Very problematic. How could you distinguish between organic matter that originated within the jar from the organic matter that had been placed into the jar at the packing plant?
Unless something incredibly unlikely happened, the organic matter placed in the jar would have evidence of billions of years of evolution whereas the the new life wouldn't. It is thus unlikely that the two would life forms would be regarded as even remotely related after molecular analysis (I know I said microscopic, but I was being wrong at the time).
A related problem with new life forming in the present day, besides being able to distinguish it from life that's already existing, is that there's a name for that new life: food! In other words, the life forms that are already present will make lunch out of new organic matter that might form spontaneously.
Absolutely - the peanut butter jar is an extremely harsh environment for the origination of life.
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