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Or perhaps any words that perfectly encapsulate the meaning of that phrase?

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You can't prove a negative

The specific/technical meaning isn't exactly the same (and it is technically incorrect), but the intended meaning basically is, and it's a commonly used phrase for that purpose.

While you can actually "prove a negative" in some contexts, this phrase is generally just used as a shorthand for "absence of evidence is not evidence of absence", with an added emphasis on the point that continuing to provide examples of something not being present will, at best, asymptotically approach a convincing case that it does not exist.

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