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Akule: Fast and Scalable Multi-Robot Motion Planning via Sparse Interaction Diffusion
Manav Vora, Gokul Puthumanaillam, Aditya Penumarti, Jane Shin, Melkior Ornik
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/14/2026, 9:43:49 AM
Summary
The paper introduces Akule, a sparse interaction diffusion framework for scalable multi-robot motion planning (MRMP). Akule augments a frozen single-robot diffusion prior with three learned modules: a reusable pairwise correction model ($R_{ heta}$), a contextual selector ($U_{ heta}$) that determines relevant interactions, and a composer ($G_{ heta}$) that combines corrections via context-dependent signed weights. This approach allows for efficient coordination of up to 256 robots, achieving up to 25x speedups and zero-shot population transfer without requiring a prescribed ground-truth interaction graph.
Entities (8)
Relation Signals (8)
Akule → consistsof → R_phi
confidence 95% · augments a frozen single-robot prior with three learned modules: a reusable pairwise correction model R_phi
Akule → consistsof → U_psi
confidence 95% · augments a frozen single-robot prior with three learned modules: ... a contextual selector U_psi
Akule → consistsof → G_omega
confidence 95% · augments a frozen single-robot prior with three learned modules: ... and a composer G_omega
Akule → uses → Diffusion Models
confidence 95% · Akule, a sparse interaction diffusion framework...
Akule → achieves → 25x speedup
confidence 90% · Akule achieves ... complete-planner speedups of up to 25x
G_omega → combines → corrections
confidence 90% · G_omega that combines their effects through context-dependent signed weights
Akule → scalesto → 256 robots
confidence 90% · successful planning with up to 256 robots
U_psi → selects → interactions
confidence 90% · U_psi ... determines which corrections to evaluate
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Abstract
Abstract:Diffusion models provide expressive trajectory priors for multi-robot motion planning (MRMP), but scaling them to large teams requires coordinating individual motions without expensive joint computation. We introduce Akule, a sparse interaction diffusion framework that augments a frozen single-robot prior with three learned modules: a reusable pairwise correction model $R_{\phi}$, a contextual selector $U_{\psi}$ that determines which corrections to evaluate, and a composer $G_{\omega}$ that combines their effects through context-dependent signed weights. This decomposition separates which interactions merit computation from how their effects should be combined into coordinated denoising updates. We train $R_{\phi}$ using matched pairwise trajectory optimizations, then learn $G_{\omega}$ and $U_{\psi}$ from physically optimized joint trajectories without requiring a prescribed ground-truth interaction graph. The resulting joint proposals are passed to a downstream search-based planner for validation and repair. Across diverse MRMP benchmarks, Akule achieves competitive planning success, complete-planner speedups of up to $25\times$, zero-shot population transfer, and successful planning with up to $256$ robots. Ablations further demonstrate that learned selection and composition reduce pre-repair conflicts and downstream repair effort relative to heuristic selection and uniform summation.
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- Source: https://arxiv.org/abs/2610.14248v1
- Canonical: https://arxiv.org/abs/2610.14248v1
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Akule: Fast and Scalable Multi-Robot Motion Planning via Sparse Interaction Diffusion Manav Vora , Gokul Puthumanaillam , Aditya Penumarti , Jane Shin and Melkior Ornik Abstract. Diffusion models provide expressive trajectory priors for multi-robot motion planning (MRMP), but scaling them to large teams requires coordinating individual motions without expensive joint computation. We introduce Akule, a sparse interaction diffusion framework that augments a frozen single-robot prior with three learned modules: a reusable pairwise correction model RϕR_φ, a contextual selector UψU_ψ that determines which corrections to evaluate, and a composer GωG_ω that combines their effects through context-dependent signed weights. This decomposition separates which interactions merit computation from how their effects should be combined into coordinated denoising updates. We train RϕR_φ using matched pairwise trajectory optimizations, then learn GωG_ω and UψU_ψ from physically optimized joint trajectories without requiring a prescribed ground-truth interaction graph. The resulting joint proposals are passed to a downstream search-based planner for validation and repair. Across diverse MRMP benchmarks, Akule achieves competitive planning success, complete-planner speedups of up to 25×25×, zero-shot population transfer, and successful planning with up to 256256 robots. Ablations further demonstrate that learned selection and composition reduce pre-repair conflicts and downstream repair effort relative to heuristic selection and uniform summation. Project website: https://akule-diff.github.io/ Keywords: Diffusion Models, Multi-Agent Path Planning, Robotics 1. Introduction Diffusion-based trajectory planners learn expressive motion priors from offline data and generate goal-conditioned trajectories by iteratively denoising entire motion sequences (Janner et al., 2022; Carvalho et al., 2023). Extending these models to multi-robot motion planning (MRMP) introduces a coordination challenge: trajectories that are individually plausible may become infeasible when planned together. Avoiding inter-robot collisions therefore requires accounting for interactions between robots while generating jointly compatible motions (Shaoul et al., 2025; Liang et al., 2025a). Existing diffusion-based MRMP methods introduce coordination at different stages. Multi-robot Multi-model planning Diffusion (MMD) retains single-robot priors and uses conflict-based search to regenerate constrained trajectories (Shaoul et al., 2025). Simultaneous MRMP Diffusion (SMD) enforces inter-robot constraints during joint sampling (Liang et al., 2025a), while Discrete-Guided Diffusion (DGD) uses discrete multi-robot paths to guide constrained generation (Liang et al., 2025b). These approaches coordinate through constrained replanning, joint optimization, or discrete guidance. Yet interaction relevance can differ across robot pairs and change as planned trajectories evolve. This motivates our question: can diffusion allocate expensive interaction reasoning to the pairs that matter during denoising? One way to achieve this is to retain a single-robot motion prior and represent the influence of another robot as a reusable pairwise interaction correction to its denoising prediction, drawing on relational and compositional modeling (Battaglia et al., 2016; Mishra et al., 2025). For a team of N robots however, evaluating such corrections densely requires considering N(N−1)N(N-1) ordered interactions at every reverse step. Simply selecting fewer interactions is not sufficient either. Corrections that are individually useful may become incompatible when several robots are coordinated simultaneously. Our key insight is to decompose interaction-aware diffusion into two distinct decisions: which pairwise corrections are worth computing, and how their effects should be combined into a coherent joint update. Both decisions depend on the surrounding team and the evolving trajectory estimates. We introduce Akule, which realizes this decomposition through three learned modules. A shared pairwise model RϕR_φ learns reusable interaction corrections; a contextual selector UψU_ψ uses inexpensive pairwise descriptors and pooled team information to choose a directed interaction graph before evaluating RϕR_φ; and a composer GωG_ω combines the selected corrections using context-dependent signed weights. Shared parameters and symmetric aggregation (Zaheer et al., 2017) make the architecture permutation equivariant: relabeling the robots correspondingly relabels its outputs, allowing the same modules to operate across variable team sizes. We train these modules through offline trajectory optimization. Matched pairwise optimizations supervise RϕR_φ, joint trajectory optimization provides composition targets for GωG_ω, and sparse-support learning trains UψU_ψ without prescribing a ground-truth interaction graph. Complete denoising rollouts further refine the learned coordination. At inference, diffusion produces a coordinated joint proposal that downstream search validates and repairs as needed. We evaluate Akule on obstacle-rich navigation and congested position-exchange benchmarks (Shaoul et al., 2025; Liang et al., 2025a; Liang et al., 2025b). Akule achieves competitive or superior planning success with complete-planner speedups of up to 25×25×, alongside zero-shot population transfer across team sizes and successful planning with up to 256256 robots. Ablations further show that learned selection and composition reduce pre-repair conflicts and downstream repair effort relative to budget-matched heuristic selection and uniform summation. Contributions. Our main contributions are: (i) A permutation-equivariant diffusion architecture that coordinates variable-sized robot teams by selectively evaluating reusable pairwise interaction corrections and composing their effects through context-dependent signed weights; (i) a trajectory-optimization-based learning procedure that supervises pairwise interaction corrections and their composition using physically optimized trajectories, without requiring a unique ground-truth interaction graph; and (i) extensive benchmark evaluations demonstrating competitive planning success with complete-planner speedups of up to 25×25×, zero-shot population transfer, and scalability to hundreds of robots. 2. Related Work Prior work addresses inter-robot coupling through classical search, constrained or conditional diffusion, and compositional models built from reusable factors. Akule instead learns reusable pairwise corrections, which ones to evaluate, and how to combine their effects in the context of the team. Diffusion trajectory models — Learning-based motion planners exploit learned sampling distributions and motion priors to accelerate planning (Ichter et al., 2018; Qureshi et al., 2019). Diffusion-based planners generate complete trajectories through iterative denoising (Janner et al., 2022; Ajay et al., 2023), learning from expert demonstrations (Chi et al., 2025), trajectories produced by classical planners or optimization (Carvalho et al., 2023; Luo et al., 2024), or navigation data (Yu et al., 2024). These approaches provide expressive single-agent motion priors, but independently generated trajectories do not account for inter-robot constraints. Classical multi-robot planning — Inter-robot collision constraints couple individual plans, making joint-space search expensive as team size grows. Decoupled and prioritized planners reduce computation but can fail on solvable instances (Erdmann and Lozano-Pérez, 1987; Silver, 2005; Sánchez and Latombe, 2002). Subdimensional expansion couples robots only when interactions require it (Wagner and Choset, 2011), while conflict-based search (CBS) resolves conflicts through constrained replanning (Sharon et al., 2012). Both exploit selective coupling, but search can remain expensive in heavily interacting scenes. Bounded-suboptimal CBS variants accelerate search while preserving cost bounds (Barer et al., 2014; Li et al., 2021). Large-neighborhood repair (Li et al., 2022), priority inheritance (Okumura et al., 2019), and lazy constraint addition (Okumura, 2023) offer other scalable strategies with differing guarantees. Discrete planning followed by continuous refinement (Hönig et al., 2018) and reactive collision avoidance (van den Berg et al., 2011) provide further alternatives. These methods exploit interaction structure during planning or execution, rather than learning reusable denoising-time corrections. Multi-agent diffusion planning — Offline multi-agent diffusion models learn coordination through cross-agent attention (Zhu et al., 2024) or sparse coordination graphs (Zeng et al., 2025), but address different control objectives from collision-constrained MRMP. For MRMP, MMD retains single-robot diffusion priors but resolves coupling through conflict-driven search and constrained regeneration (Shaoul et al., 2025). SMD enforces joint constraints through optimization during sampling (Liang et al., 2025a), while DGD uses discrete MAPF guidance to organize constrained generation and repair (Liang et al., 2025b). Other approaches use decentralized neighbor prediction (Liang et al., 2026), local conditional denoising (Lee et al., 2026), or model-based diffusion with conflict-based search (He et al., 2026). These methods account for interactions through search, joint optimization, or conditioning on other agents’ motions. Akule instead makes the evaluation and composition of reusable pairwise denoising corrections separate learned operations. Compositional diffusion and sparse interactions — Compositional diffusion constructs complex distributions from reusable scores or energy-based factors (Liu et al., 2022; Du et al., 2023). Graph-structured approaches organize local models through prescribed graphical relationships (Zhang et al., 2023; Weilbach et al., 2023). In planning, related work composes skills and task factors (Mishra et al., 2023; Mishra et al., 2025), locally valid trajectory predictions and reusable score factors (Sun et al., 2026; Mitra et al., 2026), or obstacle and map-region potentials (Luo et al., 2024; Shaoul et al., 2025). These approaches demonstrate the value of reusable components, but their factorization or composition rule is often prescribed by task structure. Sparse interaction relationships can also be learned for multi-agent coordination (Zeng et al., 2025) and humanoid control (Li et al., 2026), although these address different planning objectives. Akule instead learns a time-varying directed support and a context-dependent signed composition of pairwise corrections to a frozen motion prior. 