Unexplained unexpectedness: deus ex machina

The expression deus ex machina denotes a sudden solution to a plot or problem, which comes as a surprise to those who merely know the plot in the way it was presented. Deux ex machina literally means “god from the machine.” Sounds modern, but has its origin in ancient Greek drama [1].

Alan J. Cain nicely demonstrates the concept of deus ex machina (or lack of it) in mathematical proof, using the proof of Morley's Theorem as a case in point [2]. The deus seems to be a smart trick to short-cut a proof. He or she (dea ex machina) is disappearing as one receives retrospective explanation and reasoning. In science as well as in fiction, deus is sort of a mental construct deriving from the order or chronology in which facts and facets are presented. In a play or movie, deus may just be a redeemer from over-length.

How can the Latin phrase deus ex machina be replaced by an expression in English? If you are more attracted to the magic than to the divine, you perhaps like “out of the hat.” If you are looking for a less vivid, neutral phrase, “surprisingly, but inevitably” might work, while the inevitability leaves room for arguments.

References
[1] The Literary Encyclopedia: Deus ex machina.
[2] Alan J. Cain: Deus ex Machina and the Aesthetics of Proof. The Mathematical Intelligencer Fall 2010, 32 (3), pp. 7-11. PDF.