
Errenor |
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I consider Class DCs different and separate and would not allow this, even for archetypes. Unless there's explicit permission in the rules or something really breaks (and lack of several points in DC is not something breaking).
As for the rules, there's no explicit statement that they are different I suppose. But I think that "A class DC sets the difficulty for certain abilities granted by your character's class. This DC equals 10 plus their proficiency bonus for their class DC (typically +3 for most 1st-level characters) plus the modifier for the class's key attribute modifier" and lines like "Trained in investigator class DC" implicitly mean exactly that.

NorrKnekten |
They are different and separate but ultimately not written to cover Dual Classing similar to how other rules/elements arent written with ABP or Free Arch in mind.
Just as Errenor states you don't become trained in Class DC but rather the Alchemist Class DC and whenever an Alchemist ability/feat/feature asks for your class DC it specifically asks for your Alchemist Class DC.

Claxon |

The downside to that is, anything that relies on a class DC becomes practically worthless.
Edit: Sorry this about dual class not dedications.
My thought was about dedications. Dedications that rely on class DCs are painful because the class DC of dedications doesn't Advance. So like Alchemist you become trained in the class DC, and as far as I can see it doesn't advance. Meaning offensive items from alchemist aren't very good.
Please let me know if I've missed something on that.
Not sure how it shakes out for Dual Class rules. I think you get the abilities of each class, except where they would overlap so you take the better. But kineticist class dc and alchemist class dc are different things, they both progress as you level but are separate.

NorrKnekten |
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I wouldnt say class DC for dedications are worthless, Less potent for sure. but most dedications to my knowledge have a method of progressing their archetype class DC to atleast expert in the cases where it matter.
Alchemist has Alchemical Power
Kineticist has Expert Kinetic Control
Inventor has Brilliant Crafter
Alchemist Archetype in particular has the same class proficiency by the time they can make alchemical items use their class DC. Its only at 17 that the archetype falls behind. Quite comparable to spellcaster archetypes which lag behind 1 proficiency step.

HammerJack |
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I think you've already given something very close to the answer to that in this thread. Because a kineticist dabbling in alchemy making stringer alchemical poisons than a full time, single class Toxicologist (and other cases of being better at using a class's abilities than the class themselves) is a terrible feeling way for the system to work.
Casting can bring up some small shades of that, but they're pretty limited since spell ranks mean you really only see it get weird if a full caster is taking something like a focus spell, that can heighten fully, from a non caster class.

Finoan |

Even if you do it like that, using the different stats would not change much honestly.
It is a couple of points of difference. Maxed out attribute bonus in your primary class's casting stat and non-maxed attribute bonus in your archetype spellcasting.
Which is about the same amount of difference that you are going to get from using your Kineticist's Legendary proficiency class DC instead of only your Alchemist's Master proficiency class DC. So I think that it is fair that they are different.