3. Preliminaries and Problem Formulation Multi-robot motion planning. We consider N robots operating in a shared workspace W (Shaoul et al., 2025; Liang et al., 2025a). Robot i has state shi∈is_h^i ^i at physical time index h, with configuration qhiq_h^i determining its occupied region ℛi(qhi)R^i(q_h^i). Given start and goal states ssis_s^i and sgis_g^i, we seek trajectories τi=(s1i,…,sHi)τ^i=(s_1^i,…,s_H^i) satisfying s1i=ssis_1^i=s_s^i and sHi=sgis_H^i=s_g^i. The joint trajectory is =(τ1,…,τN) τ=(τ^1,…,τ^N). Let Ω contain joint trajectories satisfying these endpoints, static-obstacle avoidance, benchmark-specific motion constraints, and inter-robot collision avoidance: ℛi(qhi)∩ℛj(qhj)=∅, ∀i≠j,h=1,…,H.R^i(q_h^i) ^j(q_h^j)= , ∀ i≠ j,\;h=1,…,H. The objective is to find a feasible ⋆∈Ω τ ∈ . Frozen unary diffusion prior. We build on a pretrained single-robot diffusion planner (Janner et al., 2022; Shaoul et al., 2025). Let xtix_t^i denote robot i’s noisy trajectory representation at reverse-diffusion step t, and cic^i its start, goal, and environment context. The frozen denoiser fθf_θ defines xt−1i=Φt(xti,fθ(xti,t,ci),ξti),x_t-1^i= _t\! (x_t^i,f_θ(x_t^i,t,c^i), _t^i ), where Φt _t is the sampler update and ξti _t^i its stochastic component. Endpoints are hard-conditioned throughout sampling. Applying fθf_θ independently yields the unary joint prior pθunary(∣)=∏i=1Npθ(τi∣ci)p_θ^unary( τ )= _i=1^Np_θ(τ^i c^i), which supports variable team sizes but does not model inter-robot dependencies. Coordinated proposal and repair. We augment the unary reverse process to generate a joint proposal ^=() τ=P(c). A downstream conflict-based search planner ℋH then repairs remaining conflicts: sol=ℋ(^). τ^sol=H( τ). A returned solution is accepted only if sol∈Ω τ^sol∈ ; planning may also terminate without a solution. The central challenge is therefore to capture sufficient inter-robot coordination to produce high-quality proposals without requiring dense pairwise coordination computation at every reverse-diffusion step. We address this by separating which interactions should be modeled from how their effects should be composed. Figure 1. Akule architecture. At each reverse-diffusion step, the frozen unary prior produces independent intent, UψU_ψ selects a directed interaction graph, RϕR_φ computes corrections only for the selected pairs, and GωG_ω composes them using context-dependent signed weights. Repeating this update produces a joint trajectory proposal that is passed to downstream hard repair.Overview of the Akule planning architecture. A frozen unary diffusion prior generates independent robot intent. A selector chooses directed robot interactions, pairwise interaction corrections are computed only for selected pairs, and a context-aware composer combines them with signed weights. The coordinated correction is applied during reverse diffusion, producing a joint trajectory proposal that is passed to a hard repair planner. 4. Akule We now introduce Akule, which preserves the frozen single-robot diffusion prior from Sec. 3 and adds the interaction reasoning missing from independent planning. Akule consists of three learned modules: (i) A reusable pairwise model RϕR_φ that learns how one robot should alter another’s denoising prediction; (i) A contextual selector UψU_ψ that determines which pairwise corrections are worth computing; and (i) A composer GωG_ω which learns how the selected corrections should be combined. At inference, UψU_ψ selects interactions before the expensive RϕR_φ evaluations, and GωG_ω composes the admitted corrections into a coordinated update. All three modules share parameters across robots, yielding a permutation-equivariant architecture that is independent of team size. We learn them sequentially through physical trajectory optimization, in the order Rϕ→Gω→UψR_φ→ G_ω→ U_ψ (Sec. 4.4). The resulting interaction-aware prediction for robot i is (1) ϵ~i,t=ϵi,tunary+∑j≠imij,tαij,tRϕ(xti,xtj,t,ci,cj), ε_i,t= _i,t^unary+ _j≠ im_ij,t\, _ij,t\,R_φ (x_t^i,x_t^j,t,c^i,c^j ), where mij,t∈0,1m_ij,t∈\0,1\ determines whether the interaction between robots i and j is evaluated, RϕR_φ produces the corresponding pairwise correction, and αij,t∈ℝ _ij,t determines how that correction contributes in the context of the full team. Figure 1 summarizes the resulting reverse-step pipeline; the remainder of this section develops its components and learning procedure. 4.1. Reusable Pairwise Interaction Corrections The single-robot diffusion model produces ϵi,tunary=fθ(xti,t,ci) _i,t^unary=f_θ(x_t^i,t,c^i) for robot i at reverse-diffusion step t. Applied independently across robots, these predictions capture goal-directed motion but do not account for how another robot should alter the denoising update. We capture this missing interaction through a shared temporal-convolutional model RϕR_φ. For an ordered pair (i,j)(i,j), where robot j influences the update of robot i, rij,t=Rϕ(xti,xtj,t,ci,cj)r_ij,t=R_φ (x_t^i,x_t^j,t,c^i,c^j ) is a correction in the same space as the unary denoising prediction. Conceptually, considering only the interaction with robot j gives the pair-aware prediction ϵi|j,tpair=ϵi,tunary+rij,t _i j,t^pair= _i,t^unary+r_ij,t. Thus, RϕR_φ need not reproduce robot i’s goal-directed trajectory prior; it only represents how that prediction should change because robot j is present. These reusable corrections provide a common backbone for inter-robot coordination. Evaluating that backbone densely, however, would require N(N−1)N(N-1) calls to RϕR_φ at every reverse step. The first learned decision in Akule is to identify which pairwise corrections are worth computing from information available before RϕR_φ is evaluated. 4.2. Sparse Contextual Interaction Selection A useful interaction cannot always be identified from pairwise proximity alone, and the relevant interactions can change as trajectory intent sharpens through reverse diffusion (Fig. 2). For example, whether robot j should influence robot i may depend on the motion of both robots, their intended trajectories, and whether either is simultaneously interacting with other robots. For every ordered candidate pair (i,j)(i,j), we therefore construct an inexpensive pair descriptor χij,t _ij,t using: the current noisy trajectories xtix_t^i and xtjx_t^j, their planning contexts, relative trajectory geometry, unary denoising features, and the diffusion timestep. Three snapshots of the same multi-robot planning scene during reverse diffusion. Early in diffusion several directed interactions are selected because robot intent is uncertain. As trajectories become more defined, the selector removes interactions that become irrelevant and admits others that become important. Figure 2. Contextual interaction selection. As reverse diffusion sharpens trajectory intent, UψU_ψ can revise the directed interaction graph, removing interactions that become irrelevant while admitting others that become important.Three snapshots of the same multi-robot planning scene during reverse diffusion. Early in diffusion several directed interactions are selected because robot intent is uncertain. As trajectories become more defined, the selector removes interactions that become irrelevant and admits others that become important. Pair-local information is supplemented with context from the rest of the team. We use Deep-Sets-style symmetric aggregation (Zaheer et al., 2017), denoted by Pool(⋅)Pool(·), so that the resulting representation does not depend on robot ordering or population size. For candidate pair (i,j)(i,j), we form ρi,tego=Poolk≠iχik,t, ρj,tpartner=Poolk≠jχkj,t, ρtglobal=Poola≠bχab,t. _i,t^ego=Pool_k≠ i _ik,t, _j,t^partner=Pool_k≠ j _kj,t, _t^global=Pool_a≠ b _ab,t. The first two summaries describe the other candidate interactions associated with robots i and j, while ρtglobal _t^global summarizes the interaction context of the full team. A shared selector UψU_ψ uses these quantities to predict a real-valued interaction relevance logit, zij,t=Uψ(χij,t,ρi,tego,ρj,tpartner,ρtglobal).z_ij,t=U_ψ ( _ij,t, _i,t^ego, _j,t^partner, _t^global ). The sigmoid pij,t=σ(zij,t)p_ij,t=σ(z_ij,t) provides a differentiable relevance score during learning. At inference, each directed pair is either admitted or rejected using (2) mij,t=[pij,t>12]=[zij,t>0],m_ij,t=1 [p_ij,t> 12 ]=1 [z_ij,t>0 ], and the resulting mask mij,tm_ij,t defines the directed interaction graph Et=(i,j):i≠j,mij,t=1.E_t= \(i,j):i≠ j,\;m_ij,t=1 \. The graph may change across reverse diffusion steps, and the two directions of an interaction are scored independently. Crucially, RϕR_φ is evaluated only for (i,j)∈Et(i,j)∈ E_t. Because the selector uses shared pair processing, no robot identities, and symmetric set aggregation, reindexing the robots correspondingly reindexes the selected interactions. The same UψU_ψ can therefore be applied without architectural modification as N changes. 4.3. Context-Aware Signed Composition Selection determines which pairwise corrections are worth computing, but it does not determine how several simultaneously relevant corrections should be used together. As illustrated in Fig. 3, simply summing them treats every admitted interaction as equally compatible, even though corrections that are individually useful may reinforce, cancel, or conflict when considered jointly. Comparison of naive and context-aware composition for one robot. Several pairwise interaction corrections are shown. Naively summing them produces a conflicting next state, whereas the context-aware composer assigns positive and negative weights that amplify, attenuate, or reverse individual corrections to produce a coordinated next state. Figure 3. Context-aware signed composition. A blind sum of useful pairwise corrections can produce a conflicting joint update. GωG_ω uses team context to amplify, attenuate, or reverse selected corrections before the reverse update.Comparison of naive and context-aware composition for one robot. Several pairwise interaction corrections are shown. Naively summing them produces a conflicting next state, whereas the context-aware composer assigns positive and negative weights that amplify, attenuate, or reverse individual corrections to produce a coordinated next state. After RϕR_φ is evaluated on the selected edge set, we construct permutation-invariant context tEC_t^E from the admitted interactions and the current joint reverse state. This context allows the effect assigned to one pairwise correction to depend on the other corrections that are simultaneously active. A shared composer GωG_ω assigns each admitted interaction a signed coefficient, αij,t=Gω(rij,t,χij,t,tE) _ij,t=G_ω (r_ij,t, _ij,t,C_t^E ), with (i,j)∈Et.(i,j)∈ E_t. The coefficients are unconstrained in sign: depending on the surrounding interactions, a primitive can be amplified, attenuated, or applied in the opposite direction. Substituting these coefficients and the hard selection mask into Eq. (1) gives the coordinated denoising prediction ϵ~i,t ε_i,t. The standard reverse-diffusion transition from Sec. 3 is then applied to ϵ~i,t ε_i,t, with the prescribed start and goal states hard-conditioned throughout the reverse process. 4.4. Learning from Trajectory Optimization We learn coordination from physically optimized trajectories rather than requiring binary labels specifying which interactions are correct. Different supports and coefficient combinations can lead to equally valid behavior, making a single hand-designed binary graph an artificial supervision target. We therefore obtain supervision from trajectory-level physical performance rather than prescribing a particular interaction graph. Learning proceeds in the same dependency order as the model: we first learn the reusable pairwise primitive RϕR_φ, then use offline trajectory optimization to generate high-quality composition targets for GωG_ω, and finally learn which interactions can be removed through UψU_ψ. Learning the pairwise primitive. We construct the RϕR_φ supervision through pairwise trajectory optimization. For each ordered pair (i,j)(i,j), starting from the same unary estimate, we solve (3) ^0,t(γ)=argmin[Jbase()+γJijint()],γ∈0,1, x_0,t^(γ)= _ x [J_base( x)+γ J_ij^int( x) ], γ∈\0,1\, where JbaseJ_base contains the shared trajectory-quality and feasibility terms, while JijintJ_ij^int penalizes physical interaction between robots i and j. The two solves use identical initialization, endpoint constraints, and optimization settings; γ=1γ=1 gives the interaction-aware solution and γ=0γ=0 the matched null solution. Denoting their robot-i trajectories by x^0,ti,real x_0,t^i,real and x^0,ti,null x_0,t^i,null, respectively, the standard DDPM correspondence (Ho et al., 2020) gives the pairwise diffusion target (4) Δϵi←j,t⋆=−α¯t1−α¯t(x^0,ti,real−x^0,ti,null). _i← j,t =- α_t1- α_t ( x_0,t^i,real- x_0,t^i,null ). Because the two optimizations differ only in whether interaction (i,j)(i,j) is modeled, Δϵi←j,t⋆ _i← j,t isolates how robot j should modify robot i’s unary diffusion prediction. We train RϕR_φ by regressing to this target over qualified pairs in both directions, and subsequently freeze it. Offline composition targets. With RϕR_φ fixed, we next solve the multi-robot composition problem offline. For each training state, we optimize signed coefficients βij,t∈[−st,st] _ij,t∈[-s_t,s_t] in the dense composition ϵi,t=ϵi,tunary+∑j≠iβij,trij,t _i,t β= _i,t^unary+ _j≠ i _ij,tr_ij,t. The bound sts_t acts as a trust region, preventing the optimizer from arbitrarily amplifying a pairwise primitive and keeping the optimized composition within the regime represented by the frozen interaction model. Let ^() τ( β) denote the resulting composed joint trajectory. We solve (5) ⋆=argmin|βij,t|≤stJopt(^()), β = _ _ij,t ≤ s_tJ_opt ( τ( β) ), where JoptJ_opt combines swept inter-robot and obstacle clearance, trajectory regularity, path quality, trust to the unary motion prior, and workspace feasibility. Exact objective terms and weights are given in Appendix B.2. We denote the resulting optimized trajectory by ⋆ τ and use it as an offline trajectory teacher for the learned coordination modules. Because the optimization acts only through the frozen pairwise primitives, these targets remain within the representational span available to the deployed model. Learning the composer. We train GωG_ω to amortize the optimization described above. Under dense interaction support, the composer is first trained to reproduce the optimized composition through the transition-matching objective (6) ℒGtrans=∑twt∑i=1N‖ϵ~i,t−ϵi,t⋆‖22.L_G^trans= _tw_t _i=1^N \| ε_i,t- _i,t β \|_2^2. Here, wtw_t is a fixed timestep-dependent weight that compensates for the different noise scales across the diffusion schedule. Local transition matching provides a useful initialization but does not guarantee that the learned composer remains effective when its own predictions are propagated through the entire diffusion chain. We therefore refine GωG_ω through complete reverse-diffusion rollouts. If ^G τ_G is the resulting trajectory, the rollout objective is (7) ℒGroll=ℒmatch(^G,⋆)+λclearℒclear(^G)+λtransℒGtrans.L_G^roll=L_match ( τ_G, τ )+ _clearL_clear ( τ_G )+ _transL_G^trans. Here, ℒmatchL_match keeps the rollout near the optimized trajectory target, ℒclearL_clear penalizes insufficient inter-robot separation, and the small transition auxiliary stabilizes local denoising behavior. Learning the selector. Once composition is fixed, we learn which interactions can be removed without sacrificing trajectory quality. We first optimize per-edge selector logits offline using the same hard decisions employed at deployment, while the sigmoid scores provide differentiable gradients. Sparsity is encouraged through the expected mean directed degree, ℛsp(z)=1TN∑t=1T∑i≠jσ(zij,t).R_sp(z)= 1TN _t=1^T _i≠ jσ(z_ij,t). The resulting physically successful sparse supports provide bootstrap supervision for the shared selector UψU_ψ without declaring any one binary graph to be uniquely correct. The bootstrapped selector is then refined through complete reverse diffusion rollouts with fθf_θ, RϕR_φ, and GωG_ω fixed. Forward rollouts use the deployed hard decisions from Eq. (2), while gradients are propagated through the corresponding sigmoid scores using a straight-through estimator. The rollout objective is augmented by λspℛsp _spR_sp, directly trading trajectory quality against interaction density. Additional architectural details, objective definitions, optimization settings, and the equivalent implementation parameterizations are provided in Appendices A and B. 4.5. Complete Planner and Computational Cost The final interaction-aware proposal τ is passed to the downstream repair planner ℋH from Sec. 3. Akule need not eliminate every collision during diffusion; remaining conflicts are resolved through constrained search, and only validated solutions are accepted. The selector still scores all N(N−1)N(N-1) directed candidate pairs, retaining O(N2)O(N^2) candidate-scoring complexity. Its computational benefit comes from avoiding expensive RϕR_φ evaluations and composition on rejected edges. With mean selected