starting point, item descriptions.
Tumultuous Sash:
Whenever your Paradox count resets, you gain 1 - 2 Paradoxes.
Otherworldly Artifact:
Your Chaos abilities have a 45% chance to generate 1 Paradox.
It’s hard to tell for sure but I think it’s because of the order they check in. After a cast, if it’s a multiple of 8 those Paradoxes get added, then Tumultuous Sash looks to see if the Paradox count reset, then Otherworldly Artifact adds a Paradox. If it resets from that +1, the Sash does nothing.
It’s pretty easy to prove because when using the Otherworldly Artifact, my paradox count can reset to 0, with other chaos weapons the belt prevents this ever happening.
So an example series of Paradox counts is
0 0 0 1 1 4 (reset and gained +1)
1 1 4 4 6 (reset and gained +1)
1 1 2 2 3 5 6 6 (reset and gained +2)
2 3 4 5 6 6 6 7 (reset and gained +0)
0 3 3 6 8 (reset and gained +2)
2 5 6 (reset and gained +3)
3 3 7 (reset and gained +0)
0 0 0 4 5 (reset and gained +2)
2 3 4 (reset and gained +0)
0 4 5 5 5 6 6 6 7 7 7 (reset and gained +0)
0 1 2 2 3 3 4 7 7 (reset and gained +2)
2 2 2 3 3 4 7 (reset and gained +0)
So in summary that’s
+0 * 5 ; +1 * 2 ; +2 * 4 ; +3 * 1
Which is 13 total paradoxes in 12 resets, at least 1 of which is from the Otherworldly Artifact. If Otherworldly was assumed to only apply after Tumultuous Sash the combination should be doing
+1 27.5%
+2 50%
+3 22.5%
Which is the distribution it shows, when it gets paradoxes at all. (implying 45% of the +2’s have an Otherworldly) so probably 10 paradoxes from belt, 3 from Otherworldly.
Consequently instead of an average of 1.5 paradoxes from the belt per reset, it’s only making around 0.8.
I’m pretty sure I’ve seen the Tumultuous Sash fail to proc after my Paradoxes have reset while using other Chaos Foci than the Otherworldly Artifact. But it’s rare and I could never discern a pattern for why this had happened. Is it possible Resonance Cascade or even Butterfly Effect could also be a factor in this phenomenon in the same way you suspect is happening with the Artifact?
Can’t be Resonance Cascade cause that cannot generate the 8th paradox. Could definitely be Butterfly Effect though. Harder to prove cause 10% is a lot less likely to interfere than 45%.
The scope of the problem Onevia has reported is greater than was at first suspected. It seems that any Paradoxes that are not generated by dealing damage divisible by eight that happen to be the cause of the Paradox bar resetting will fail to proc Tumultuous Sash. Abilities for which I’ve tested that exhibit this behavior include Butterfly Effect, Paradox Control, Incongruity, Reality Fracture, and the Otherworldly Artifact. I suspect this would hold true for Anomaly as well. Resonance Cascade was proven to be an exception, as its effect is inactive once you’ve amassed seven Paradoxes, and therefore cannot be a cause for resetting your Paradoxes.
It’s simple to replicate for yourself. Equip Tumultuous Sash and test each effect individually. Once you are attacking mobs with a Chaos ability and see your Paradox bar reset, stop attacking and look to see if 1 - 2 Paradoxes are generated from Tumultuous Sash. If the bar remains empty, it means Tumultuous Sash has failed to trigger. You may need to repeat the experiment for a minute or two with Butterfly Effect and the Otherworldly Artifact before a failure is observed, since these effects won’t always be the cause for a Paradox reset.