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Clark is a gnome Bard, and took First World Magic to gain a Electric Arc as an innate spell.

On level 7, he is Expert in occult spell DCs. Innate spells seemingly do not care about your tradition:

You're always trained in spell attack rolls and spell DCs for your innate spells, even if you aren't otherwise trained in spell attack rolls or spell DCs. If your proficiency in spell attack rolls or spell DCs is expert or better, apply that proficiency to your innate spells, too. You use your Charisma modifier as your spellcasting ability modifier for innate spells unless otherwise specified.

Does the lack of "appropriate tradition" in the highlighted section mean that innate spells always use your best DC, if the spellcasting ability is the same?

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Not necessarily

A character's occult spellcasting DC and innate spellcasting DC are calculated separately. The same is true for spellcasting of other traditions (such as arcane, primal, or divine).

Their spellcasting DC is calculated based on a proficiency bonus and a spellcasting ability modifier. The tradition (or lack of tradition, in the case of innate spells) determines which spellcasting ability modifier to use. From the Core Rulebook section on Spell Attack Rolls:

Your spell DC is calculated using the following formula.

Spell DC = 10 + ability modifier used for spellcasting + proficiency bonus + other bonuses + penalties

As in the rules you quoted, by default, a character applies their trained proficiency bonus (level + 2) for their innate spellcasting. If they have a better proficiency bonus (such as expert) for their class's spells, then this proficiency also applies to their innate spell bonuses and DCs.

In your example, the bard counts their Charisma modifier toward both their occult spellcasting DC and their innate spellcasting DC. If they're an expert spellcaster, then their expert proficiency bonus (level + 4) applies to both DC calculations too. But the other modifiers may be different. For example, maybe they have an item that adds a bonus to the DCs of occult spells, but not innate spells. The proficiency bonus is the same, but the rest of the equation could be different.

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    \$\begingroup\$ Sorry, the answer made me realize I formulated the question badly. Could you please update? \$\endgroup\$
    – András
    Jun 6 at 12:33
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Yes, that's right. Innate spells always use your highest proficiency. It doesn't matter what tradition they come from.

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    \$\begingroup\$ This looks like it could really use some more details, including references to the relevant rules, so that it explains why it is correct \$\endgroup\$
    – Someone_Evil
    Jun 6 at 0:00
  • \$\begingroup\$ But where are my manners, Welcome to RPG.SE! Take the tour if you haven't already and see the help center or ask us here in the comments (use @ to ping someone) if you need more guidance. Good Luck and Happy Gaming! \$\endgroup\$
    – Someone_Evil
    Jun 6 at 0:00

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