degree d¯t=|Et|/N d_t=|E_t|/N, only Nd¯tN d_t corrections are evaluated rather than N(N−1)N(N-1), saving the fraction Savingst=1−d¯tN−1Savings_t=1- d_tN-1 of dense pairwise-correction evaluations. Sec. 5 evaluates how this sparsity, proposal quality, and complete-planner cost evolve with team size. 5. Experiments We ask whether sparse interaction selection preserves planning quality, whether one coordination model transfers across team sizes, whether the learned proposal reduces complete-planner computation, and whether these properties hold across map families and very large teams. Benchmarks. Canonical Weave (Shaoul et al., 2025) is a high-congestion position-exchange task (N∈10,20,28N∈\10,20,28\); Basic, Dense, Shelf, and Room (Liang et al., 2025a) are obstacle maps of increasing clutter (N∈3,…,18N∈\3,…,18\, 25 instances per cell); Highways and Conveyor (Shaoul et al., 2025) are the MMD obstacle environments (N∈3,…,20N∈\3,…,20\, 10 tasks per N); and ScaledWeave scales the Weave geometry to very large teams (N∈40,…,256N∈\40,…,256\, 10 scenes per N). In every family, Akule is trained or adapted only at the largest population (N=28N=28, 1818, 2020, and 256256, respectively) and evaluated zero-shot at all smaller populations. Baselines. MPD (Carvalho et al., 2023) applies the same frozen single-robot prior independently and relies entirely on downstream repair, isolating the effect of coordination; MMD (Shaoul et al., 2025) couples single-robot diffusion with conflict-based search (we use its strongest variant, xECBS, rerun on our panels); SMD (Liang et al., 2025a) and DGD (Liang et al., 2025b) enforce inter-robot constraints during joint sampling and via discrete guidance, respectively. On the map families, baseline values are the published DGD-benchmark results and are not same-hardware timings. Protocol and metrics. Akule and MPD share the same hard-repair planner ℋH (MMD’s obstacle-aware xECBS) and differ only in the proposal. We report success (validated collision-free plans within the time limit), planning latency (wall-clock time including repair, averaged over all runs with timeouts counted at the limit), and quality (mean path length and acceleration over successful plans). Full benchmark, timing, and hardware details are in Appendices C and F. 5.1. Quantitative Results Table 1. Canonical Weave: complete-planner results on matched panels and the three planners’ saved trajectories on one N=28N=28 task. Scarlet brackets mark selected collisions. N Method Success Plan (s) Quality (P/A) 10 MPD 512/512 3.973 1.781 / 1.454 10 MMD-xECBS 512/512 12.180 1.811 / 1.504 10 Akule 512/512 3.128 1.811 / 1.593 20 MPD 256/256 26.673 1.828 / 1.942 20 MMD-xECBS 256/256 33.424 1.842 / 1.848 20 Akule 256/256 7.106 1.910 / 2.447 28 MPD 512/512 77.602 1.877 / 2.214 28 MMD-xECBS 512/512 82.909 1.889 / 2.196 28 Akule 512/512 29.944 1.949 / 2.813 MPD [FAIL] MMD-xECBS Akule Table 2. Projected-diffusion map families: MPD, MMD, SMD, and DGD are published DGD-benchmark values. Below, one N=18N=18 task per map family, the same task for SMD and Akule; scarlet brackets mark selected collisions. Basic Dense Shelf Room N Method Success Time (s) Quality (P/A) Success Time (s) Quality (P/A) Success Time (s) Quality (P/A) Success Time (s) Quality (P/A) 3 MPD 23/25 9.300 1.110 / 0.006 22/25 8.300 1.140 / 0.008 8/25 9.100 1.150 / 0.007 3/25 8.800 1.120 / 0.007 3 MMD 25/25 13.100 1.120 / 0.006 15/25 19.100 1.100 / 0.006 17/25 23.700 1.160 / 0.006 15/25 28.200 1.220 / 0.007 3 SMD 25/25 76.600 1.100 / 0.005 25/25 81.200 1.120 / 0.006 25/25 88.000 1.140 / 0.007 25/25 72.600 1.170 / 0.008 3 DGD 25/25 6.500 1.160 / 0.008 25/25 6.600 1.170 / 0.007 25/25 31.500 1.270 / 0.008 25/25 17.100 1.360 / 0.008 3 Akule 25/25 0.570 1.253 / 0.004 25/25 2.184 1.271 / 0.009 25/25 0.587 1.437 / 0.003 25/25 1.064 1.224 / 0.005 6 MPD 19/25 9.100 1.120 / 0.006 1/25 9.200 1.110 / 0.008 0/25 [FAIL] 0/25 [FAIL] 6 MMD 25/25 27.100 1.120 / 0.006 10/25 36.500 1.150 / 0.006 8/25 48.500 1.180 / 0.007 6/25 55.500 1.160 / 0.007 6 SMD 25/25 254.300 1.100 / 0.006 25/25 287.300 1.130 / 0.007 25/25 274.300 1.160 / 0.008 25/25 181.000 1.190 / 0.007 6 DGD 25/25 12.200 1.210 / 0.011 25/25 12.300 1.260 / 0.012 25/25 64.600 1.300 / 0.011 25/25 38.900 1.440 / 0.013 6 Akule 25/25 2.042 1.301 / 0.007 25/25 4.067 1.268 / 0.010 25/25 1.449 1.430 / 0.006 25/25 2.553 1.256 / 0.007 9 MPD 18/25 9.900 1.080 / 0.008 0/25 [FAIL] 0/25 [FAIL] 0/25 [FAIL] 9 MMD 25/25 41.400 1.120 / 0.006 7/25 51.400 1.160 / 0.006 2/25 61.600 1.160 / 0.007 2/25 142.400 1.150 / 0.007 9 SMD 25/25 504.000 1.120 / 0.006 25/25 582.100 1.130 / 0.008 24/25 607.800 1.170 / 0.009 24/25 239.400 1.220 / 0.008 9 DGD 25/25 17.400 1.260 / 0.013 25/25 18.300 1.300 / 0.014 25/25 80.400 1.310 / 0.013 25/25 59.900 1.490 / 0.014 9 Akule 25/25 3.397 1.300 / 0.008 25/25 8.557 1.289 / 0.011 25/25 3.547 1.436 / 0.008 25/25 4.740 1.289 / 0.010 12 MPD 13/25 8.700 1.100 / 0.009 0/25 [FAIL] 0/25 [FAIL] 0/25 [FAIL] 12 MMD 25/25 56.000 1.120 / 0.006 2/25 62.500 1.150 / 0.006 1/25 95.100 1.220 / 0.008 1/25 122.600 1.140 / 0.008 12 DGD 24/25 25.000 1.280 / 0.014 25/25 32.900 1.350 / 0.016 25/25 131.500 1.330 / 0.013 25/25 84.000 1.500 / 0.015 12 Akule 25/25 6.110 1.302 / 0.009 25/25 15.513 1.291 / 0.012 25/25 6.644 1.464 / 0.011 25/25 11.941 1.304 / 0.012 15 MPD 3/25 10.300 1.090 / 0.009 0/25 [FAIL] 0/25 [FAIL] 0/25 [FAIL] 15 MMD 25/25 70.900 1.120 / 0.006 1/25 72.400 1.170 / 0.007 0/25 [FAIL] 0/25 [FAIL] 15 DGD 24/25 41.400 1.310 / 0.014 24/25 37.000 1.350 / 0.017 23/25 187.000 1.360 / 0.014 25/25 115.500 1.520 / 0.016 15 Akule 25/25 8.991 1.336 / 0.011 25/25 38.405 1.315 / 0.013 25/25 13.931 1.484 / 0.012 24/25 15.946 1.327 / 0.013 18 MPD 2/25 9.400 1.090 / 0.008 0/25 [FAIL] 0/25 [FAIL] 0/25 [FAIL] 18 MMD 24/25 86.300 1.130 / 0.006 2/25 87.100 1.180 / 0.007 0/25 [FAIL] 0/25 [FAIL] 18 DGD 24/25 65.700 1.330 / 0.015 19/25 47.800 1.370 / 0.017 21/25 324.800 1.380 / 0.015 25/25 219.000 1.550 / 0.016 18 Akule 25/25 12.058 1.333 / 0.011 25/25 37.381 1.335 / 0.017 22/25 76.923 1.487 / 0.013 24/25 66.399 1.372 / 0.017 Basic Dense Shelf Room SMD Akule SMD [FAIL] Akule SMD Akule SMD [FAIL] Akule Table 3. Highways and Conveyor (10 tasks per N; all methods share the environment-specific unary prior and Akule is adapted at N=20N=20 only), with one N=20N=20 task per environment shown below. Scarlet brackets mark selected collisions. Highways Conveyor N Method Success Plan (s) Quality (P/A) Success Plan (s) Quality (P/A) 3 MPD 4/10 2.152 1.935/0.038 4/10 2.287 3.019/0.093 3 MMD 10/10 2.642 1.846/0.038 10/10 2.923 3.157/0.103 3 Akule 10/10 0.905 1.841/0.039 10/10 1.058 3.125/0.104 6 MPD 0/10 [FAIL] 1/10 4.510 2.882/0.094 6 MMD 10/10 5.276 1.905/0.037 10/10 5.818 2.987/0.102 6 Akule 10/10 1.392 1.873/0.038 10/10 2.284 3.284/0.125 9 MPD 0/10 [FAIL] 0/10 [FAIL] 9 MMD 10/10 8.358 1.978/0.041 10/10 10.402 3.353/0.124 9 Akule 10/10 3.072 2.045/0.045 10/10 4.712 3.483/0.138 12 MPD 0/10 [FAIL] 0/10 [FAIL] 12 MMD 10/10 11.102 1.972/0.042 10/10 14.013 3.135/0.113 12 Akule 10/10 4.244 2.020/0.046 10/10 6.935 3.424/0.137 15 MPD 0/10 [FAIL] 0/10 [FAIL] 15 MMD 10/10 15.022 2.188/0.047 10/10 18.913 3.370/0.129 15 Akule 10/10 7.209 2.143/0.049 10/10 11.822 3.670/0.156 20 MPD 0/10 [FAIL] 0/10 [FAIL] 20 MMD 10/10 22.434 2.146/0.047 6/10 42.965 3.856/0.179 20 Akule 10/10 15.384 2.179/0.051 6/10 38.151 4.113/0.209 Highways, N=20N=20 MPD [FAIL] MMD Akule Conveyor, N=20N=20 MPD [FAIL] MMD Akule Table 4. ScaledWeave (10 scenes per N): the Weave geometry scaled to very large teams, with Akule trained at N=256N=256 and evaluated unchanged at smaller N. N Method Success Plan (s) 40 MPD 8/10 219.7 40 MMD-xECBS 10/10 176.4 40 Akule (sparse) 10/10 32.1 64 MPD 0/10 >>240 64 MMD-xECBS 0/10 >>240 64 Akule (sparse) 10/10 38.7 100 MPD 0/10 >>240 100 MMD-xECBS 0/10 >>240 100 Akule (sparse) 10/10 48.5 128 MPD 0/10 >>240 128 MMD-xECBS 0/10 >>240 128 Akule (sparse) 10/10 68.7 256 MPD 0/10 >>240 256 MMD-xECBS 0/10 >>240 256 Akule (sparse) 8/10 143.4 Canonical Weave. All three planners solve every panel (1280/1280, Table 1), so the comparison is one of computation and quality. Akule is 2.82.8–4.7×4.7× faster than MMD-xECBS and up to 3.8×3.8× faster than MPD across N∈10,20,28N∈\10,20,28\, with path lengths within 4% of MMD-xECBS and acceleration RMS 6–32% higher. On the N=28N=28 task shown, the independent MPD proposal is natively infeasible, while MMD-xECBS and Akule both complete the exchange. Projected-diffusion map families. With one model trained at N=18N=18, Akule succeeds on 595/600 instances across Basic, Dense, Shelf, and Room (Table 2), against 112/600 for MPD, 238/600 for MMD, and 584/600 for DGD; SMD solves 298/300 where reported (N≤9N≤ 9) but is 50–170× slower at N=9N=9. At N=18N=18, Akule is 1.31.3–5.4×5.4× faster than DGD (published values, not same-hardware). The hardest cells remain hard: success dips to 22–24/25 on Shelf and Room at N≥15N≥ 15, and DGD is marginally faster on Dense at N=15N=15. Table 2 also shows one N=18N=18 task per map: SMD fails on Dense and Room, while Akule reaches every goal. Obstacle environments and ScaledWeave. On Highways and Conveyor (Table 3), a model adapted only at N=20N=20 matches MMD’s success at every population (60/60 and 56/60; MPD: 4/60 and 5/60) while planning 1.11.1–3.8×3.8× faster, with path length within 7% and acceleration within 17% of MMD at N=20N=20. On ScaledWeave (Table 4), Akule solves every scene through N=128N=128 and 8/10 at N=256N=256, whereas MPD and MMD-xECBS time out on every scene from N=64N=64; at N=40N=40, the only population all methods complete, Akule is 5.5×5.5× faster than MMD-xECBS. 5.2. Key Findings and Ablation Studies Tables 1–5 support the central claim of this work: coordination can be allocated selectively inside the denoising process without giving up complete-planner success. Four key findings (KF) explain why, each paired with a figure that isolates one mechanism. (KF#1) Sparse selection preserves planning quality, and the share of pairwise work it removes grows with the team. Fig. 4a compares sparse Akule with the dense variant that evaluates every ordered pair. The sparse planner is consistently faster, and the fraction of RϕR_φ evaluations it skips grows with N, plateauing on Highways and Conveyor. Local congestion affects the pairwise denoising update, not team size; therefore, most of its N(N−1)N(N-1) dense corrections are close to zero, and the selector learns to recognize them using cheap pairwise descriptors. As the team grows, irrelevant pairs multiply faster than relevant ones, increasing the discarded share while the trajectory remains largely unchanged. The exception is extreme congestion (Shelf at N=18N=18, Table 5), where borderline pairs are dropped, and the sparse planner loses a few time-limited instances. (a) KF#1 (b) KF#2 (c) KF#3 Figure 4. (a) Sparse against dense Akule on 44 matched tasks: (i) fraction of RϕR_φ evaluations skipped, (i) complete planning time. (b) Success against population relative to the single training population (Ntrain=28N_ train=28 on Weave, 1818 on the map families, 2020 on Highways and Conveyor). (c) Pre-repair collisions per scene with and without Akule’s corrections, on matched scenes and noise before any repair (39 populations, 2,000 paired scenes); dotted guides mark 50/75/90% reductions. (KF#2) One coordination model transfers zero-shot across team sizes. Fig. 4b plots success against population for every family, normalized by the single training population; everything to the left of NtrainN_ train is zero-shot. Akule’s curves stay flat where MMD’s fall away. Nothing in the architecture sees N; the selector and composer operate on a set of pairwise descriptors and a permutation-invariant summary of the team, so a smaller team is simply a smaller set whose interactions are sub-configurations of those encountered in a crowded scene at the training population. Training at the largest feasible population is therefore the right default: it exposes the modules to the densest interactions they will meet, and the smaller teams come for free. (KF#3) The learned proposal hands repair far fewer conflicts, and that is where the speed-up comes from. Fig. 4c counts sampled collisions in the unary prior’s joint proposal before and after Akule’s corrections and before any smoothing or repair, on matched scenes and noise. Across all populations and all seven environments, every condition remains well below the diagonal by roughly the same fraction. Conflict-based repair pays for each remaining conflict with a constraint tree branch and low-level replans, so with an identical backend, this explains the speed-ups over MPD and MMD (Tables 1, 3); in the ablation (Table 5), variants leaving more conflicts take proportionally longer. The reduction is smallest where the prior already yields few conflicts (Weave, N=10N=10) and on Conveyor, whose narrow lanes leave corrections little room. (KF#4) The same formulation covers different map families and scales to hundreds of robots. The learned modules are shared across robots and conditioned on each robot’s own planning context, so a new map changes the unary prior; they correct not the coordination mechanism, carrying from free-space exchanges to cluttered maps, shelves, rooms, and the MMD environments (Tables 2 and 3). On ScaledWeave (Fig. 5), from 40 to several hundred robots, planning time grows more slowly than the team and per-robot time stays roughly constant, while the search-based baselines hit the time limit from the second population onward. Pair scoring, the quadratic part, is cheap; the expensive evaluation and composition scale with the selected degree, and the discarded share keeps rising with N (Fig. 4a). Conflict-based search faces a conflict count that grows faster than N and lacks such a lever. Figure 5. ScaledWeave (10 scenes per N): (a) complete planning time, with MPD and MMD-xECBS timing out on every scene from N=64N=64; (b) planning time per robot. Ablation Studies We ablate interaction selection and composition on Canonical Weave (N=28N=28, 30 instances) and Shelf (N=18N=18, 25 instances), using the largest population in each benchmark family. All variants share the frozen unary prior, pairwise model RϕR_φ, diffusion schedule, and downstream repair planner. Ablation variants. Dense Akule: evaluates all directed pairwise corrections using the learned composer. Raw sum: replaces GωG_ω with uniform weights (α=1α=1), retaining UψU_ψ. Calibrated sum: selects a constant coefficient from 0,0.25,0.5,0.75,1,1.5,2\0,0.25,0.5,0.75,1,1.5,2\ using full diffusion rollouts on held-out validation instances, prioritizing inter-robot conflicts on Weave and obstacle feasibility on Shelf (α=1α=1 and α=0α=0, respectively). Geometric selector: ranks pairs by Euclidean distance between their start positions. Collision-risk selector: ranks pairs by minimum time-aligned separation along the current step’s estimated denoised unary trajectories. Both heuristics select the global top-KtK_t directed pairs, where KtK_t matches the rounded mean interaction count of UψU_ψ at each diffusion step. Results. Table 5 shows that learned composition reduces pre-repair conflicts and downstream repair relative to uniform summation, while learned selection outperforms both geometric and collision-risk heuristics at comparable interaction budgets. On Weave, Akule produces 32.132.1 conflicts compared with 94.294.2 for uniform summation and 59.259.2/63.063.0 for the heuristic selectors, with similar trends on Shelf. Relative to dense Akule, sparse selection avoids 29.4%29.4\% and 80.0%80.0\% of pairwise corrections on Weave and Shelf, respectively, while reducing planning latency. All