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I had this same question after reading in the ability damage section that Dex ability damage affects: Armor Class, attack rolls that rely on Dexterity (usually ranged), weapon damage rolls that rely on Dexterity (such as operative weapons), initiative checks, and reflex saves.
However I could not find anywhere in the core rule books where it states that Operatives or weapons with the operative special weapon property actually get Dex to damage.