Examples of this theory can be earliest found in Alan's book.
The sentence is meant to say that the theory can for the first time be found in Alan's book. Is the use of "earliest" idiomatic?
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It makes sense, but let me say it this way: I had to read it a few times to understand what it says. It's generally fairly awkward, stuffy, and formal and I wouldn't describe it as idiomatic. Certainly it is grammatically correct, though. I would personally switch it and say:
Examples of this theory can be found earliest in Alan's book.
Examples of this theory appear first in Alan's book.