Temperature reshapes the curve. Top-k cuts the guest list by headcount. Top-p cuts by share of the probability, so it adapts: few tokens when the model is sure, many when it is not. Top-k cannot. Most bad sampling configs come from treating the three as interchangeable.
Each row feeds the next, in the order many libraries use: temperature, then top-k, then top-p. Some runtimes let you reorder the chain. Move a slider and only its own row and the rows below it change.
The 20 logits per context are hand-set examples. Everything after them (softmax, the cuts, renormalizing, every sample) is computed live. Probability bars use a square-root scale so small values stay visible.