variants solve every Weave instance, although dense Akule achieves slightly higher success on Shelf (24/2524/25 versus 22/2522/25), illustrating the efficiency–reliability trade-off under greater congestion. Table 5. Ablation of UϕU_φ and GωG_ω on Weave and Shelf. Conf. is mean pre-repair collision pair-times, and RftR_ ft save is the fraction of pairwise corrections avoided relative to dense. Method Success Conf. Repair (s) Plan (s) Quality (P/A) R_ ft save Canonical Weave, N=28N=28 Akule 30/30 32.133 22.901 29.944 1.949/2.813 29.369% Akule (dense) 30/30 32.797 25.163 43.571 1.950/2.824 0% Raw/calibrated sum (α=1α=1) 30/30 94.200 43.573 50.756 1.935/2.602 29.740% Geometric selector 30/30 59.167 32.766 39.929 1.940/2.719 29.381% Collision-risk selector 30/30 62.967 34.948 40.113 1.949/2.794 29.381% Shelf, N=18N=18 Akule 22/25 24.800 73.632 76.923 1.487 / 0.013 80.0% Akule (dense) 24/25 19.200 71.051 81.495 1.512 / 0.014 0.0% Raw sum (α=1α=1) 21/25 61.960 83.179 86.835 1.556 / 0.016 81.8% Calibrated sum (α=0α=0) 20/25 152.800 103.958 107.744 1.517 / 0.018 79.6% Geometric selector 21/25 68.160 88.469 92.077 1.505 / 0.015 79.8% Collision-risk selector 21/25 43.240 79.909 82.403 1.504 / 0.013 79.8% 6. Conclusion and Limitations We introduced Akule, which augments a frozen single-robot diffusion prior with reusable pairwise corrections, learned sparse interaction selection, and context-aware signed composition for MRMP. Trained from physically optimized trajectories, these modules coordinate variable-sized robot teams without prescribed interaction graphs. Across diverse MRMP benchmarks, Akule achieves competitive planning success with 25×25× computational savings, zero-shot population transfer, and successful planning with up to 256256 robots. Ablations demonstrate the benefits of learned selection and composition in reducing the coordination burden before hard repair. Limitations and Future Work. Although sparse selection avoids expensive pairwise correction evaluations, the selector still scores O(N2)O(N^2) candidate interactions. Hierarchical candidate screening could reduce this overhead while preserving context-dependent selection. Moreover, composed updates remain within the span of learned pairwise corrections, potentially limiting coordination that requires more complex higher-order interactions. Extending the correction basis to capture such interactions is a promising direction. Finally, learned proposals do not guarantee feasibility, and downstream repair can dominate computation in highly congested settings. Incorporating repair directly in learning, alongside adaptive sparsity, could improve the trade-off between computational efficiency and planning success. References Ajay et al. (2023) A. Ajay, Y. Du, A. Gupta, J. B. Tenenbaum, T. S. Jaakkola, and P. Agrawal Is conditional generative modeling all you need for decision making?. In 11th International Conference on Learning Representations (ICLR), External Links: Link Cited by: §2. Barer et al. (2014) M. Barer, G. Sharon, R. Stern, and A. Felner Suboptimal Variants of the Conflict-Based Search Algorithm for the Multi-Agent Pathfinding Problem. Proceedings of the International Symposium on Combinatorial Search 5 (1), p. 19–27 (en). External Links: ISSN 2832-9163, Link, Document Cited by: §2. Battaglia et al. (2016) P. Battaglia, R. Pascanu, M. Lai, D. J. Rezende, and K. kavukcuoglu Interaction networks for learning about objects, relations and physics. In Proceedings of the 30th International Conference on Neural Information Processing Systems (NIPS), Cited by: §1. Carvalho et al. (2023) J. Carvalho, A.T. Le, M. Baierl, D. Koert, and J. Peters Motion planning diffusion: learning and planning of robot motions with diffusion models. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Cited by: §A.1, §1, §2, §5. Chi et al. (2025) C. Chi, Z. Xu, S. Feng, E. Cousineau, Y. Du, B. Burchfiel, R. Tedrake, and S. Song Diffusion policy: visuomotor policy learning via action diffusion. The International Journal of Robotics Research 44 (10-11), p. 1684–1704. External Links: Document, Link, https://doi.org/10.1177/02783649241273668 Cited by: §2. Du et al. (2023) Y. Du, C. Durkan, R. Strudel, J. B. Tenenbaum, S. Dieleman, R. Fergus, J. Sohl-Dickstein, A. Doucet, and W. S. Grathwohl Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC. In Proceedings of the 40th International Conference on Machine Learning, p. 8489–8510. External Links: ISSN 2640-3498 Cited by: §2. Erdmann and Lozano-Pérez (1987) M. Erdmann and T. Lozano-Pérez On multiple moving objects. Algorithmica 2 (1), p. 477–521 (en). External Links: ISSN 1432-0541, Link, Document Cited by: §2. He et al. (2026) Z. He, Y. Shaoul, and J. Li Model-Based Diffusion Optimal Control for Multi-Robot Motion Planning. In Proceedings of Robotics: Science and Systems, Sydney, Australia. External Links: Document Cited by: §2. Ho et al. (2020) J. Ho, A. Jain, and P. Abbeel Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, Vol. 33, p. 6840–6851. Cited by: §A.4, §4.4. Hönig et al. (2018) W. Hönig, J. A. Preiss, T. K. S. Kumar, G. S. Sukhatme, and N. Ayanian Trajectory Planning for Quadrotor Swarms. IEEE Transactions on Robotics 34 (4), p. 856–869. External Links: ISSN 1941-0468, Link, Document Cited by: §2. Ichter et al. (2018) B. Ichter, J. Harrison, and M. Pavone Learning sampling distributions for robot motion planning. In 2018 IEEE International Conference on Robotics and Automation (ICRA), Vol. , p. 7087–7094. External Links: Document Cited by: §2. Janner et al. (2022) M. Janner, Y. Du, J. B. Tenenbaum, and S. Levine Planning with diffusion for flexible behavior synthesis. In 39th International Conference on Machine Learning (ICML), Cited by: §1, §2, §3. Lee et al. (2026) H. Lee, X. Wang, T. Kim, and D. Panagou Distributed model-based diffusion for scalable multi-robot trajectory optimization. External Links: 2607.20992, Link Cited by: §2. Li et al. (2026) B. Li, R. Zhang, H. Liang, J. Zhang, J. Zhang, X. Chen, L. Xu, J. Yu, and J. Wang InterAgent: physics-based multi-agent command execution via diffusion on interaction graphs. Note: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) External Links: 2512.07410, Link Cited by: §2. Li et al. (2022) J. Li, Z. Chen, D. Harabor, P. J. Stuckey, and S. Koenig MAPF-LNS2: Fast Repairing for Multi-Agent Path Finding via Large Neighborhood Search. Proceedings of the AAAI Conference on Artificial Intelligence 36 (9), p. 10256–10265 (en). External Links: ISSN 2374-3468, Link, Document Cited by: §2. Li et al. (2021) J. Li, W. Ruml, and S. Koenig EECBS: A Bounded-Suboptimal Search for Multi-Agent Path Finding. Proceedings of the AAAI Conference on Artificial Intelligence 35 (14), p. 12353–12362 (en). External Links: ISSN 2374-3468, Link, Document Cited by: §2. Liang et al. (2025a) J. Liang, J. K. Christopher, S. Koenig, and F. Fioretto Simultaneous multi-robot motion planning with projected diffusion models. In Proceedings of the 42nd International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 267, p. 37162–37180. External Links: Link Cited by: §C.1, §1, §1, §1, §2, §3, §5, §5. Liang et al. (2025b) J. Liang, S. Koenig, and F. Fioretto Discrete-guided diffusion for scalable and safe multi-robot motion planning. External Links: 2508.20095, Link Cited by: §C.3, §1, §1, §2, §5. Liang et al. (2026) J. Liang, S. Koenig, and F. Fioretto Simulation-informed diffusion for decentralized multi-robot motion planning. External Links: 2605.27697, Link Cited by: §2. Liu et al. (2022) N. Liu, S. Li, Y. Du, A. Torralba, and J. B. Tenenbaum Compositional visual generation with composable diffusion models. In Proceedings of the European Conference on Computer Vision (ECCV), p. 437–454. Cited by: §2. Luo et al. (2024) Y. Luo, C. Sun, J. B. Tenenbaum, and Y. Du Potential based diffusion motion planning. In Proceedings of the 41st International Conference on Machine Learning, R. Salakhutdinov, Z. Kolter, K. Heller, A. Weller, N. Oliver, J. Scarlett, and F. Berkenkamp (Eds.), Proceedings of Machine Learning Research, Vol. 235, p. 33486–33510. External Links: Link Cited by: §2, §2. Mishra et al. (2025) U. A. Mishra, Y. Chen, and D. Xu Generative Factor Chaining: Coordinated Manipulation with Diffusion-based Factor Graph. In Proceedings of The 8th Conference on Robot Learning, p. 1461–1472. External Links: ISSN 2640-3498 Cited by: §1, §2. Mishra et al. (2023) U. A. Mishra, S. Xue, Y. Chen, and D. Xu Generative Skill Chaining: Long-Horizon Skill Planning with Diffusion Models. In Proceedings of The 7th Conference on Robot Learning, p. 2905–2925. External Links: ISSN 2640-3498 Cited by: §2. Mitra et al. (2026) S. Mitra, E. Yuceel, N. Giles, and A. Pai Factored diffusion policies: compositionally generalized robot control with a single score network. External Links: 2605.22596, Link Cited by: §2. Okumura et al. (2019) K. Okumura, M. Machida, X. Défago, and Y. Tamura Priority inheritance with backtracking for iterative multi-agent path finding. In Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), p. 535–542. Cited by: §2. Okumura (2023) K. Okumura LaCAM: search-based algorithm for quick multi-agent pathfinding. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 37, p. 11655–11662. Cited by: §2. Qureshi et al. (2019) A. H. Qureshi, A. Simeonov, M. J. Bency, and M. C. Yip Motion planning networks. In 2019 International Conference on Robotics and Automation (ICRA), Vol. , p. 2118–2124. External Links: Document Cited by: §2. Sánchez and Latombe (2002) G. Sánchez and J. Latombe On Delaying Collision Checking in PRM Planning: Application to Multi-Robot Coordination. The International Journal of Robotics Research 21 (1), p. 5–26. External Links: Link, Document Cited by: §2. Shaoul et al. (2025) Y. Shaoul, I. Mishani, S. Vats, J. Li, and M. Likhachev Multi-robot motion planning with diffusion models. In International Conference on Learning Representations, Cited by: §A.1, §A.6, §1, §1, §1, §2, §2, §3, §3, §5, §5. Sharon et al. (2012) G. Sharon, R. Stern, A. Felner, and N. Sturtevant Conflict-Based Search For Optimal Multi-Agent Path Finding. Proceedings of the AAAI Conference on Artificial Intelligence 26 (1), p. 563–569 (en). External Links: ISSN 2374-3468, Link, Document Cited by: §2. Silver (2005) D. Silver Cooperative Pathfinding. Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment 1 (1), p. 117–122 (en). External Links: ISSN 2334-0924, Link, Document Cited by: §2. Sun et al. (2026) T. Sun, U. A. Mishra, J. Lu, D. Xu, and I. Armeni Energy-based compositional diffusion planning. In Proceedings of the 43rd International Conference on Machine Learning, External Links: Link Cited by: §2. van den Berg et al. (2011) J. van den Berg, S. J. Guy, M. Lin, and D. Manocha Reciprocal n-Body Collision Avoidance. In Robotics Research, C. Pradalier, R. Siegwart, and G. Hirzinger (Eds.), Berlin, Heidelberg, p. 3–19 (en). External Links: ISBN 978-3-642-19457-3, Document Cited by: §2. Wagner and Choset (2011) G. Wagner and H. Choset M*: a complete multirobot path planning algorithm with performance bounds. In 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, p. 3260–3267. Cited by: §2. Weilbach et al. (2023) C. D. Weilbach, W. Harvey, and F. Wood Graphically structured diffusion models. In Proceedings of the 40th International Conference on Machine Learning (ICML), Cited by: §2. Yu et al. (2024) X. Yu, S. Zhang, X. Song, X. Qin, and S. Jiang Trajectory diffusion for ObjectGoal navigation. In Advances in Neural Information Processing Systems, A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang (Eds.), Vol. 37, p. 110388–110411. External Links: Document Cited by: §2. Zaheer et al. (2017) M. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. Salakhutdinov, and A. J. Smola Deep sets. In Advances in Neural Information Processing Systems, Vol. 30, p. 3391–3401. Cited by: §1, §4.2. Zeng et al. (2025) X. Zeng, H. Su, Z. Wang, and Z. Lin Graph diffusion for robust multi-agent coordination. In Proceedings of the 42nd International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 267, p. 74157–74174. External Links: Link Cited by: §2, §2. Zhang et al. (2023) Q. Zhang, J. Song, X. Huang, Y. Chen, and M. Liu DiffCollage: Parallel Generation of Large Content with Diffusion Models. In 2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), p. 10188–10198. External Links: ISSN 2575-7075, Document Cited by: §2. Zhu et al. (2024) Z. Zhu, M. Liu, L. Mao, B. Kang, M. Xu, Y. Yu, S. Ermon, and W. Zhang MADiff: offline multi-agent learning with diffusion models. In Advances in Neural Information Processing Systems, Vol. 37. External Links: Document Cited by: §2. Appendix Appendix A Architecture and Implementation This supplement specifies the implementation behind the sparse interaction model, its training dependencies, and the boundaries of the reported measurements. We retain the main paper’s notation: fθf_θ is the frozen unary diffusion prior, RϕR_φ the pairwise correction model, UψU_ψ the selector, and GωG_ω the composer. A directed pair (i,j)(i,j) means that robot j influences robot i. All three interaction modules share parameters across robots. Physical trajectory support h and reverse-diffusion index t are distinct: the standard trajectory has H=64H=64 supports, while the reverse chain has T=25T=25 transitions, indexed 24,…,024,…,0. A.1. State representation and frozen unary prior The grouped trajectory tensor is B×64×N×4B× 64× N× 4. Its physical channels are two positions and two stored displacements. They are not interchangeable with velocities in metres per second. The canonical codec is an unclipped affine transformation, xphys=s⊙xnorm+bx^phys=s x^norm+b, with s and b fixed by the pretrained normalizer. Highways and Conveyor use a velocity bridge: the native prior’s velocity channels are multiplied by Δt=5/64 t=5/64 to obtain displacement channels and divided by Δt t when encoded again. The conditioning tensor contains the start and goal four-channel states, hence eight endpoint values per robot. The start and goal positions and prescribed endpoint motion channels are reinstated after the reverse updates. The canonical unary prior is the released MPD temporal U-Net (Carvalho et al., 2023; Shaoul et al., 2025). It maps 64×464× 4 noisy single-robot states to equally sized noise predictions. The base width is 32 and the channel multipliers are (1,2,4,8)(1,2,4,8), giving widths (32,64,128,256)(32,64,128,256). Each down stage has two residual temporal blocks, followed by downsampling except at the deepest stage; the bottleneck has two residual blocks, and three up stages concatenate encoder skip features. A residual block contains two kernel-5 convolutions with GroupNorm and Mish, an additive time projection, and a channel-matching residual connection. Downsampling uses a kernel-3 stride-2 convolution; upsampling uses a kernel-4 stride-2 transposed convolution. The output applies a kernel-5 block and a 1×11× 1 convolution to four channels. The 32-dimensional time encoder uses a sinusoidal embedding and an MLP. The frozen configuration uses epsilon prediction and the 25-step exponential variance schedule. Highways and Conveyor load their own released unary checkpoints and normalizers with the same four-level width configuration. For SMD maps, the frozen unary model adds goal-conditioned spatial features to the temporal backbone. A 64×6464× 64 raster contains occupancy, clipped signed distance, and start/goal Gaussian heatmaps with standard deviation 0.070.07 m. Five spatial levels have widths (16,24,32,48,64)(16,24,32,48,64), two kernel-3 convolutions with SiLU per level, and stride-2 convolutions between levels. Bilinear samples