring earlier steps on ties. Evaluation uses raw step-15,00015,000 weights from one seed-00 run. Inference requires no reward-gradient or projection iterations. D.6.5 Baselines Baselines share the frozen prior, native DDPM sampler, classifier-free guidance scale, scenes, decoder, and audit. Selection and guidance use CκbaseC_κ^base, with weights (100,1,1,500,5,5,1,200,10)(100,1,1,500,5,5,1,200,10) in the order of eq. 132. These retain exactly the cost normalisations above; in particular, the baseline goal weight is 11, rather than the training value 5050. Best-of-256. Best-of-256 generates 256256 independent complete four-person assemblies and returns the one with minimum baseline cost. Reference samples the same frozen MDM component priors independently and is used as a qualitative reference. DPS guidance. DPS differentiates the joint baseline reward of the clean-motion estimate through the frozen denoiser to the current noisy state: vℓ v_ ≔∇Xℓrκbase(φ(ℓ(Xℓ))), _X_ r_κ^base\! ( (D_ (X_ )) ), (145) X~ℓ+1 X_ +1 ≔stopgrad(Fℓb(Xℓ,Ξℓ)+ηgvℓ). \! (F_ ^b(X_ , _ )+ _gv_ ). DPS sets Xℓ+1=X~ℓ+1X_ +1= X_ +1. DPS, PCD, and PCD++ all use ηg=0.1 _g=0.1 at every transition, including the deterministic final one. This coefficient directly multiplies the reward gradient: there is no ςℓ2 _ ^2 factor and no separate application of the control-training multiplier λ=100λ=100. All guidance gradients are with respect to state variables, with denoiser parameters frozen and no gradient propagation across outer diffusion steps. Unlike the learned control in eq. 127, this constant guidance also changes the deterministic endpoint; the subsequent PCD projections are not Gaussian mean shifts either. These baselines therefore do not inherit the learned controller’s quadratic path-KL interpretation. PCD and PCD++. The PCD baseline combines eq. 145 with a root-goal projector enforcing the final-window destination condition. The projector changes planar root increments and the corresponding stored joint-displacement channels while keeping headings, heights, and poses fixed. It minimises normalised-feature change within this restricted parameterisation, but does not enforce collision clearance or the kinematic-validity audit. PCD++ uses the same guidance update and replaces that projector with a joint optimisation of the proposed state. All 263263 normalised feature channels of each active person and frame are free variables in this solve. Feasibility is checked on a freshly recomputed clean estimate at the candidate’s next noise level, rather than on the clean estimate used for the preceding guidance update. At the final transition, the candidate is already a clean motion and K=IdD_K=Id. Joint goal and collision constraints. Using the planar roots and signed capsule clearances defined above, set cg,i(Y) c_g,i(Y) ≔minf∈i‖pi(f,Y)−gi‖2−εg, _f _i\|p_i(f;Y)-g_i\|_2- _g, (146) c,ijabτ(Y) c_c,ijabτ(Y) ≔δc−dijab(τ,Y), _c-d_ij^ab(τ;Y), Vκ(Y) V_κ(Y) ≔max0,maxicg,i(Y),maxi<j,a,bτ∈ijc,ijabτ(Y). \! \0,\ _ic_g,i(Y),\ _ subarrayci<j,\,a,b\\ τ _ij subarrayc_c,ijabτ(Y) \. Here ijT_ij contains the checked times when both people are active; in the reported fixed-length experiment, ij=T_ij=T for every pair. We use εg=0.5m _g=0.5\,m and δc=0 _c=0, so PCD++ requires