🤖 AI Summary
This work addresses the lack of verifiable guarantees—such as convergence, interpretability, and bounded interaction rounds—in multi-agent large language model reasoning. The authors model collective reasoning as a nonlinear dynamical system over a communication graph and, for the first time, apply Koopman operator theory to construct a linear representation from interaction trajectories. Spectral analysis of this representation yields three machine-verifiable certificates: convergence deadlines, identification of cohesive cliques with interpretable validity, and an auditable basis for compressed messages. Experiments demonstrate that convergence rounds are predicted accurately in 96% of configurations (log-scale correlation of 0.93), attributions are exact, and decision-relevant information is preserved with 99.7% fidelity using only 8 out of 32 spectral coordinates. Certificates trained on 15 debates remain fully valid across 60 leave-one-out tests and are computable within minutes on a CPU.
📝 Abstract
Orchestrated collectives of large language model (LLM) agents that debate and vote are an emerging form of computational intelligence: the intelligent behaviour resides in the \emph{interaction}, not in any single agent. They improve task accuracy, yet remain black boxes at the system level: there is no principled test of convergence, no bound on the rounds needed, and no faithful account of what drove a decision. This paper develops a novel framework based on Koopman operator theory and validates its theoretical guarantees on multi-agent consensus dynamics. Treating the collective as one nonlinear dynamical system on a communication graph, we read its essential behaviour off the spectrum of its Koopman transfer operator, an exact linear representation of the nonlinear dynamics estimated from interaction traces. The spectrum yields three machine-checkable certificates: the sub-dominant eigenvalue $λ_2$ fixes the intrinsic timescale of reasoning and yields a convergence deadline computable \emph{before} the debate runs; its eigenvector names the coherent factions the collective reasons in, and $|λ_2|$ certifies when that explanation is valid; and the leading spectral coordinates form a compressed, auditable message basis. On an attention-consensus model, the deadline tracks observed convergence with log--log correlation $0.93$ and bounds it in 96\% of 24 configurations; attribution is exact whenever the spectrum certifies metastability; eight of 32 coordinates preserve the decision at 99.7\% fidelity; and a certificate learned from 15 debates held on 60/60 held-out debates. The study runs in minutes on a CPU, making spectral certification a practical layer for trustworthy collective reasoning.