dais, surely?

Why? As Faldo has pointed out, the two words are not related. The Indo-European root for the Persian root daiz- is *dheigh-, and in Latin that initial dh corresponds to an f (cf. Latin fumus 'smoke', Greek θυμος (thumos) 'mind, soul', Sanskrit dhuma 'smoke' < PIE *dheu-m-).

Ceci n'est pas un seing.