nonnegative capsule clearance. The feasible set is ℱκ≔Y:Vκ(Y)=0,pi(0;Y)=sifor all i.F_κ \Y:V_κ(Y)=0,\ p_i(0;Y)=s_i\ for all i \. (147) Starts are already enforced by the decoder. For a guided proposal X~ℓ+1 X_ +1, the projection targets minimise12‖Z−X~ℓ+1‖22subject toφ(ℓ+1(Z))∈ℱκ. Zminimise 12\|Z- X_ +1\|_2^2 to (D_ +1(Z)) _κ. (148) The distance is Euclidean in the active normalised motion features, not in decoded joint coordinates. The implemented solver approximates this constrained problem; it does not certify a nearest feasible point. Penalty and numerical guards. The inner objective uses the squared-hinge penalty κ(Y,ωg,ωc)≔ _κ(Y; _g, _c) ∑i[cg,i(Y)+ωg]+2 _i[c_g,i(Y)+ _g]_+^2 (149) +∑i<j∑τ∈ij∑a,b=115[c,ijabτ(Y)+ωc]+2. + _i<j _τ _ij _a,b=1^15[c_c,ijabτ(Y)+ _c]_+^2. Unlike the soft collision cost, this penalty uses unnormalised sums of constraint violations. The small interior guards tighten the penalty only; acceptance uses the original constraints in eq. 146. At inner iterate m, let J(m)J^(m) denote the active decoded world joints in Y^(m)=φ(ℓ+1(Z(m))) Y^(m)= (D_ +1(Z^(m))), and let ϵmach _mach be the machine epsilon of their data type. The detached coordinate scale and guards are Sm S_m ≔stopgrad[max(1,‖J(m)‖∞,maxi‖gi‖∞)], \! [ \! (1,\|J^(m)\|_∞, _i\|g_i\|_∞ ) ], (150) ωc(m) _c^(m) ≔max(10−7,64ϵmachSm), (10^-7,64 _machS_m), ωg(m) _g^(m) ≔min(10−3εg,ωc(m)). (10^-3 _g, _c^(m)). The infinity norms denote the largest absolute coordinate. These guards are recomputed at every inner iterate and held fixed when differentiating that iterate’s objective: ℒm(Z)≔12‖Z−X~ℓ+1‖22+ρκ(φ(ℓ+1(Z)),ωg(m),ωc(m)).L_m(Z) 12\|Z- X_ +1\|_2^2+ρ\,P_κ\! ( (D_ +1(Z)); _g^(m), _c^(m) ). (151) Projection settings and failure handling. PCD++ permits at most M=100M=100 inner Adam updates per outer transition, with step size ηp=0.01 _p=0.01, initial penalty ρ0=10 _0=10, and cap ρmax=107 _ =10^7. Adam uses (β1,β2)=(0.9,0.999)( _1, _2)=(0.9,0.999), ϵ=10−8ε=10^-8, and zero weight decay, with fresh optimiser state at every outer transition. Every five inner updates, the penalty is multiplied by 1010, up to its cap, if the maximum violation has not decreased by more than 5%5\% since the previous reference check. The solver checks all 15×1515× 15 capsule pairs for each pair of people at the 596596 dense times, and checks goals in each person’s final five native frames. It accepts the first verified feasible iterate, with at most M updates and M+1M+1 checks per transition. The accepted state is Z(m)Z^(m), not its denoised estimate, so it remains at the correct noise level for the next DDPM transition. Non-finite states, objectives, residuals, or gradients terminate the affected assembly with failure. Failure to find a feasible iterate within the budget also terminates that assembly, without an infeasible fallback. Failed assemblies count as failures in all binary success metrics. Algorithm 2 gives the complete procedure. Each completed outer transition uses one guidance backward pass and at most M inner-optimisation backward passes, giving at most K(1+M)=5050K(1+M)=5050 backward passes per assembly under the reported settings. Early acceptance or failure reduces this count; feasibility checks additionally require forward computations. This counts