«وَرَقَشْنَا الْحِكَايَةَ عَلَى مَا سَرَدَهَا»
We wrote down the story as he had narrated it.
Raqashnā relates to writing or marking; sardahā refers to relating the account in sequence. The expression ʿalā mā suggests preserving the form of the narrated account.
Recording words and confirming the events behind them are different actions. The gathering preserves Abū Zayd’s testimony as speech, but writing it down does not independently prove his alleged family history.