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Look at the following sentence:

Evidence exists that these pursuits can positively impact participants.

I don't think that the above sentence has the correct syntax. Per correct syntax, it would be

Evidence that these pursuits can positively impact participants exists.

I know some may even want to put the above as "There is evidence that these pursuits can positively impact participants," but I want to restrict the discussion to the first two and see whether my judgment is correct.

Any different view from that of mine? Is my judgment correct?
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Both forms are fine; it is the author's style choice. The first places emphasis on the fact of positive impact, while the second places emphasis on the existence of the evidence.
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Thanks Micawber. If you could help me on how you judge the emphasis here it would be helpful. Going further on this discussion, there is a composition rule that I've read that states "keep related words together." Moreover, you cannot separate the subject and the verb. Going by these, the clause "that these pursuits can positively impact participants" is something that defines "evidence" and hence should be kept together (to put it simply, the subject here should be "evidence that these pursuits can positively impact participants"). Probably, these are the reasons why I chose the second version as correct. But I'm not sure if the first version is wrong. Is my understanding correct? Also, based on what you say, I assume the original version (the first one) is not wrong.
If you could help me on how you judge the emphasis here it would be helpful.-- English emphasis normally falls at or near the end of the sentence. In normal composition, previously known or assumed information comes first and new information comes later.

there is a composition rule that I've read that states "keep related words together."-- To be accurate it must read: Keep related words together in the reader's comprehension. Any manner of structures may intervene if the sentence remains clear and comprehensible and unambiguous within the greater context.

Moreover, you cannot separate the subject and the verb.-- Again, many words can separate subject from verb: adverbials, appositives, relative clauses. The accurate rule is: You cannot separate the subject from the verb by an odd number of commas.