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During a recent game PFS scenario, we had a disagreement as to the wording of Aeon Stone (Preserving) which says: "This red crystalline star covers you in a faint aura when you are subject to lingering wounds. You gain resistance 3 to persistent damage. At the end of any turn where the persistent damage can't overcome this resistance, end that condition. The resonant power allows you to cast stabilize as a primal innate cantrip."
There were 2 different interpretations of how this could be handled, both around "can't overcome" to cause the end of a condition. The situation was a player was taking a d8 persistent of bleed, and the GM rolled 3 bleed that round, and thus the player took no damage at the end of the turn.
First, the persistent damage was unable to overcome the resistance at the end of that turn since it was 3 or less, so the condition does end.
Second, the player didn't take any damage this turn; however, the player could have taken damage greater than the 3 resistance (since a d8), so the persistent damage continues.
This was a GM call at the table, but trying to determine how to rule it going forward in PFS.

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The item’s language seems clear to me:
“At the end of any turn where the persistent damage can’t overcome this resistance, end that condition.”
It isn’t conditional on whether or not the damage could overcome the resistance, but whether or not it can’t. The GM rolled a 3 - the persistent damage could not overcome the resistance, and thus, the condition ends. Rolling less than the resistance would mean, at the end of that turn, it can’t overcome the resistance.
I believe that is the right call. Otherwise, the item is entirely worthless against any random die roll for persistent damage where the size (d4, d6, etc) is greater than the resistance. Under the second interpretation, a d8 persistent damage would always continue if the resistance is 8 or less, for example.