Large language models (LLMs) remain limited in multi-agent planning because independently generated plans can create coordination failures such as spatial collisions, resource contention, and temporal deadlocks. We introduce Tensor-Coord, a multilinear algebra framework that represents the joint plan of N agents as a third-order tensor \(T \in R^{N \times H \times A}\) over agents, timesteps, and actions. Canonical Polyadic (CP) and Tucker decompositions are used to identify latent coordination structure. The minimal epsilon-approximate CP rank R* defines a computable coordination complexity measure, with \(CC(Pi)=(R*-N)/N\). We prove that R*=N is necessary and sufficient for plan independence. The residual \(E=T-T_{R*}\) defines a conflict score over agent pairs, timesteps, and actions, localizing failures without domain-specific rules. Tucker factors provide interpretable agent roles, temporal phases, and action clusters that are converted into natural language constraints for iterative LLM replanning. Experiments on multi-robot delivery tasks across Easy (2 agents, 5x5 grid), Medium (3 agents, 5x5 grid), and Hard (4 agents, 5x5 grid) settings show convergence to conflict-free plans in 100% of 2-agent cases within 1.4 iterations on average, 80% of 3-agent cases within 3.2 iterations, and 60% of 4-agent cases within 4.0 iterations. CP rank scaled approximately linearly as \(R*(N) = 3.9N + 0.5\), supporting its use as a predictor of coordination complexity.

翻译：大型语言模型（LLMs）在多智能体规划中仍存在局限性，因为独立生成的规划可能导致协调失败，例如空间碰撞、资源争用和时间死锁。我们提出张量坐标（Tensor-Coord），一个多重线性代数框架，将N个智能体的联合规划表示为关于智能体、时间步和动作的三阶张量 \(T \in R^{N \times H \times A}\)。利用标准多路分解（CP）和塔克（Tucker）分解来识别潜在协调结构。最小ε近似CP秩R*定义了一种可计算的协调复杂度度量，其中 \(CC(Pi)=(R*-N)/N\)。我们证明R*=N是规划独立性的充要条件。残差 \(E=T-T_{R*}\) 定义了智能体对、时间步和动作上的冲突分数，无需领域特定规则即可定位失败。塔克因子提供可解释的智能体角色、时间阶段和动作聚类，这些被转化为自然语言约束，用于迭代式LLM重新规划。在简易（2个智能体，5x5网格）、中等（3个智能体，5x5网格）和困难（4个智能体，5x5网格）设置下的多机器人配送任务实验中，2智能体案例在平均1.4次迭代内100%收敛至无冲突规划，3智能体案例在平均3.2次迭代内80%收敛，4智能体案例在平均4.0次迭代内60%收敛。CP秩近似线性缩放为 \(R*(N) = 3.9N + 0.5\)，支持其作为协调复杂度预测器的有效性。