(Jeffro) And even if 1 Esdras came after, it was obviously understood then that the 'foundations' at Ezra 4:12 referred to the foundations of the temple. It's never specified as the foundations of anything other than the temple.
Ezra makes no suggestion in 4:12 that the foundations relate to the temple. We can infer from Ezra that the foundations of the city or walls are meant.
In contrast with Ezra, 1 Esdras adds that Darius granted permission for the city to be rebuilt (4:47-57). Ezra indicates Darius only supported the construction of the temple.
1 Esdras comments that in the 2nd year after they returned, "the builders built the temple of the Lord" (5:56-58). Josephus apparently understood it thus:
"... and the temple, by the great diligence of those that had the care of it, was finished sooner than any one would have expected. And when the temple was finished [the priests and Levites celebrated/mourned the loss of the previous temple] ..." (Ant. XI.4.2)
But the temple wasn't finished then, and Josephus later resumes the narrative about how it came to be rebuilt in Darius' time. There's a contradiction here.
1 Esdras has the period between the end of Cyrus' reign and Darius' reign as 2 years (5:73). Josephus has it as 9 years (Ant. XI.2.2)
FWIW, The New Bible Commentary (1970, 3rd ed.) remarks on p. 395:
"It [Esdras A/1 Esdras] probably reflects a better Hebrew text than the Massoretic text but the translation is very free and its historical and chronological details are unreliable, and so the biblical Ezra is to be preferred. The Jewish historian Josephus bases his work upon 1 Esdras."
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(Jeffro) Actually, Josephus took Ezra's 'Artaxerxes' to be Cambyses ...
Oops. Yes, you're right. Ezra makes reference to Ahasuerus but 1 Esdras does not. Interestingly, 1 Esdras isn't in chronological order anyway.
Cyrus - 2:1-16
Artaxerxes - 2:16-30
Darius - 2:30-5:6
Cyrus - 5:7-73
Darius - 6:1-7:15
Artaxerxes - 8:1-end
So all-in-all, it doesn't make sense to me to use 1 Esdras (even via Josephus) to argue that Ezra 4 keeps strict chronological sequence and that v. 6-23 referred to the years between Cyrus and Darius.
... which is indeed probably in error, but still in the correct period, prior to Darius. Ezra's 'Artaxerxes' - the recipient of the letter - would more likely refer to Bardiya, after complaints were sent to Ezra's 'Ahasuerus', who would more logically refer to Cambyses.
Well, I've already discussed why I find those identifications unsatisfactory.