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 \emph{once} for a partition of the arms, the partition becomes a positive-definite correlation matrix $\Sigma$ through an RBF kernel on cluster ranks, and the per-round posterior sample is drawn with covariance $\Sigma$ while the Beta posteriors are updated from real rewards only, so the LLM shapes \emph{how} the sampler moves, not \emph{what} it believes. We give a self-contained Bayesian regret bound for the idealized Gaussian sampler whose information gain splits into a $K\log T$ term from the $K$-cluster structure and a ridge term that grows to $d\log T$: 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 \emph{shape} (a random partition, or a plain tempering of the sampling noise, reproduces it), while belief injection at matched oracle quality never helps.