τοῦ ὁρισθέντος υἱοῦ θεοῦ ἐν δυνάμει: appointed, not made Son

Says ὁρισθέντος (horisthentos) is the aorist passive participle of ὁρίζω. Elsewhere in the NT the verb means "determine, appoint, fix." In Acts 10:42 and 17:31 it is used of the man God "appointed" judge. Abbott-Smith gives "determine, appoint, designate" and cites Rom 1:4. "Declared" (BSB) is a softer rendering.

Uncertain What ἐν δυνάμει modifies. - (a) It may go with υἱοῦ θεοῦ: "appointed Son-of-God-in-power." - (b) It may go with the participle: "powerfully declared."

The word order and the contrast with 1:3 (weakness κατὰ σάρκα) favor (a). The team prefers (a).

Says ἐξ ἀναστάσεως νεκρῶν: "from/by the resurrection of the dead." It is temporal ("from the time of") or causal ("on the basis of"), and most likely both.

Implies The verb's usage favors "appointed" or "designated," that is, installed in a position. Rom 1:3 has already called him "his Son" before the clause about David. 1:4 therefore describes his installation in messianic power and rule at the resurrection. It does not describe the start of his Sonship, so eternal Sonship is untouched (8:3, 32; Gal 4:4).

SBLGNT, RP2018 and TR all print ὁρισθέντος. Tischendorf (vol. 2, p. 364) reports προορισθέντος ("predestined") only from patristic and Latin witnesses (Marcellus in Eusebius, the Theodotians in Epiphanius; Latin praedestinatus: d e g vg, Irenaeus in Latin, Ambrosiaster, Augustine), with no Greek manuscript. It is best explained as an interpretive adjustment and carries no weight against ὁρισθέντος.