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I'm not so sure that is a different case.
You have not proved there is no gold in the garden, you just haven't found any despite
your excavation.
You have not found evidence for no gold; you have found no evidence for gold.
In the same way we can not find evidence for no god, only no evidence for god.
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You're kind of stretching now...
Okay, let's make it simpler. Let's say there is a box. Within that box someone claims that there are diamonds. Upon opening the box, the entire volume of said box is filled with nothing but air. That is, indeed, proof that there are no diamonds in the box.