Nobody Learns the Words: Lyric Recall as an Ill-Posed Inverse Problem

Everyone can sing the first verse of a song, and almost nobody can sing the second. We report that this is because the words to a song are never learned at all. Singing is a smearing operation: a melody spreads the syllables of a verse into one continuous signal, and what a singer stores is the melody—a few dozen numbers—rather than the words. The words are recovered on the spot by undoing the smear. Recovery is exact while the undoing stays stable, and because each successive verse composes the melody’s operator with itself, the difficulty of undoing it does not accumulate but squares. Across 12,846 recorded sing-throughs, syllable accuracy falls from $0.94$ at verse one to $0.11$ at verse two and then stops falling—a cliff, not a decay, which no account based on forgetting predicts. A four-arm randomised trial isolates the mechanism: recovery survives transposition into an unfamiliar key, which changes the melody but not its invertibility, and collapses when the melody is withheld. The songs that outlive their authors are precisely those tuned so that exactly one verse can be recovered; the rest are unsingable and are not transmitted. The mumble that replaces verse two is not a failure of memory but a correctly regularised estimate. You have never known the words; you have only ever solved for them.