joint backward evaluations, not separate passes for each person; at the final transition, inner gradients pass through the decoder but not through a denoiser, since K=IdD_K=Id. The solver does not certify convergence or global distance optimality. Feasibility covers only the stated goal constraints and capsule proxy at checked times, not continuous-time clearance or the complete kinematic-validity audit. Algorithm 2 PCD++ with a DDPM rollout and joint Tweedie projection 1: frozen DDPM maps Fℓbℓ=0K−1\F_ ^b\_ =0^K-1, with ςK−1=0 _K-1=0; clean-estimate maps ℓ=0K\D_ \_ =0^K, with K=IdD_K=Id; assembly φ , task κ, and baseline reward rκbaser_κ^base; constraint residual VκV_κ and guarded penalty κP_κ; guidance strength ηg _g; maximum inner updates M; Adam step size ηp _p; penalty parameters ρ0,ρmax _0, _ 2: Conventions: optimise only active coordinates; keep padding zero. Any non-finite state, objective, residual, or gradient returns ProjectionFailure. 3: Sample X0i∼(0,I)X_0^i (0,I) and independent rollout noise Ξℓi∼(0,I) _ ^i (0,I), for all i,ℓi, , on active coordinates 4: for ℓ=0,…,K−1 =0,…,K-1 do 5: 1. Native DDPM proposal and DPS guidance 6: X^1|ℓ←ℓ(Xℓ),Y^1|ℓ←φ(X^1|ℓ) X_1 _ (X_ ), Y_1 ← ( X_1 ) 7: vℓ←∇Xℓrκbase(Y^1|ℓ)v_ ← _X_ r_κ^base( Y_1 ) # Differentiate through the frozen denoiser with respect to its input 8: X~ℓ+1←stopgrad(Fℓb(Xℓ,Ξℓ)+ηgvℓ) X_ +1 (F_ ^b(X_ , _ )+ _gv_ ) # Also guide at ℓ=K−1 =K-1; no variance or training-reward multiplier 9: 2. Joint optimisation of the proposed state 10: Z(0)←X~ℓ+1,ρ←ρ0Z^(0)← X_ +1, ρ← _0; initialise fresh Adam state 11: for m=0,…,Mm=0,…,M do 12: Y^(m)←φ(ℓ+1(Z(m))) Y^(m)← (D_ +1(Z^(m)) ) # Fresh clean estimate at the candidate’s next noise level; K=IdD_K=Id 13: qm←Vκ(Y^(m))q_m← V_κ( Y^(m)) 14: if qm=0q_m=0 then 15: Xℓ+1←stopgrad(Z(m))X_ +1 (Z^(m)) # Keep the corrected state, not its denoised estimate 16: break 17: end if 18: if m=Mm=M then 19: return ProjectionFailure # Terminate this assembly; no infeasible fallback 20: end if 21: if m=0m=0 then 22: qref←qmq_ref← q_m 23: else if mmod5=0m 5=0 then 24: if qm≥0.95qrefq_m≥ 0.95\,q_ref then 25: ρ←min(10ρ,ρmax)ρ← (10ρ, _ ) 26: end if 27: qref←qmq_ref← q_m 28: end if 29: Compute detached guards ωg(m),ωc(m) _g^(m), _c^(m) from Y^(m) Y^(m) using eq. 150 30: Define ℒm(Z):=12‖Z−X~ℓ+1‖active2+ρκ(φ(ℓ+1(Z)),ωg(m),ωc(m)) _m(Z):= 12\|Z- X_ +1\|_active^2+ρ\,P_κ\! ( (D_ +1(Z)); _g^(m), _c^(m) ) 31: Gm←∇Zℒm(Z)|Z=Z(m)G_m← . _ZL_m(Z) |_Z=Z^(m) # Hold the guards fixed; one backward pass per inner update 32: Z(m+1)←AdamStep(Z(m),Gm,ηp)Z^(m+1)← AdamStep(Z^(m),G_m, _p) 33: end for 34: end for 35: return XK,φ(XK)X_K,\ (X_K) # At most K(1+M)K(1+M) backward passes; 50505050 for the reported settings D.6.6 Additional Experimental Evaluations Evaluation uses 2424 fixed scenes drawn from three training action families: walking, jogging, and crouched walking. Thirteen scenes are homogeneous and eleven mix families. Every method is evaluated with five sampling seeds, giving 120120 attempts per method. These seeds measure sampling variability for one controller, not training variability or held-out layout generalisation. Main figures compare uncontrolled MDM, PCD++, and CMDS; the appendix comparison also includes Best-of-256, DPS, and PCD. Matched views use the