from all five levels are concatenated and projected from 184 channels to 32 local features. Mean and maximum pooling of the deepest level produce 128 features projected to a 32-dimensional global token. Four exact local geometry features augment the local vector to 36 channels: scaled support clearance, its two-dimensional unit gradient, and scaled clearance at the following segment midpoint (the final support reuses its own clearance). Learned local and global projections inject this context into each temporal residual block. These parameters are adapted before coordination training and remain frozen during R/G/UR/G/U training and evaluation. A.2. Clean-state implementation of the pairwise correction The main paper expresses coordination as an additive epsilon correction. The implementation instead predicts a smooth physical clean-position correction and converts the composed clean state through the usual diffusion posterior. Let btib_t^i denote the decoded, endpoint-conditioned unary clean estimate. The 29-channel pair sequence concatenates (8) χRij,t(h)=[bti(h),btj(h),ei,ej,ptj(h)−pti(h),∥ptj(h)−pti(h)∥,t/24,h/63], splitχ^R_ij,t(h)= [&b_t^i(h),b_t^j(h),e_i,e_j,p_t^j(h)-p_t^i(h),\\ &\|p_t^j(h)-p_t^i(h)\|,t/24,h/63 ], split where ei,ej∈ℝ8e_i,e_j ^8. Thus the descriptor is computed from unary clean estimates of the noisy reverse state, not by directly feeding the normalized noisy states to every interaction module. RϕR_φ applies a 1×11× 1 temporal convolution from 29 to 64 channels and SiLU, followed by four residual kernel-5 convolutions at dilations 1,2,4,81,2,4,8, each with SiLU. Adaptive average pooling produces 12 control locations; a 1×11× 1 head maps 64 channels to two position components. The decoder uses a cubic B-spline matrix constructed with 16 controls, retaining its 12 interior columns. Removing the first and last two controls enforces zero position correction and endpoint derivative. The deployed primitive subtracts the same network’s null-partner prediction: (9) dij,t=Rϕ(bti,btj,ei,ej,t)−Rϕ(bti,0,ei,0,t),dij,t∈ℝ64×2.d_ij,t=R_φ(b_t^i,b_t^j,e_i,e_j,t)-R_φ(b_t^i,0,e_i,0,t), d_ij,t ^64× 2. This centering is part of the forward computation. An “interaction evaluation” in the reported savings counts an admitted directed primitive; it does not count the real and null network branches as two separate graph edges. A.3. Selector descriptor and contextual pooling The selector builds a 27-channel sequence for every candidate directed pair. It contains scaled ego and partner states (eight values), relative position and displacement (four), normalized separation minus one and an approach dot product (two), five geometric scene summaries for each endpoint robot (ten), a support-state scalar (one), and physical and reverse time (two). The position scale is 0.950.95 m, the displacement scale is 0.5Δt0.5 t, and relative distance is scaled by 0.1050.105 m. At each support, the five scene summaries are nearest normalized separation, summed and maximum inverse-quadratic proximity, and a two-dimensional proximity-weighted relative direction. The support-state input is zero for one-shot deployed scoring. An elementwise asinhasinh limits feature dynamic range without clipping. A shared linear layer 27→3227→ 32 with SiLU and a kernel-5 convolution 32→3232→ 32 with SiLU encode the sequence. Within eight blocks of eight supports, mean and maximum pooling yield 64 features. These are concatenated with 24 transformed endpoint features (ei,ej,ej−ei)(e_i,e_j,e_j-e_i) and mapped 88→4888→ 48 with SiLU. Mean and maximum pooling over the eight blocks, followed by 96→4896→ 48 with SiLU, yields an edge token hijh_ij. The abstract PoolPool in the main paper is implemented by learned-query attention pooling. Four shared inducing queries and four attention heads summarize (i) outgoing edges of the ego, (i) incoming edges of the partner, and (i) all directed edges. Self-edges are masked. Each four-query output is flattened and mapped 192→48→48192→ 48→ 48 with SiLU. A residual context update 192→48→48192→ 48→ 48 lets each edge incorporate its three set summaries; pooling is then repeated. The final contextual head is 192→48→1192→ 48→ 1 with SiLU. The frozen selector also has a shared local residual head 48→192→148→ 192→ 1, whose output is added to the contextual logit. The extra per-agent density head supported by the code is not enabled in the canonical checkpoint. Deployment admits precisely zij,t>0z_ij,t>0, with no fixed degree cap. A.4. Composer and signed temporal weights GωG_ω encodes each admitted edge using the 29 pair features, the two residual components divided by 0.10.1, and the gate, giving 32 input channels. Its temporal stack has width 112, a kernel-1 entry convolution, and four residual kernel-5 layers at dilations 1,2,4,81,2,4,8. SiLU follows each convolution. Adaptive average pooling produces eight temporal blocks. Two residual relational layers, each 336→112→112336→ 112→ 112 with SiLU, combine an edge token with normalized incident-edge sums at both robots. Gate-weighted aggregation excludes rejected interactions. For each ego, the subsequent set representation concatenates mean, maximum, and degree-square-root-normalized sums of 112-dimensional tokens; three corresponding residual-magnitude summaries; admitted degree divided by N−1N-1; available degree divided by N; and population divided by 100. This 342342-dimensional vector maps to 112 features with SiLU. A node MLP 14→112→11214→ 112→ 112 consumes the four block-averaged state values, eight endpoint values, and two time values. Scene mean/max node pooling and population yield a 225→112225→ 112 scene token. The edge head concatenates the edge token, two node tokens, two set tokens, one scene token, and two time scalars. It maps 674→112→1674→ 112→ 1 with SiLU. For each admitted pair it outputs eight coefficients (10) βij,t,k=sttanh(ℓij,t,k),wij,t(h)=∑k=18Bhk(8)βij,t,k. _ij,t,k=s_t ( _ij,t,k), w_ij,t(h)= _k=1^8B^(8)_hk _ij,t,k. B(8)B^(8) is the saved cubic B-spline basis, and st>0s_t>0 is a fixed, training-calibrated trust bound. The main paper’s scalar αij,t _ij,t suppresses this dependence on physical support. The deployed update is (11) δpi(h) δ p_i(h) =∑j≠imij,twij,t(h)dij,t(h), = _j≠ im_ij,t\,w_ij,t(h)d_ij,t(h), (12) b~t b_t =bt+[δp,Dδp], =b_t+[δ p,Dδ p], where DδpDδ p is the forward support difference with zero hard endpoint channels. Differencing the weighted sum accounts for variation of w along the trajectory. The codec then encodes b~t b_t and applies the DDPM posterior. In normalized coordinates the corresponding epsilon change is −α¯t/(1−α¯t)δx0- α_t/(1- α_t)\,δ x_0 (Ho et al., 2020); physical position corrections must first pass through the codec and displacement construction. Table 6. Frozen canonical interaction modules. Parameter counts exclude fixed spline bases and scale buffers. Module Output per directed pair Parameters RϕR_φ (width 64) 64×264× 2 correction 84,226 UψU_ψ (width 48) one logit 87,938 GωG_ω (width 112) eight signed coefficients 509,601 A.5. Equivariance and computational boundary The specifications in Table 6 are independent of population. No learned module uses a robot’s array index as a feature. Shared pair encoders, masked set attention, symmetric reductions, and scatter-add composition commute with a simultaneous permutation of robots and their endpoints. The resulting trajectory update and directed mask therefore permute with the input. This is architectural permutation equivariance, up to floating-point reduction order. It does not imply rotation invariance, and the physical axes remain meaningful. The selector still scores N(N−1)N(N-1) candidates (the vectorized encoder also forms masked diagonal tokens). Sparse admission saves evaluations of RϕR_φ and the admitted-edge composer; it does not make candidate scoring subquadratic. If no edge is admitted, composition returns a zero correction. The graph is recomputed at every interaction-aware reverse step, without a monotonic sparsification constraint. A.6. Complete planner and repair interface For Weave and SMD the complete proposal algorithm is: (1) sample the saved initial noise and condition endpoints; (2) for t=24,…,0t=24,…,0, predict unary clean states, encode selector descriptors, admit directed edges, evaluate centered pair corrections on those edges, compose signed corrections, and apply the conditioned posterior with the prescribed noise; (3) decode the final proposal; (4) perform the benchmark’s static cleanup and hard repair; and (5) validate the returned complete trajectory. The posterior noise is zero at t=0t=0. For Highways and Conveyor, step (2) is replaced by independent candidate generation and selection followed by a single t=0t=0 interaction correction. Section C specifies this distinction. Akule supplies a root trajectory to downstream conflict-based search (Shaoul et al., 2025). The repair backend detects remaining conflicts, branches on constraints, and regenerates affected robot trajectories. SMD additionally uses targeted static repair with the native geometry and endpoint adapter; low-level RRT fallbacks are part of that implementation. Successful proposal generation alone is not complete-planner success, and the learned model provides no collision-free guarantee before repair. Appendix B Learning and Teacher Generation Learning follows the dependency order unary prior →Rϕ→Gω→Uψ→ R_φ→ G_ω→ U_ψ. A later stage consumes frozen earlier modules. Physical trajectory teachers define useful coordination outcomes, not a uniquely correct interaction graph. We describe the optimization and supervision separately from checkpoint selection: a training stage can be executed without its last checkpoint being the deployed model. B.1. Matched pair optimization and residual supervision For a common unary clean estimate b, both pair optimizations start from the same positions, preserve the same endpoints, and use the same base objective. The interaction solve includes pair clearance; the null solve sets its weight to zero. The position variables are optimized by Adam at 0.010.01 for 180 updates with projection to the workspace. Motion channels are reconstructed as forward position differences. The best observed objective is retained separately for each example. In the pair-target configuration, (13) J=50C+0.2QGP+0.2L+0.05Qtrust+100Qbox,J=50C+0.2Q_GP+0.2L+0.05Q_trust+100Q_box, with the 50C50C term removed in the null solve. Specifically, (14) C C =1N∑h,i<j[(0.110−‖pi(h)−pj(h)‖)+0.105]2, = 1N _h,i<j [ (0.110-\|p_i(h)-p_j(h)\|)_+0.105 ]^2, (15) L L =1Nℓ0∑h,i‖pi(h+1)−pi(h)‖, = 1N _0 _h,i\|p_i(h+1)-p_i(h)\|, (16) Qtrust Q_trust =meanh,i,d[(p−bp)2]0.01, = mean_h,i,d[(p-b_p)^2]0.01, (17) Qbox Q_box =1N∑h,i,d[(|pid(h)|−0.95)+0.05]2. = 1N _h,i,d [ (|p_id(h)|-0.95)_+0.05 ]^2. The GP regularizer uses v=w/Δtv=w/ t, ep=ph+1−ph−Δtvhe_p=p_h+1-p_h- t\,v_h, and ev=vh+1−vhe_v=v_h+1-v_h. It averages over robots the sum over supports and coordinates of (18) 12Δt3ep2−12Δt2epev+4Δtev2, 12 t^3e_p^2- 12 t^2e_pe_v+ 4 te_v^2, then divides by the fixed GP scale. The canonical residual checkpoint records ℓ0=1.7466968662 _0=1.7466968662 and GP scale 43.941084270243.9410842702. These differ from the composition-teacher scales below; they must not be exchanged. If prealp^real and pnullp^null are the optimized positions, the stored regression target is (19) y=Π(preal−pnull),Π=BintBint†,y= (p^real-p^null), =B_intB_int , where BintB_int contains the 12 interior cubic-spline columns. The noise-space target in the main paper follows by applying the codec and the DDPM clean/noise correspondence. In practice, supervision is mean squared position error on the centered primitive, divided by the stored scale 1.415236620232463×10−41.415236620232463× 10^-4. This avoids claiming that a physical displacement target is already a normalized epsilon target. Both pair directions are supervised. The canonical R optimizer is Adam with learning rate 10−410^-4, β=(0.9,0.999)β=(0.9,0.999), ϵ=10−8ε=10^-8, and zero weight decay. Its replay fraction is zero. The checkpoint records update 6,000 and freezes the unary prior, codec, G, and U during this residual adaptation. The SMD residual corpus uses 16 training and two selection scenes per family over Empty, Basic, Dense, Shelf, and Room. Five unary reverse states are collected per scene. At each state up to eight unordered pairs are sampled from each minimum-distance stratum: below 0.1100.110 m, [0.110,0.250)[0.110,0.250) m, and at least 0.2500.250 m. Actual retained counts depend on pair qualification. The selected target policy is obstacle-agnostic: the matched pair optimization has an empty obstacle list, while the source unary trajectories remain map-conditioned. This separates the learned pair correction from the obstacle prior; it does not claim that all target trajectories satisfy the original map’s obstacles. The trainer uses batches of 64: six examples from each of ten family/near–far bins and four rotating extra examples. Adam uses 10−410^-4, with gradient norm clipping at 5 and rejection of nonfinite loss/gradients; checkpoints are saved every 500 updates. The selected SMD residual is update 6,500, selected with an additional disjoint composition panel rather than pair MSE alone. B.2. Offline dense composition teachers The canonical teacher corpus contains 25,000 qualified geometries partitioned into 20,000 training, 2,500 selection, and 2,500 shadow geometries, with eight deterministic noise realizations per geometry. A frozen dense parent proposal is produced by a 25-step rollout. At its terminal clean-state interface, the optimizer adjusts eight signed coefficients for each directed frozen pair field. This is the concrete implementation of the main paper’s trajectory-space composition optimization; it is not a separate free-trajectory replacement of the learned proposal. The parameterization is β=s0qβ=s_0q with q∈[−1,1]q∈[-1,1]. The 25 trust scales are calibrated on training states alone using a 99.599.5th percentile coefficient-magnitude rule and a 1.251.25 safety multiplier; the terminal scale is s0=2.0591923185s_0=2.0591923185. The complete scale vector is included in the compact supporting data. No scale is fit to final-test outcomes. The optimizer uses float32 composition and gradients, transported to a float64 SciPy L-BFGS-B solver. Four penalty levels are 2×1032× 10^3, 2×1052× 10^5, 2×1072× 10^7, and 2×1092× 10^9, each with at most 80 iterations and 320 objective evaluations, ftol=10−10 ftol=10^-10, gtol=10−8 gtol=10^-8, and at most 30 line-search steps. For positions p(q)p(q) it minimizes (20) Jλ(q)=λCswept,0.120+0.2QGP+0.2L+0.05Qtrust+1000Qbox+0.01Qguide, splitJ_λ(q)=&λ C_swept,0.120+0.2Q_GP+0.2L\\ &+0.05Q_trust+1000Q_box+0.01Q_guide, split where Cswept,0.120C_swept,0.120 sums squared clearance deficits divided by 0.012N0.01^2N at exact closest points of synchronized linear support intervals. Qguide=mean[(p−pguide)2]/0.01Q_guide=mean[(p-p^guide)^2]/0.01. The GP and length scales here are 97.055659488497.0556594884 and 1.87587272801.8758727280. The guide is secondary, and the teacher always remains a composition of the frozen pair fields. The Weave guide is generated with ORCA and refined by 480 Adam updates at 0.010.01. Restart order is fixed: the initial signed fit, zero coefficients, a committed robust start, and, when needed, a guide fit and three seeded perturbations. A swept-constraint projection can refine an otherwise numerically stable candidate. Qualification requires sampled and swept separation of at least 0.1200.120 m, valid endpoints and workspace, finite values, and the stored motion envelope. Path, GP, acceleration, and jerk cannot exceed 1.101.10 times their