Dashed line shows the Top-p cumulative cutoff. Green highlight denotes sampled token.
Large Language Models are probabilistic next-token predictors parameterized over token co-occurrence manifolds, not relational database query engines. Hallucination is not a "software glitch", but a fundamental property of statistical autoregression under finite context attention.
When evaluating $P(w_{t+1}|w_{\le t}) = \frac{\exp(z_i / T)}{\sum_j \exp(z_j / T)}$, high temperatures or wide nucleus sampling allow tokens from the tail of the distribution to be chosen. Even if a factual token has 65% probability and a confabulatory token has 18%, repeated non-greedy sampling will eventually trigger a factual divergence.
Transformers condition each new step on all previous steps: $P(S) = \prod_{t=1}^N P(w_t | w_{<t})$. Once a single hallucinatory token $w_k$ enters the context, the model conditions subsequent generations on its own mistake. The probability of hallucinating token $k+1$ conditional on token $k$ being false jumps from 15% to over 90%.
In long sequences, attention weights dilute across hundreds of tokens (the "Attention Sink" phenomenon). When prompt attention weight drops below a critical threshold, the transformer's dense feed-forward memory priors override external instructions, reverting to the statistical average of its training corpus.