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Imagine caring as a value, for example, my care™ about this topic is 2, whereas yours is 5.
In this interpretation, "I don't care" implies that the care™ value of the speaker is 0, by the same logic, saying "I care" implies that their care™ value is greater than 0.
With that in mind, "I could care less" implies there is a care™ value lower than the one they currently hold. Meanwhile, "I couldn't care less" implies the opposite, there is no care™ value lower, which is only true for 0 (AKA "I don't care").