Coppola proves that sigmoid transformers implement weighted loopy belief propagation regardless of weight initialization. Constructively, he shows transformers can execute exact belief propagation on knowledge bases, producing provably correct posterior probabilities at every node for acyclic cases. The weight assignment is unique — sigmoid transformers that compute exact posteriors must have exactly those weights.
The architectural decomposition is clean: attention is AND, the feed-forward network is OR, and the strict alternation between them matches Pearl's variable elimination algorithm. Each attention head selects which parent variables to condition on; the FFN combines evidence from different conditioning contexts. The transformer's depth corresponds to the number of message-passing rounds in belief propagation.
The hallucination result follows directly. Verifiable inference requires a finite concept space — a discrete set of variables whose joint distribution the network can represent exactly. Without explicit concepts, the transformer performs approximate inference on an implicit, ungrounded graphical model. The approximation produces plausible but unverifiable outputs — hallucinations. This is not a failure of scale or training but a structural consequence of operating without a concept vocabulary that bounds the inference.
The structural point: the connection between transformers and Bayesian networks is not metaphorical. Sigmoid attention literally computes messages in a belief propagation algorithm, and the weights that produce correct posteriors are determined by the graphical model's structure. Hallucination is what happens when this message-passing runs on a graph whose nodes don't correspond to defined concepts — the messages are computed correctly, but the graph they traverse is the wrong one.