initial-reference values. Correction RMS is bounded by the larger of 1.51.5 times its reference value and that value plus 0.030.03 m. At most 20% of coefficients may saturate; float32/float64 composition disagreement must have RMS at most 0.0010.001 m and maximum at most 0.0030.003 m. Qualified candidates are ordered by the original physical objective, guide fit, restart name, and penalty phase. Failed attempts do not enter the qualified corpus. SMD teacher generation uses the same frozen-field principle with the map-specific physical contract. Its final training/selection manifest contains 204/26 qualified geometries per family, or 1,020/130 in total, with eight noise realizations. Geometry, obstacle, unary, residual, and teacher hashes are stored together. Thus these corpus sizes describe qualified examples and should not be confused with the number of attempted optimizations or the evaluation panel sizes. B.3. Composer transition bootstrap A teacher-consistent reverse chain is constructed from the optimized clean trajectory and saved noise. For each sampled step, the unary-only next state is subtracted from both the predicted and teacher next states. Let Δyt y_t and Δyt⋆ y_t be these physical transition differences, separated into position and displacement channels c∈p,wc∈\p,w\. The implemented loss is (21) ℒGtrans=[wt∑c∈p,wmean[(Δyt,c−Δyt,c⋆)2]max(10−6,mean[(Δyt,c⋆)2])].L_G^trans=E [w_t _c∈\p,w\ mean[( y_t,c- y_t,c )^2] (10^-6,mean[( y_t,c )^2]) ]. The denominator is detached. This is the normalized physical posterior-transition implementation of the abstract denoising-matching loss. The saved residual-confidence timestep weights are normalized to mean one. Timesteps are stratified over three portions of the 25-step schedule. The composer is initialized freshly with a normally distributed final-head weight (standard deviation 0.010.01) and zero bias. AdamW uses learning rates 3×10−53× 10^-5 for the temporal/relational trunk and 10−410^-4 for the remaining context/head parameters, weight decay 10−510^-5, and gradient norm clipping at 5. A training update uses one geometry/noise condition. Fixed 16-scene sentinel and 128-scene selection panels monitor the canonical fit; the full corpus’s shadow split is not used for checkpoint selection. B.4. Complete-rollout refinement The next stage propagates the composer’s own outputs through all 25 reverse steps. Its loss is (22) ℒGroll= _G^roll= Huber0.05(pG,p⋆)+λcC0.120(pG)+10−3ℒGtrans, _0.05(p_G,p )+ _cC_0.120(p_G)+10^-3L_G^trans, (23) C0.120(p) C_0.120(p) =1N∑h,i<j[(0.120−‖pi(h)−pj(h)‖)+0.01]2, = 1N _h,i<j [ (0.120-\|p_i(h)-p_j(h)\|)_+0.01 ]^2, with λc=3.9508342635550774×10−6 _c=3.9508342635550774× 10^-6. AdamW is reset and both learning rates are divided by ten. Autograd checkpointing limits memory, while gradients pass through the reverse states and frozen modules’ computations; frozen parameters receive no optimizer updates. Nonfinite gradients fail the update, and the trainable gradient norm is clipped at 5. The canonical schedule cycles through 4,096 training geometries; the selected checkpoint is refinement update 3,290. The SMD recipe also evaluates transition and rollout candidates, but its final repair-based selection retains transition-stage update 14,500. It would be inaccurate to infer a deployed rollout-refined composer solely from the existence of a refinement script. Selection prioritizes native complete-planner success, then repair and total time, then constraint-tree, low-level, and RRT workload on held-out scenes. B.5. Support optimization, selector bootstrap, and refinement With the unary, residual, and composer frozen, an offline tensor of edge logits is optimized with Adam at 0.030.03 for at most 50 updates. Off-diagonal logits initialize to 0.010.01. The objective adds expected mean directed degree to the physical teacher-matching and clearance loss: (24) ℒU=Huber0.05(pU,p⋆)+λcC0.120(pU)+λs1TN∑t,i≠jσ(zij,t),L_U=Huber_0.05(p_U,p )+ _cC_0.120(p_U)+ _s 1TN _t,i≠ jσ(z_ij,t), where λs=1.885424111516031×10−5 _s=1.885424111516031× 10^-5. Hard masks are used forward, with (25) mST=sg([z>0]−σ(z))+σ(z)m^ST=sg\! (1[z>0]-σ(z) )+σ(z) for backpropagation. Dense fields provide gradient credit to currently rejected edges during optimization; inference evaluates admitted fields only. The gate multiplies a residual contribution once, avoiding an accidental squared straight-through gate. Candidate supports are checked at updates 0,5,10,25,500,5,10,25,50 against goal completion and a 10% motion-quality envelope. The support optimizer seeks a useful physical/sparsity trade-off, not an identifiable ground-truth graph. The canonical target manifest records 256 attempted conditions and 130 valid targets. The first 128 valid conditions give 128×25=3,200128× 25=3,200 supervised states. Selector bootstrap uses a Huber loss with threshold 0.10.1, separately averaged over retained and removed outgoing edges and eligible robots, weighting retained edges twice as strongly. AdamW uses 5×10−45× 10^-4, weight decay 10−510^-5, batches of eight, and gradient clipping at 5. Three sweeps give 1,200 optimizer updates; this is the deployed canonical selector checkpoint. SMD generates 26 valid support conditions per family (130 total), from 154 attempts. Its selector additionally undergoes complete-rollout straight-through refinement with the unary, R, and G frozen. It uses AdamW at 5×10−55× 10^-5, weight decay 10−510^-5, and clip norm 5. Update 1,000 is selected from the fixed validation frontier through complete-planner evaluation. The refinement uses the same physical loss and degree penalty. The shadow panel is consulted only after freezing the selected checkpoint. B.6. Map adaptation and available compute measurements The Highways/Conveyor adaptation interface operates at N=20N=20 on terminal selected-candidate states. The saved training contract specifies 128 training and 32 validation scenes per map, a shared composer without environment-ID features, bounded coefficient teachers with 240 Adam steps at 0.0250.025 and two starts, and a native adherence guard. Transition bootstrap uses the canonical optimizer for 4,000 updates and terminal-rollout refinement for 400; validation is separate from the final ten-scene-per-population panels. This terminal interface has posterior coefficients c1=1,c2=0c_1=1,c_2=0 and zero posterior noise. It must not be interpreted as training or applying 25 interaction corrections to these map candidates. The final batched map pipeline continues the shared composer for 1,500 updates (1,000 transition updates followed by 500 physical-refinement updates) on 128 qualified training scenes per map. It freezes composer update 1,500 and selector update 200. Selector adaptation uses the same balanced Huber loss, AdamW at 5×10−45× 10^-4, decay 10−510^-5, clip norm 5, and batches containing four Highways and four Conveyor states. The budget is 1,000 updates with checks every 100 and stopping after two nonimproving checks; complete-planner selection uses five disjoint validation tasks per map. The frozen residual remains unchanged. End-to-end training and teacher-generation compute is not uniformly recorded, so we do not provide a total training-cost claim. Two canonical checkpoint records contain partial elapsed times: 3,820.15 s through the selected G refinement checkpoint and 11.89 s for the three-sweep U fit. These exclude prior training, teacher generation, and other selection runs, and are not a matched hardware comparison. The SMD unary-intervention replay’s 130.14 s for 600 paired scenes is an evaluation measurement, not training cost. Appendix C Experimental Protocol The benchmarks share the sparse-interaction decomposition but use different proposal interfaces and native validation contracts. Table 7 summarizes the evaluated grids. Population transfer means reusing a frozen coordination model at the stated test population; it does not imply an unchanged environment-specific unary prior or identical downstream planning software across benchmark families. Table 7. Evaluation populations and proposal interfaces. Scene counts are per population, except the three explicitly listed Weave counts. The 39-condition intervention study comprises the first seven environments (2,000 paired scenes). ScaledWeave is a separate ten-scene-per-population family. Source: frozen scene and runtime manifests. Environment Evaluated populations N Scenes per N Interaction interface Canonical Weave 10, 20, 28 512, 256, 512 25 reverse steps Highways 3, 6, 9, 12, 15, 20 10 one correction at t=0t=0 Conveyor 3, 6, 9, 12, 15, 20 10 one correction at t=0t=0 SMD Basic / Dense / Shelf / Room 3, 6, 9, 12, 15, 18 25 per environment 25 reverse steps ScaledWeave 40, 64, 100, 128, 256 10 separate scaled geometry and physical clock C.1. Scenes, geometry, and physical representation Canonical Weave. Robots exchange opposite sides of the fixed obstacle-free square [−1,1]2[-1,1]^2 m. The construction starts on a perturbed angular ring and snaps a dominant coordinate to the square-side boundary; the goal reflects that coordinate. Robot radius is 0.050.05 m, and the frozen center bound is 0.950.95 m. The coordination model is trained at N=28N=28 and transferred to N=10,20N=10,20. Weave’s fixed construction is retained rather than enlarged for higher populations. The N=28N=28 panel is the predeclared first 512 executed conditions of a 1,024-condition frozen manifest. The smaller-population panels contain 512 and 256 scenes, respectively. Learned/unary comparisons share deterministic noise; the native MMD path uses the recorded compatible seed convention. Highways and Conveyor. These use the released MMD obstacle geometry, disk radius 0.050.05 m, and workspace [−1,1]2[-1,1]^2 m. Obstacles are rounded boxes with the exact saved centers, full sizes, and corner radii, not rectangular approximations inferred from images. Each of the six populations has ten frozen start/goal tasks per map. Adaptation is performed at N=20N=20 where applicable; the evaluation does not fit a separate coordination network at each smaller N. The environment-specific single-robot prior produces 64 independent candidates per robot. Its frozen runtime uses 64 supports, 20 guide steps with guidance starting at reverse index 13, one final noise-free step, and three noising/three denoising steps for constrained local inference. The supporting runtime configuration includes the exact four-channel normalizer limits. Akule batches candidate generation across robots and selects candidates before applying one terminal interaction correction. The reference MPD/MMD paths may generate robot candidates sequentially. In particular, MMD’s roots already incorporate soft inter-robot guidance; they are not independent unary trajectories. Consequently, the map experiments are not 25-step interaction rollouts. Their priors use the diffusion schedule during candidate generation, but U/R/GU/R/G operate once on the selected candidates at t=0t=0. Candidate batching and guidance change sampler cost independently of interaction selection. Comparisons of complete planning time evaluate these implemented pipelines, while the matched unary-intervention study isolates changes to the same selected candidates. SMD maps. Basic, Dense, Shelf, and Room use the released instance obstacle circles (Liang et al., 2025a). Shelf and Room partitions are represented by the actual circle unions, not substituted rectangles. A shared map-conditioned unary and the frozen R/G/UR/G/U stack are trained/adapted using N=18N=18 contexts and transferred across the six test populations. Each final cell contains 25 held-out instances with radius 0.050.05 m. Training also includes Empty, but the seven-environment intervention table reports the four named obstacle families. The figure data include exact layout and scene identities. SMD representative baseline reruns are kept distinct from any published aggregate reference, and a displayed example is not itself a 25-scene estimate. All seven intervention environments record 64 trajectory supports at Δt=5/64=0.078125 t=5/64=0.078125 s, with nominal duration 5 s. The first-to-last support span is 63Δt63 t, which is distinct from the nominal horizon. Endpoints remain conditioned throughout the 25-step exponential diffusion schedule in the full-rollout implementation. ScaledWeave. The square workspace has half-extent s(N)s(N) metres, where (26) s(N)=max(1,1.001(0.150)N20.8(2π)(1−0.005)).s(N)= \! (1, 1.001(0.150)N 20.8(2π)(1-0.005) ). Construction radius is 0.8s0.8s and the snapped coordinate is 0.87s0.87s. Angular phase is sampled within ±0.005± 0.005 radians and per-robot jitter within ±0.0025(2π/N)± 0.0025(2π/N). The minimum required endpoint separation is 0.1500.150 m; the disk radius remains 0.050.05 m. This preserves boundary packing rather than constant areal density. The saved proposal study uses nominal duration 5s(N)5s(N) seconds and physical timestep 5s(N)/645s(N)/64 seconds, with 64 supports and 25 reverse steps. Positions and displacements are mapped through the similarity frame; the physical disk radius remains fixed. C.2. Success, failure, and latency Complete-planner success. A run must return a complete team solution and meet its benchmark’s final native validation and endpoint requirements within the applicable time budget. On canonical Weave, the frozen contract requires all goals within 0.050.05 m and zero sampled pair-time conflicts at strict physical separation below 0.1000.100 m. Native MMD search also has its own conflict predicate; search-node conflict counts are not interchangeable with the physical pair-time diagnostic. The released SMD sampled pair/obstacle predicate compares squared distance with the squared sum of radii minus 0.0010.001. Goal and endpoint checks also apply. This is distinct from a strict disk-overlap diagnostic and from a swept-path validation. We retain native success as the SMD benchmark contract and label physical diagnostics separately. No sampled check proves continuous collision freedom between supports. Swept checks used for teacher qualification are a stronger and separately stated requirement. Latency. Let M be the attempted scene count, ℓ its planning limit, and tst_s its recorded planning time. The reported all-attempt mean is (27) t¯=1M∑s=1Mℓ,run s reaches its configured timeout,ts,otherwise, including terminal failures. t= 1M _s=1^M cases ,&run s reaches its configured timeout,\\ t_s,&otherwise, including terminal failures. cases Canonical Weave uses a 240 s native limit (255 s external watchdog); SMD Akule complete-planner evaluation uses 900 s. Highways/Conveyor main-panel reporting uses a 60 s budget; longer diagnostic requests are not additional successful attempts under that budget. ScaledWeave artifacts must retain their own configured budget and proposal/complete-planner scope, rather than inheriting the map limit. Failure remains in the denominator. Model loading, compilation, warmup, serialization, and visualization are excluded where the native benchmark timer excludes them; proposal generation, static repair, and hard search belong to online planning. New backend measurements in Section H use an explicitly fixed 240 s contract. Trajectory quality. Mean path length averages summed Euclidean support-to-support distances over robots and then successful scenes. Canonical acceleration RMS is the root mean square norm of second position differences divided by Δt2 t^2; jerk analogously uses third differences divided by Δt3 t^3. SMD’s reported per-step acceleration statistic is the mean norm of second position differences without this physical-time rescaling. Native MMD quality follows its released evaluator