Hearing Has a Computable Ceiling

Spoken agents are built and evaluated on three assumptions: that the voice carries information the words do not; that a model's loss with and without audio measures what the audio adds; and that speech synthesized from a label can stand in for a person. We prove a ceiling on what hearing can add and test all three assumptions against it. Let $I$ be the information, in nats, that the voice carries about the speaker's hidden state beyond her words: no policy gains more than $\sqrt{I/2}$ from hearing in one decision, and a turn-by-turn version bounds a whole conversation. Both bounds have sharp constants and are machine-checked in Lean 4. Because $I$ is a difference of two log-losses, a system can estimate it from its own conversations, given labels of the user's state and no transcript-reading judge. On television dialogue, the words already carry $92$ to $95$ percent of what the voice says about the speaker's emotion. The with-and-without comparison overstates what audio adds by the text model's shortfall, six-fold for a bag-of-words text model, and audio from the wrong utterance reproduces most of two off-the-shelf models' apparent gain. Speech synthesized from an emotion label, using instructions we wrote, retains little of what the speaker's own delivery carries. One off-the-shelf model collects a third of the ceiling on dialogue, the other nothing detectable.