Semantic Residual: A Paired-Oracle Decomposition of LLM Multi-Agent Systems under Lossy Channels

LLM-based multi-agent systems (LLM-MAS) coordinate well on reliable channels but fail in poorly understood ways when communication is lossy. We measure these failures against a content-blind reference: synchronous push gossip on idempotent payloads, which admits classical bounds in diameter, conductance, and percolation. Our \emph{paired-Oracle} protocol runs an LLM agent and a deterministic gossip Oracle on identical scheduler, topology, and channel realisation, so any outcome difference is attributable to per-node decision logic. On Distributed Common-Free-Slot (\textsc{DCFS}), a set-union task isomorphic to all-to-all rumour spreading, three signatures of a \emph{semantic residual} emerge as channel reliability is swept: a phase-boundary shift $\Delta p^* \approx 0.35$ identified directly from paired trials; a $16$--$46\%$ rate of consensus on wrong answers despite complete knowledge, a regime classical gossip cannot enter; and a $3.6\times$--$5.1\times$ round-count overhead at perfect channels. A mediation analysis shows that coverage and failure-mode labels are jointly a sufficient statistic for accuracy --- no within-stratum residual remains --- so the LLM--Oracle gap is fully captured by a $p$-dependent stochastic mapping over four failure modes. This decomposition supports a four-tier mitigation hierarchy whose tier-wise gains match the externalised stratum's marginal share, and a five-parameter scaling law over an eight-model panel with leave-one-model-out RMSE $0.08$.