and velocity convention. These differently normalized quantities should not be compared across benchmark families as though they were one physical statistic. Undefined quality or makespan for failed runs is omitted, never set to zero. Makespan, when available, is sustained goal arrival computed from saved supports, not a physical robot execution measurement. C.3. Frozen models and baseline scope The supporting provenance data identify the four canonical checkpoints and four SMD checkpoints by SHA256; changing population does not change those checkpoint identities. Map evaluation also pins the environment-specific priors, normalizers, and adapted composer/selector files in its final model manifest. Baseline names refer to executable pipelines: independent MPD proposals, native MMD with its specified CBS-family search, and simultaneous projected diffusion (SMD). A saved MPD proposal with collisions may be displayed even when MPD plus subsequent repair succeeds; captions identify that stage. The main manuscript's SMD-map MPD, MMD, SMD, and DGD aggregate references come from the published DGD benchmark (Liang et al., 2025b); their acceleration formula is not supplied by that aggregate table. The displayed SMD trajectories are local representative reruns, not reconstructed published trajectories. Published-only aggregates are not treated as locally paired reruns or used for hardware speed ratios. The supplementary component timings and conflict counts come from the frozen local panels and explicitly state their scope. Appendix D Additional Quantitative Results D.1. Matched unary intervention We compare Akule with its own unary process before any smoothing, static cleanup, or hard repair. This controls for the input scene and unary prior, and avoids attributing differences between candidate-generation pipelines to interaction learning. Canonical Weave compares saved unary-only and interaction-aware 25-step rollouts. Highways/Conveyor compare the same selected independent candidate trajectories before and after the terminal correction. SMD uses matched 25-step proposal-only replays with identical initial and posterior noise tensors and frozen checkpoints; no hard repair is invoked. The replay verification records agreement with the frozen Akule pair-conflict, residual-evaluation, and native static-invalid counts for all 600 SMD cases. For positions pi(h)p_i(h) in metres, collision pair-times are (28) C(p)=∑h=063∑1≤i<j≤N[∥pi(h)−pj(h)∥2<0.100].C(p)= _h=0^63 _1≤ i<j≤ N1\! [\|p_i(h)-p_j(h)\|_2<0.100 ]. Distances are computed from float64 positions at the recorded supports. An unordered pair contributes once per colliding support. This is neither a unique-pair count nor an interpolated collision-duration measure, and it excludes obstacle contacts. Equality at 0.1000.100 m does not count. The intervention is paired by scene identity and inputs; the full-rollout arms additionally share noise. Complete-planner failures remain eligible because the quantity measured is the pre-repair proposal. Table 8 gives the largest population in each of seven environments. Table 9 preserves all 39 conditions and all 2,000 paired scenes, with no missing pairs. The reported percentage is a reduction in the mean count, not the average of per-scene percentages (which would be undefined on zero-conflict unary scenes). Table 8. Largest-population unary intervention results. Mean physical collision pair-times before postprocessing; reduction and F/E/M follow Table 9. Environment N Scenes Unary Akule F/E/M Weave 28 512 115.613 32.133 72.2% 512/0/0 Highways 20 10 311.600 85.700 72.5% 10/0/0 Conveyor 20 10 285.300 122.900 56.9% 10/0/0 SMD Basic 18 25 129.960 28.120 78.4% 25/0/0 SMD Dense 18 25 138.240 31.160 77.5% 25/0/0 SMD Shelf 18 25 152.800 24.800 83.8% 25/0/0 SMD Room 18 25 143.160 20.400 85.8% 25/0/0 Table 9. All 39 evaluated population conditions; no population or paired scene is omitted. Mean unordered physical collision pair-times before smoothing, static cleanup, and hard repair. Reduction is 100(1−C¯Akule/C¯unary)100(1- C_Akule/ C_unary). F/E/M counts scenes with fewer/equal/more conflicts after interaction. Provenance: frozen paired-intervention export; 64 supports, strict distance <0.100<0.100 m. Environment N Pairs Unary Akule Reduction F/E/M Canonical Weave 10 512 8.195 6.410 21.8% 430/49/33 Canonical Weave 20 256 72.379 12.082 83.3% 256/0/0 Canonical Weave 28 512 115.613 32.133 72.2% 512/0/0 Highways 3 10 4.400 0.500 88.6% 7/3/0 Highways 6 10 16.200 2.900 82.1% 9/1/0 Highways 9 10 66.100 17.700 73.2% 10/0/0 Highways 12 10 120.300 31.400 73.9% 10/0/0 Highways 15 10 157.800 42.600 73.0% 10/0/0 Highways 20 10 311.600 85.700 72.5% 10/0/0 Conveyor 3 10 2.600 1.300 50.0% 5/5/0 Conveyor 6 10 15.800 7.400 53.2% 9/0/1 Conveyor 9 10 53.400 23.500 56.0% 10/0/0 Conveyor 12 10 87.700 37.800 56.9% 10/0/0 Conveyor 15 10 145.200 64.400 55.6% 10/0/0 Conveyor 20 10 285.300 122.900 56.9% 10/0/0 SMD Basic 3 25 1.520 0.320 78.9% 6/19/0 SMD Basic 6 25 14.040 3.560 74.6% 22/3/0 SMD Basic 9 25 34.040 7.440 78.1% 25/0/0 SMD Basic 12 25 57.080 12.520 78.1% 25/0/0 SMD Basic 15 25 95.880 17.960 81.3% 25/0/0 SMD Basic 18 25 129.960 28.120 78.4% 25/0/0 SMD Dense 3 25 3.320 1.760 47.0% 9/16/0 SMD Dense 6 25 13.120 2.760 79.0% 19/6/0 SMD Dense 9 25 30.280 5.000 83.5% 24/1/0 SMD Dense 12 25 55.120 11.040 80.0% 25/0/0 SMD Dense 15 25 92.600 22.120 76.1% 25/0/0 SMD Dense 18 25 138.240 31.160 77.5% 25/0/0 SMD Shelf 3 25 2.760 0.560 79.7% 14/11/0 SMD Shelf 6 25 12.080 1.760 85.4% 23/2/0 SMD Shelf 9 25 28.720 4.160 85.5% 25/0/0 SMD Shelf 12 25 65.120 10.280 84.2% 25/0/0 SMD Shelf 15 25 104.560 16.280 84.4% 25/0/0 SMD Shelf 18 25 152.800 24.800 83.8% 25/0/0 SMD Room 3 25 3.440 1.320 61.6% 11/13/1 SMD Room 6 25 12.760 1.400 89.0% 21/4/0 SMD Room 9 25 30.960 2.640 91.5% 25/0/0 SMD Room 12 25 54.560 5.760 89.4% 25/0/0 SMD Room 15 25 94.520 10.760 88.6% 25/0/0 SMD Room 18 25 143.160 20.400 85.8% 25/0/0 Every condition has a positive mean reduction. At the largest populations, reductions range from 56.9% on Conveyor to 85.8% on Room. This does not imply per-scene dominance: Weave N=10N=10 has 33/512 scenes with increased conflicts, Conveyor N=6N=6 has 1/10, and Room N=3N=3 has 1/25. The weaker mean improvements—21.8% at Weave N=10N=10 and 47.0% at Dense N=3N=3—are retained. The table supports a proposal-stage coordination benefit under the stated pairing, but does not by itself establish obstacle feasibility, complete success, or a fixed latency improvement. D.2. Extended proposal-stage measurements Table 10 separates independent candidates, interaction-corrected proposals, static cleanup, and roots entering hard search. Static cleanup can increase pair conflicts because it repairs obstacles one robot at a time. On Highways N=20N=20, for example, Akule’s mean increases from 85.7 after interaction to 86.6 after static cleanup. MMD’s soft-guided roots contain fewer residual conflicts than Akule’s corrected map proposals: 13.7 versus 85.7 on that condition. At Highways N=3,6N=3,6, all ten MMD roots are conflict-free. These results are consistent with MMD spending computation on sequential soft coordination before hard search. Table 10. Map proposal-stage mean collision pair-times. Akule uses ten scenes per cell; the Conveyor N=20N=20 MMD root mean uses its six available complete roots. Map N Unary Corrected After static MMD root Roots Highways 3 4.4 0.5 0.4 0.0 10/10 Highways 6 16.2 2.9 2.8 0.0 10/10 Highways 9 66.1 17.7 18.6 1.2 10/10 Highways 12 120.3 31.4 30.0 2.7 10/10 Highways 15 157.8 42.6 42.6 4.2 10/10 Highways 20 311.6 85.7 86.6 13.7 10/10 Conveyor 3 2.6 1.3 1.1 0.4 10/10 Conveyor 6 15.8 7.4 7.1 1.5 10/10 Conveyor 9 53.4 23.5 26.0 7.0 10/10 Conveyor 12 87.7 37.8 36.7 9.2 10/10 Conveyor 15 145.2 64.4 68.1 16.8 10/10 Conveyor 20 285.3 122.9 145.4 34.0 6/10 Conveyor N=20N=20 has only six complete MMD roots. The other four attempts return incomplete teams and cannot supply full-team root conflict counts. Their planning failures and recorded computation remain part of complete-planner accounting, while root comparisons explicitly restrict to available complete roots. We do not impute missing agents, zero conflicts, or ten complete MMD proposals. The independent unary candidates used in Table 9 are all available and are a different baseline from these soft-guided roots. Appendix E Ablation Details The final ablation panel uses Canonical Weave N=28N=28 (30 instances) and Shelf N=18N=18 (25 instances). The variants are inference-time substitutions with frozen unary and pairwise models, the same diffusion schedule, and the same downstream repair interface. They are not separately retrained neural architectures. Table 11 reproduces the final manuscript’s ablation results, including success, pre-repair conflicts, repair and complete latency, successful-trajectory quality, and directed-interaction savings. Table 11. Ablation of selection and composition on Canonical Weave (N=28N=28, 30 instances) and Shelf (N=18N=18, 25 instances). Conf. is mean pre-repair collision pair-times; Repair and Plan are repair and complete-planner latency (Plan over all runs); Quality is path length / acceleration over successful plans (Weave: physical RMS; Shelf: per-step second difference); RftR_ ft save is the fraction of pairwise corrections avoided relative to dense Akule. Source: the current manuscript’s final ablation table. Method Success Conf. Repair (s) Plan (s) Quality (P/A) R_ ft save Canonical Weave, N=28N=28 Akule 30/30 32.133 22.901 29.944 1.949/2.813 29.369% Akule (dense) 30/30 32.797 25.163 43.571 1.950/2.824 0% Raw/calibrated sum (α=1α=1) 30/30 94.200 43.573 50.756 1.935/2.602 29.740% Geometric selector 30/30 59.167 32.766 39.929 1.940/2.719 29.381% Collision-risk selector 30/30 62.967 34.948 40.113 1.949/2.794 29.381% Shelf, N=18N=18 Akule 22/25 24.800 73.632 76.923 1.487 / 0.013 80.0% Akule (dense) 24/25 19.200 71.051 81.495 1.512 / 0.014 0.0% Raw sum (α=1α=1) 21/25 61.960 83.179 86.835 1.556 / 0.016 81.8% Calibrated sum (α=0α=0) 20/25 152.800 103.958 107.744 1.517 / 0.018 79.6% Geometric selector 21/25 68.160 88.469 92.077 1.505 / 0.015 79.8% Collision-risk selector 21/25 43.240 79.909 82.403 1.504 / 0.013 79.8% Composition substitutions. Full Akule uses learned hard selection and signed contextual composition. Dense Akule opens every non-self directed edge while retaining the learned composer. Raw summation retains the learned selector and gives every admitted primitive coefficient one. Calibrated summation retains the selector and uses a single constant from 0,0.25,0.5,0.75,1,1.5,2\0,0.25,0.5,0.75,1,1.5,2\, selected on disjoint validation scenes by complete proposal rollouts. Weave validation lexicographically minimizes mean pre-repair pair conflicts, then acceleration and path length. The first criterion is smallest at α=1α=1 (94.6875 validation pair-times), narrowly ahead of α=0.75α=0.75 (95.21875), so calibrated and raw summation are the same intervention and share measurements. Shelf first minimizes obstacle-invalid robots, then pair conflicts, acceleration, and path length. The α=0α=0 validation candidates have zero obstacle-invalid robots, whereas every positive grid coefficient has a positive mean. The rule therefore selects α=0α=0 even though its pair-conflict mean is 165.5769. This is a physical-feasibility-first validation decision, not a choice minimizing complete-planner latency. A zero coefficient removes the composed correction but does not imply that an unoptimized implementation skipped computing its admitted residuals. Selector substitutions. For each reverse step separately, let (29) Kt=round(1||∑s∈|Et(s)|)K_t=round\! ( 1|V| _s |E_t^(s)| ) be the learned selector’s mean directed edge count on the held-out validation panel. Both heuristic selectors rank the complete set of directed non-self candidates globally and admit the top KtK_t; they do not select KtK_t partners for every robot. Geometric selection ranks by physical start-position distance. Collision-risk selection ranks by minimum time-aligned separation along the current step’s estimated clean unary trajectories. Both use the learned composer on the resulting support. Separate KtK_t values preserve the learned mean budget profile over reverse time; geometric rankings can remain fixed while KtK_t changes. Calibration uses 32 Weave validation scenes and 26 Shelf validation scenes, disjoint from the evaluation scenes. The supporting data contain all scalar scores and per-step KtK_t values. An evaluation outcome does not change the calibration or frozen model. Matching mean counts does not force per-scene graph equality and does not imply that the heuristic has the same repair workload. On the final manuscript panel, raw summation increases Weave conflicts from 32.133 to 94.200 and Shelf conflicts from 24.800 to 61.960. Dense composition also illustrates an efficiency–success trade-off: dense Shelf solves 24/25 instances versus 22/25 for sparse Akule, while sparse selection avoids 80.0% of directed residual evaluations. The benefits of sparsity should therefore be assessed jointly with success and downstream repair, rather than from residual savings alone. The calibrated Shelf unary-like proposal has higher complete latency than full Akule under this panel’s reported results. Appendix F Interaction and Computational Analysis F.1. Dynamic support, additions, and removals At a reverse step the mean selected directed degree is |Et|/N|E_t|/N, and the fraction of pairwise evaluations avoided is 1−|Et|/[N(N−1)]1-|E_t|/[N(N-1)]. Across a full rollout we sum actual admitted evaluations and divide by TN(N−1)TN(N-1). For maps, the dense comparison is N(N−1)N(N-1) because interaction occurs once. Edge additions and removals between consecutive reverse steps are normalized by the possible directed support N(N−1)N(N-1); the symmetric difference divided by the union supplies a complementary turnover statistic. Figure 6 shows the frozen Weave profiles. Support is not monotonically pruned. At N=28N=28, mean degree falls from 22.388 at t=24t=24 to 6.596 at t=23t=23, then grows to the full directed degree 27 at t=1,0t=1,0. The first transition adds 3.90% and removes 62.39% of possible directed edges; 76.35% of the union changes. Every profiled Weave trajectory, at all three populations, has at least one degree increase during denoising. Thus the savings arise from selectively reduced support at intermediate steps, not from a progressively vanishing graph. Highways and Conveyor have a single observed interaction step, so additions/removals over reverse time are undefined for those proposal interfaces. Figure 6. Frozen Weave interaction profiles over reverse steps 24→024→ 0. Left: mean selected directed degree; dashed lines show dense degree N−1N-1. Right: mean added and removed fractions of possible directed edges at N=28N=28. Each curve uses the full corresponding panel (512/256/512 scenes). Source: saved interaction-degree and edge-transition exports. F.2. Signed coefficient usage Table 12 distinguishes negative spline coefficients from negative evaluated temporal weights. These differ because a support’s weight interpolates several controls. On Weave N=28N=28, 42.96% of admitted coefficients and 44.46% of evaluated weights are negative. The corresponding median negative magnitudes are 0.276 and 0.296. The map interface also uses negative values. Such statistics establish that the trained model uses sign reversal; they do not establish that sign freedom is necessary. That causal claim would require a controlled retrained nonnegative-composer comparison, which is not supplied by an inference-time absolute-value or clipping intervention. Table 12. Signed-composer