same sampling seed, identical scene conditions, and a common camera. Capsule poses and footpaths show body clearance and goal progress, with faint intermediate poses and emphasised starts and endpoints. Uncontrolled MDM supplies the visual prior reference; table 15 aggregates the full evaluation. Figure 22 extends the qualitative comparison to all evaluated baselines. Figure 22: Extended qualitative comparison of four-person motion generation. We compare uncontrolled diffusion, Best-of-256, DPS, PCD, PCD++, and CMDS on the same four-person task shown in the main paper. The prompts are “A person moves forward at a jogging pace.” for Person 1 (∙ ) and Person 4 (∙ ), and “A person walks forward.” for Person 2 (∙ ) and Person 3 (∙ ). Top-down insets show starts and target regions. Solid curves show foot trajectories; intermediate poses are faded, while start and final poses are opaque. Red crosses mark inter-person collisions. In this example, uncontrolled diffusion, Best-of-256, DPS, and PCD do not satisfy the full joint goal-and-collision constraint, whereas PCD++ and CMDS reach all goals without inter-person collisions. Table 15: Four-person locomotion coordination. All methods use the same 2424 scenes with walking, jogging, and crouched walking, evaluated over five sampling seeds (120120 attempts per method). Entries are mean ± sample standard deviation across the five seed summaries. Binary rates weight all 2424 attempts per seed equally, including generation failures as zero successes; continuous geometry and time use available generated outputs. Timing excludes failed-attempt costs. Bold denotes the best mean, including ties. Method Joint success ↑ (%) Collision-free ↑ (%) Peak body overlap ↓ (cm) All-person goals ↑ (%) Goal error ↓ (m) Kinematic validity ↑ (%) Time ↓ (s/assembly) Best-of-256 0.0±0.00.0\, [rgb]0.5,0.5,0.5 ± 0.0 16.7±5.916.7\, [rgb]0.5,0.5,0.5 ± 5.9 8.860±1.0868.860\, [rgb]0.5,0.5,0.5 ± 1.086 0.0±0.00.0\, [rgb]0.5,0.5,0.5 ± 0.0 2.456±0.1462.456\, [rgb]0.5,0.5,0.5 ± 0.146 83.3±6.683.3\, [rgb]0.5,0.5,0.5 ± 6.6 69.895±0.29269.895\, [rgb]0.5,0.5,0.5 ± 0.292 DPS 0.0±0.00.0\, [rgb]0.5,0.5,0.5 ± 0.0 61.7±10.861.7\, [rgb]0.5,0.5,0.5 ± 10.8 3.335±1.0493.335\, [rgb]0.5,0.5,0.5 ± 1.049 0.0±0.00.0\, [rgb]0.5,0.5,0.5 ± 0.0 3.751±0.2843.751\, [rgb]0.5,0.5,0.5 ± 0.284 57.5±6.857.5\, [rgb]0.5,0.5,0.5 ± 6.8 8.566±0.0188.566\, [rgb]0.5,0.5,0.5 ± 0.018 PCD 4.2±2.94.2\, [rgb]0.5,0.5,0.5 ± 2.9 8.3±2.98.3\, [rgb]0.5,0.5,0.5 ± 2.9 19.002±1.86519.002\, [rgb]0.5,0.5,0.5 ± 1.865 100.0±0.0100.0\, [rgb]0.5,0.5,0.5 ± 0.0 0.500±0.0000.500\, [rgb]0.5,0.5,0.5 ± 0.000 27.5±4.827.5\, [rgb]0.5,0.5,0.5 ± 4.8 9.985±0.0649.985\, [rgb]0.5,0.5,0.5 ± 0.064 PCD++ 75.0±6.675.0\, [rgb]0.5,0.5,0.5 ± 6.6 98.3±2.398.3\, [rgb]0.5,0.5,0.5 ± 2.3 0.000±0.0000.000\, [rgb]0.5,0.5,0.5 ± 0.000 98.3±2.398.3\, [rgb]0.5,0.5,0.5 ± 2.3 0.487±0.0010.487\, [rgb]0.5,0.5,0.5 ± 0.001 75.0±6.675.0\, [rgb]0.5,0.5,0.5 ± 6.6 145.335±4.460145.335\, [rgb]0.5,0.5,0.5 ± 4.460 CMDS 90.0±6.390.0\, [rgb]0.5,0.5,0.5 ± 6.3 90.0±6.390.0\, [rgb]0.5,0.5,0.5 ± 6.3 0.740±0.3510.740\, [rgb]0.5,0.5,0.5 ± 0.351 100.0±0.0100.0\, [rgb]0.5,0.5,0.5 ± 0.0 0.158±0.0080.158\, [rgb]0.5,0.5,0.5 ± 0.008 100.0±0.0100.0\, [rgb]0.5,0.5,0.5 ± 0.0 0.541±0.0020.541\, [rgb]0.5,0.5,0.5 ± 0.002