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A natural 20 is automatically a hit, and threatens to be a critical, then you re-roll for confirming the critical (for which it has to be higher than the enemies AC to a maximum of 20).

Now if you're wielding a keen rapier you threaten a critical anywhere between 15-20 on the attack roll. What if that wouldn't be a hit though?

For example you roll a 15, attack bonuses are +4, enemies AC is 20. If it was a normal roll that would definitively be a miss, but it was a potentially critical roll; yet it wasn't a natural 20. Does it hit?

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No, it does not hit. From the Pathfinder Reference Document (emphasis mine):

Increased Threat Range: Sometimes your threat range is greater than 20. That is, you can score a threat on a lower number. In such cases, a roll of lower than 20 is not an automatic hit. Any attack roll that doesn't result in a hit is not a threat.

To threaten a critical hit with that rapier, you need to roll 15 or above and hit normally.

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