What do we learn from "mi'Piv Yikra Elai"?
Malbim: (a) He said the matters in the way of Nevu'ah. Kisvei ha'Kodesh are written only from a Sefer (Megilah 8b)! You are forced to say that his recital was the recital of Hashem via Nevu'ah, like it says in Bava Basra (15a) 'Hashem says and Moshe writes 1 .' (b) Baruch did not add anything. What Yirmeyah said, he wrote.
This is unlike Malbim above (verse 4), citing Mahari, wrote that Megilas Kinos was not written via Nevu'ah, only with Ru'ach ha'Kodesh. How does Malbim learn from Moshe? This is like Rashi (15a), who says that all the Nevi'im wrote based on their Rebbi's (Hashem's) mouth. Sifri (Bamidbar 153) says that other Nevi'im prophesized with "Ko [Amar Hashem]", but only Moshe prophesized "Zeh ha'Davar [Asher Tvivah Hashem]!" The Meforshim on "ba'Chalom Adaber Bo" (Bamidbar 12:6) imply that other Nevi'im did not write the way Moshe did. Also, Chizkiyahu and his contemporaries wrote Sefer Yeshayah, and Elazar or Pinchas finished Sefer Yehoshua (15a), and we do not find that they were Nevi'im! Would a Sefer be partially dictation from Hashem, and partially not?! (PF)
Why does it say "Al ha'Sefer"? At the time he wrote, it was not yet a Sefer!
Radak: At the end (after he finished writing) it was a Sefer, so even now it is called a Sefer.
What do we learn from "Al ha'Sefer ba'Deyo"?
Malbim: This teaches that it is part of Kisvei ha'Kodesh, which require a Sefer and ink.