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The paper formalizes and quantifies compositional incoherence in multi-component LLM agents, where individually coherent components produce globally incoherent joint probability estimates. They introduce the compositional residual eps* to measure this incoherence and show it can be predicted using a Rayleigh-quotient approach. They find that incoherence is prevalent in a four-LLM panel and that intuitive LLM-side mitigations are ineffective, highlighting the need for explicit coherence mechanisms.
Even when individual LLM agents are locally coherent, composing their probabilistic claims can lead to surprisingly large violations of basic probability axioms, causing significant regret in betting scenarios.
Multi-component LLM agents assemble probabilistic claims from components that each see only part of a joint problem; the composition can violate basic probability axioms even when every component is locally coherent. We formalise this locally coherent, globally incoherent failure via the compositional residual eps*, the L2 distance from the composed quote to the joint coherent polytope, computable at runtime from system output and the declared cross-component coupling constraints. A product-structure dichotomy characterises when local coherence suffices, and a Rayleigh-quotient prediction matches the observed residual within 7% on three of four relation classes. A hierarchical Boyle-Dykstra projection repairs the composition deterministically; an anytime-valid e-process gives sequential coherence monitoring. Across 1,876 ensemble cliques on a four-LLM mid-tier panel (frontier-panel rerun in Section 5.5), eps*>0 on 33-94% of cliques, translating to +0.115 nats per bet of regret on 1,770 resolved bets under the proportional allocation rule (the gain collapses to +0.006 under bettors that themselves coherentise). Three intuitive LLM-side mitigations(retrieval, partition-aware prompting, aggregator-LLM) each fail or regress.