Logits and softmax

The exact math that turns a model's final hidden vector into a probability for every possible next token. Defines a logit as a raw, unnormalized score; derives where logits come from (the output projection, one matrix–vector product); explains why only differences between logits matter (softmax is shift-invariant); defines softmax and proves its outputs are positive and sum to one; shows the numerically stable form used in practice and works a full example by hand. Then the consequences: why a one-point logit change moves probability exponentially, exactly what temperature does (with its two limits), why the top-probability token isn't always the one chosen, why a high probability is not a claim of truth, and how to read logprobs from an API to classify, score confidence, and debug.