Je ne peux les laisser vous attraper.

This is the literary style, leaving out the “pas”, right? I take it in speech it would be “Je peux pas les laisser vous attraper”.

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You are correct.
Literary: leaving out the pas
Correct: using both “ne” and “pas”
Day to day speech and text messages (incorrect but most frequent): leave out the “ne”

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(bad post, wrong thinking, sorry)