Collective Counterfactual Planning: Coordination, Consent, and Verification under Representational Constraints

AI agents can each make locally valid decisions yet jointly produce an invalid result. We call this the global coherence problem: a failure of shared state, not merely of intelligence. Our Observation-Aliasing Impossibility Theorem gives the exact boundary. A policy can guarantee a valid action exactly when all worlds producing the same observation share an admissible action. If k indistinguishable worlds require pairwise-disjoint actions, the best randomized worst-case success is 1/k; more reasoning, roles, messages, or samples cannot recover the missing distinction. A stronger model can reason better within its context, but it cannot see beyond it. The smallest sufficient context is NP-hard to find, yet counterexample-guided refinement finds a sufficient one and recertifies it as rules change. We give local-to-global runtime semantics X = (H, C, G, F; D): topology H records overlapping scopes; category C governs state-changing actions; groupoid G keeps reversible translations; sheaf F tests whether local views glue into one world; and history D keeps only what alters legal futures. Each preserves the needed information as state is split, translated, or changed. Models propose; the harness owns shared state and governs commit. Ten studies test the boundary. On a revision benchmark, a frontier model scores 40/40 when the deciding event is visible; hidden, every arm is consistent with chance (1/3); one restored fact returns 40/40. On TeamBench, ordinary teams exceed a shared budget in 5/5 runs, a visible live count leaves 4/5, and commit enforcement leaves 0/5. In tau2-bench Telecom, current-state checks score 0.07 after silent reverts; the harness scores 1.00. Refinement keeps a sufficient context through rule changes at 28% of the all-field cost. Solvers that already own the state tie it, as predicted. Local intelligence cannot substitute for missing global state.

Embodied language-model agents must coordinate interdependent actions while each holds private task information. In dialogue-based coordination, models generate messages and interpret replies to decide what each agent should do, spending inference before the team acts. We propose GenCoord, a decentralized protocol that makes the division of work itself the message. A proposal pairs the sender's skill path with its partner's continuation in Short DSL, a compact language for expressing joint work. Compatible paths form commitments linking the sender's delivery to the partner's work after verified receipt. We check fulfillment separately from original-task success. Complete requests also let a deterministic receiver proceed without another model call. In Minecraft, trained local 0.8B agents match the task success of controlled free-form communication while reducing median commitment time by 68.2%. The same-field full-key JSON control approaches free-form output length and delay, indicating that the amount generated matters even when messages are structured. With the same prompted DeepSeek model, both protocols complete nearly all tasks, but dialogue generates 2.8 times as many output tokens. Multistep commitments improve held-out-template success from 91.3% to 98.1% with fewer decisions. These results point to a design principle for embodied cooperation: make the division of work the message, and spend inference on choosing and revising the work.

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.