Structure, Not Belief: Correlated Thompson Sampling from LLM-Derived Covariance in Combinatorial Semi-Bandits

Combinatorial Thompson sampling (CTS) draws independent posterior samples for every arm, so its exploration dynamics ignore any relation among arms. We study a minimal change to those dynamics: an LLM is queried once for a partition of the arms, the partition becomes a positive-definite correlation matrix Σ through an RBF kernel on cluster ranks, and the per-round posterior sample is drawn with covariance Σ while the Beta posteriors are updated from real rewards only, so the LLM shapes how the sampler moves, not what it believes. We give a self-contained Bayesian regret bound for the idealized Gaussian sampler whose information gain splits into a K T term from the K-cluster structure and a ridge term that grows to d T: the sqrt d/K improvement over independent sampling is a finite-horizon transient, exact only as the within-cluster correlation tends to one. The correlated sampler reduces regret by 19% over CTS on 16 synthetic Bernoulli families at T=2, 500 (6-7% at T=25, 000 with data-adaptive kernels) and by 41% on the Microsoft MIND-small news benchmark (d=200 real articles), while pseudo-observation warm starts give nothing. An LLM-free ablation with a simulated oracle of controlled quality shows that on unstructured instances the gain is a property of the kernel shape (a random partition, or a plain tempering of the sampling noise, reproduces it), while belief injection at matched oracle quality never helps.