Prediction Is Not Output: One Head Cannot Do Both, Two Heads Can

A language model uses one output head for two jobs: predicting what other people write, and producing its own replies. Where the text states a fact correctly, the two jobs ask for the same output. Where the people in the text are wrong, they do not: a good prediction reproduces their mistakes, and a good reply does not. We prove that one head trained on both jobs cannot do both well. Its best possible output is a weighted mixture of the two targets, which is wrong for both, whether the replies are trained by supervised fine-tuning, by reinforcement learning or by KL-regularized RLHF. We test this with a small transformer in a synthetic world where every target is known. A single head trained first to predict and then to answer, as language models are, forgets how to predict. A single head trained on both jobs at once answers correctly, but then expects other speakers to answer correctly 99% of the time, although only half of them do; when answering is given less weight, the same head instead answers the way the other speakers do, mistakes included. A model with two heads on a shared trunk, one trained to predict and one trained to answer, keeps both jobs right in every setting we tried. The price is speed: facts that appear only in plain statements reach the separate answering head more slowly. Author contributions and use of AI: the author conceived the study, set its question and its claims, and directed the experiments. The experiment code, the runs, the figures and the text were produced by an AI assistant (Claude, Anthropic) under the author's direction. Also posted on aiXiv: https://aixiv.science/abs/aixiv.261108.000002