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Source Core Rulebook pg. 449 4.0
Sometimes you will be called on to attempt a basic saving throw. This type of saving throw works just like any other saving throw—the “basic” part refers to the effects. For a basic save, you’ll attempt the check and determine whether you critically succeed, succeed, fail, or critically fail like you would any other saving throw. Then one of the following outcomes applies based on your degree of success—no matter what caused the saving throw.
Critical Success You take no damage from the spell, hazard, or effect that caused you to attempt the save.
Success You take half the listed damage from the effect.
Failure You take the full damage listed from the effect.
Critical Failure You take double the listed damage from the effect.
Impulse Kineticist Overflow Primal Wood
Source Rage of Elements pg. 34
A fusillade of jagged splinters flies from you. Creatures in a 30-foot cone take 1d4 piercing damage and 1d4 persistent bleed damage with a basic Reflex save against your class DC.
Level (+2) Each type of damage increases by 1d4.
So to expand the text of Hail of Splinters
Critical Success: No Damage
Success: Half Damage (1d4/2 piercing) (1d4/2 persistent bleed)
Failure: 1d4 Piercing / 1d4 Persistent Bleed
Critical Failure: 1d4*2 Piercing / 1d4*2 Persistent Bleed.
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Just wanted to check there isn't an exception for persistent damage and nope the rules are clear:
Source Core Rulebook pg. 451 4.0
Persistent damage is a condition that causes damage to recur beyond the original effect. Like normal damage, it can be doubled or halved based on the results of an attack roll or saving throw. Unlike with normal damage, when you are subject to persistent damage, you don't take it right away. Instead, you take the specified damage at the end of your turns, after which you attempt a DC 15 flat check to see if you recover from the persistent damage. See the Conditions Appendix on pages 618–623 for the complete rules regarding the persistent damage condition.
So have fun making those baddies bleed!