statistics on admitted edges. Fractions count negative coefficients/weights; medians summarize their negative magnitudes. Environment N <β<0 <w<0 Med. |β−||β^-| Med. |w−||w^-| Weave 10 41.57% 45.51% 0.291 0.305 Weave 20 43.05% 45.06% 0.281 0.299 Weave 28 42.96% 44.46% 0.276 0.296 Highways 20 30.88% 27.20% 0.144 0.133 Conveyor 20 41.38% 34.03% 0.139 0.124 F.3. Sampler and interaction computation Table 13 reports exclusive CUDA-event intervals from frozen-proposal replay. The profiling contract uses serialized GPU work and checks proposal agreement rather than selecting replays for favorable timing. The Weave component intervals include its 25-step process; map unary intervals include candidate generation and guidance, while interaction is terminal. Host overhead and downstream search are outside these intervals. Table 13. Akule CUDA-event component times (ms) on RTX 5090. Weave sums 25 steps; maps use one correction after candidate generation. Savings are relative to dense R evaluation. Environment N Scenes Unary U R G R save Weave 10 512 64.20 38.48 12.80 27.73 4.85% Weave 20 256 65.43 39.34 20.36 27.32 19.20% Weave 28 512 64.18 39.08 31.09 36.73 29.37% Highways 20 10 806.03 2.06 1.38 1.82 59.39% Conveyor 20 10 919.07 2.09 1.78 2.05 38.71% Akule’s map sampler batches independent candidates, while reference pipelines may generate candidates sequentially and MMD additionally performs soft coordination. A sampler-cost difference therefore cannot be assigned solely to U/R/GU/R/G, and saved CUDA measurements on different GPUs are not matched wall-clock measurements. The replay environment is RTX 5090 for the Akule component table; original baseline/search timings retain their own hardware provenance. We do not add event intervals to older repair timers to synthesize a new complete latency. F.4. Downstream search workload Table 14 reports actual saved hard-repair timers, completed constraint-tree expansions, and generated hard constraints. At Weave N=28N=28, Akule has 24.27 mean expansions, compared with 49.74 for MPD and 41.19 for MMD-xECBS. The two children of a normal conflict expansion add two hard constraints, explaining the relationship between these columns; generated constraint points are a separate quantity. Partial or interrupted expansions can break exact proportionality. The map findings are less favorable for downstream burden. On Highways N=20N=20, Akule has 11.3 mean CT expansions versus MMD’s 4.1. On the six Conveyor N=20N=20 tasks with complete MMD roots, these means are 53.5 and 33.67, respectively. These observations accompany MMD’s lower root conflict counts and should remain visible alongside Akule’s batching and proposal-generation advantages. For missing roots, we report coverage rather than inventing a comparable search input. Reduction against an independent unary proposal, sampler cost, and downstream search cost are three distinct observations. Table 14. Mean downstream repair workload. Weave uses full panels, Highways ten tasks, and Conveyor six matched complete-root tasks. CT counts tree expansions; constraints count generated branch constraints. Timers retain their recorded hardware. Env. / N Method Scenes CT Constraints Repair (s) Weave 10 MPD 512 5.00 10.00 3.900 Weave 10 MMD-xECBS 512 4.63 9.25 3.502 Weave 10 Akule 512 3.94 7.88 2.977 Weave 20 MPD 256 21.44 42.88 26.583 Weave 20 MMD-xECBS 256 14.16 28.31 15.588 Weave 20 Akule 256 7.20 14.40 6.936 Weave 28 MPD 512 49.74 99.47 77.496 Weave 28 MMD-xECBS 512 41.19 82.38 57.567 Weave 28 Akule 512 24.27 48.55 29.738 Highways 20 MMD-xECBS 10 4.10 8.20 3.771 Highways 20 Akule 10 11.30 22.60 14.115 Conveyor 20 MMD-xECBS 6 33.67 66.50 24.693 Conveyor 20 Akule 6 53.50 104.67 42.192 Appendix G Qualitative Results The following comparisons cover all 39 evaluated population conditions in the seven environments. Each comparison uses the same saved task across methods, with consistent robot colors and spatial limits. Discs mark starts, stars mark goals, and wheat-colored regions show obstacles. A blue box highlights Akule. Native failures are labeled [FAIL]; a drawable trajectory alone does not establish complete-planner success. Scarlet brackets and an impact symbol mark up to three spatially separated conflicts in failed baseline panels. Callouts use simultaneous saved supports and the reference renderer’s pair/obstacle predicates; geometric crossings at different times are not marked. Panels with no eligible conflict receive no callout. These annotations illustrate individual interactions; aggregate results are reported in Section D. The SMD examples use the 13th instance of each population’s 25-scene panel. Reused largest-population high-fidelity panels retain their original display treatment; other panels are rendered directly from saved supports. Scene IDs appear in each caption, and the supporting manifest maps every panel to its input array or retained asset. G.1. Canonical Weave The canonical Weave examples show opposing flows in the fixed workspace at all three evaluated populations. The MPD panels show independent proposals before repair; MMD and Akule show saved downstream outputs. Figure 7 presents the corresponding comparisons. Canonical Weave, N=10N=10 MPD [FAIL] MMD-xECBS Akule Canonical Weave, N=20N=20 MPD [FAIL] MMD-xECBS Akule Canonical Weave, N=28N=28 MPD [FAIL] MMD-xECBS Akule Figure 7. Canonical Weave at N=10,20,28N=10,20,28. Independent MPD proposals are shown before repair. Same task across methods at each population. Scene IDs, in population order: 510001, 510000, 105. The shared plotting conventions are defined in Section G. G.2. Highways Highways illustrates the interaction of several robot streams around the repeated rounded-box obstacles. Each population uses one shared start/goal task across MPD, MMD, and Akule. Figure 8 presents the corresponding comparisons. Highways, N=3N=3 Highways, N=6N=6 MPD [FAIL] MMD Akule MPD [FAIL] MMD Akule Highways, N=9N=9 Highways, N=12N=12 MPD [FAIL] MMD Akule MPD [FAIL] MMD Akule Highways, N=15N=15 Highways, N=20N=20 MPD [FAIL] MMD Akule MPD [FAIL] MMD Akule Figure 8. Highways at N=3,6,9,12,15,20N=3,6,9,12,15,20. Same task across methods at each population. Scene IDs, in population order: highways_n3_trial02, highways_n6_trial04, highways_n9_trial00, highways_n12_trial05, highways_n15_trial03, highways_n20_trial06. The shared plotting conventions are defined in Section G. G.3. Conveyor Conveyor compares the saved trajectories around its directional corridor geometry, with the same task and robot colors across methods. Figure 9 presents the corresponding comparisons. Conveyor, N=3N=3 Conveyor, N=6N=6 MPD [FAIL] MMD Akule MPD [FAIL] MMD Akule Conveyor, N=9N=9 Conveyor, N=12N=12 MPD [FAIL] MMD Akule MPD [FAIL] MMD Akule Conveyor, N=15N=15 Conveyor, N=20N=20 MPD [FAIL] MMD Akule MPD [FAIL] MMD Akule Figure 9. Conveyor at N=3,6,9,12,15,20N=3,6,9,12,15,20. Same task across methods at each population. Scene IDs, in population order: conveyor_n3_trial00, conveyor_n6_trial02, conveyor_n9_trial06, conveyor_n12_trial07, conveyor_n15_trial05, conveyor_n20_trial05. The shared plotting conventions are defined in Section G. G.4. Basic Basic illustrates coordination around scattered circular obstacles across the six evaluated populations. Figure 10 presents the corresponding comparisons. =N=3 =N=6 =N=9 SMD Akule =N=12 =N=15 =N=18 SMD Akule Figure 10. Basic at N=3,6,9,12,15,18N=3,6,9,12,15,18. Same task across methods at each population. Scene IDs, in population order: 81003012, 81006012, 81009012, 81012012, 81015012, 81018012. The shared plotting conventions are defined in Section G. G.5. Dense Dense illustrates the more constrained obstacle field; native failure labels and selected callouts remain visible on unsuccessful baseline outputs. Figure 11 presents the corresponding comparisons. =N=3 =N=6 =N=9 SMD Akule =N=12 =N=15 =N=18 SMD Akule Figure 11. Dense at N=3,6,9,12,15,18N=3,6,9,12,15,18. Same task across methods at each population. Scene IDs, in population order: 82003012, 82006012, 82009012, 82012012, 82015012, 82018012. The shared plotting conventions are defined in Section G. G.6. Shelf Shelf illustrates how the recorded trajectories pass between the obstacle columns. The plots retain the evaluated circle-union geometry. Figure 12 presents the corresponding comparisons. =N=3 =N=6 =N=9 SMD Akule =N=12 =N=15 =N=18 SMD Akule Figure 12. Shelf at N=3,6,9,12,15,18N=3,6,9,12,15,18. Same task across methods at each population. Scene IDs, in population order: 83003012, 83006012, 83009012, 83012012, 83015012, 83018012. The shared plotting conventions are defined in Section G. G.7. Room Room illustrates the alternative passages around its circle-union partitions at each evaluated population. Figure 13 presents the corresponding comparisons. =N=3 =N=6 =N=9 SMD Akule =N=12 =N=15 =N=18 SMD Akule Figure 13. Room at N=3,6,9,12,15,18N=3,6,9,12,15,18. Same task across methods at each population. Scene IDs, in population order: 84003012, 84006012, 84009012, 84012012, 84015012, 84018012. The shared plotting conventions are defined in Section G. Appendix H Additional Studies and Limitations H.1. Safety-preserving smoothing Saved-path smoothing can improve the regularity of a completed trajectory without regenerating its proposal or repeating hard search. Table 15 reports the existing Weave study at N=10,20,28N=10,20,28: a candidate is accepted only when it passes the study’s validation gate, and otherwise the original repaired trajectory is retained. Native validated success is preserved, while mean path length, acceleration RMS, and jerk RMS decrease. Length is in metres; acceleration and jerk use the physical timestep convention of Section C. These saved-path measurements are separate from online proposal and search timings. The table reproduces the recorded summary and original planning fields; post-smoothing execution/response-to-completion measurements are not reported. Table 15. Saved-path smoothing on canonical Weave. Cells show before/after values. Native success is unchanged; rejected smoothing candidates retain the original trajectory. Plan includes the recorded smoothing overhead after processing. N Accepted Success Plan (s) Length Accel. Jerk 10 118/512 512/512 3.128/3.131 1.811/1.808 1.593/1.525 33.54/31.79 20 103/256 256/256 7.106/7.113 1.910/1.887 2.447/2.112 54.19/45.74 28 200/512 512/512 29.944/29.957 1.949/1.933 2.813/2.573 62.50/56.44 H.2. Alternative downstream planners Akule’s proposal interface can be combined with different CBS-family backends. We evaluate this interface on a fixed small panel of 20 canonical Weave N=28N=28 tasks, using the first 20 frozen final-test conditions and unchanged pretrained models. Table 16 distinguishes newly executed ECBS measurements from existing CBS/xECBS records on those condition indices. Each attempt has a 240 s native planning limit and a 255 s external watchdog. We do not train or tune the coordination model for the substituted backend. Success requires the same complete-team, endpoint, and sampled physical collision checks as the canonical benchmark. The comparison tests compatibility and sensitivity to the repair backend. It is a small-panel study, not a replacement for the full Weave benchmark. Historical and newly collected timers retain their own execution provenance; the table does not establish a matched-hardware latency speedup between cohorts. Mean latency includes all 20 attempts and assigns 240 s to timeouts. Successful-run quality is reported only where a valid result exists. Table 16. Alternative repair backends on the first 20 frozen Weave N=28N=28 test conditions. H denotes existing frozen records and N a new RTX 5090 run; cohort timings are not a matched-hardware speed comparison. Latency includes all attempts (240 s for a timeout); root conflicts count physical pair-times. CT is mean constraint-tree expansions. Path length (m) and acceleration RMS (m/s2) average successful cases only; dashes denote no successful cases. Source: compact per-attempt backend records and their source hashes. Backend Proposal Cohort Success Timeouts Latency (s) Root conflicts CT Length Accel. CBS MMD H 0/20 20 240.00 124.40 68.20 – – CBS Akule H 0/20 20 240.00 31.60 90.05 – – ECBS MMD H 19/20 1 167.96 80.30 51.35 1.920 2.164 ECBS Akule N 20/20 0 77.45 31.60 28.75 1.977 2.897 xECBS MMD H 20/20 0 84.73 80.30 41.75 1.890 2.215 xECBS Akule H 20/20 0 28.90 31.60 23.45 1.954 2.873 Plain CBS times out on all 20 tasks for both proposal interfaces. Newly executed Akule–ECBS solves 20/20, with mean all-attempt latency 77.45 s and 28.75 CT expansions; the existing Akule–xECBS subset also solves 20/20. These outcomes show that useful learned proposals do not remove dependence on the repair backend. Akule’s mean physical root conflict count is 31.60 in each backend panel, whereas the native baseline’s root construction depends on its planner configuration. H.3. DropRegion and reproducibility The available released DropRegion artifacts include trained network weights and model arguments, but the environment-specific trajectory dataset required by the official inference loader is absent. The loader reads dataset metadata and successful trajectory tensors to construct a four-channel limits normalizer. The saved network and diffusion-schedule tensors do not recover these data-dependent minima and maxima. Thus the available checkpoint alone does not establish the original mapping between physical position/velocity channels and normalized diffusion inputs and outputs. A reproducible matched evaluation requires the original inference dataset or the corresponding normalization statistics and metadata. Substituting another map’s limits or estimating new limits would change the pretrained model’s inference protocol. We therefore omit DropRegion. This is an unavailable normalization dependency, not evidence of infeasibility or of Akule’s performance on that environment. H.4. Method limitations Candidate scoring. The selector evaluates directed candidates before hard admission. Its pointwise descriptors and contextual processing therefore remain quadratic in robot count, even when expensive pairwise corrections are evaluated sparsely. Savings in RϕR_φ evaluations do not imply a linear-time complete sampler or planner. Restricted correction span. The composer combines frozen pairwise correction directions through a finite temporal basis. It cannot express every joint multi-robot update, and signed coefficients do not remove this representational restriction. Our observed sign statistics describe the learned solution; a retrained nonnegative model would be needed to assess the necessity of signed weights. Offline optimization and transfer. Training depends on optimized, qualified pair and composition targets. Their cost and acceptance criteria affect the available supervision, and several supports can meet the same physical objectives. There is no uniquely correct interaction graph. Transfer to different populations and maps depends on the prior, normalization, geometry, and training contexts; the evaluated grids do not establish arbitrary out-of-distribution performance. Repair and validation. Interaction-aware proposals can retain robot–robot or robot–obstacle conflicts and still require static cleanup and hard search. The map measurements show that a faster batched proposal need not reduce subsequent search effort relative to a soft-guided MMD root. Complete planning also remains sensitive to backend choice and time budgets. Finally, sampled success and ideal-tracking trajectory quality do not certify continuous-time safety, dynamic feasibility under unmodeled robot physics, or execution robustness.