> If the program was trained on a thousand data points saying that the capital of Connecticut is Moscow, the model would encode this “truthfulness” information about that fact, despite it being false.
This isn't true.
You're conflating whether a model (that hasn't been fine tuned) would complete "the capital of Connecticut is ___" with "Moscow", and whether that model contains a bit labeling that fact as "false". (It's not actually stored as a bit, but you get the idea.)
Some sentences that a model learns could be classified as "trivia", and the model learns this category by sentences like "Who needs to know that octopuses have three hearts, that's just trivia". Other sentences a model learns could be classified as "false", and the model learns this category by sentences like "2 + 2 isn't 5". Whether a sentence is "false" isn't particularly important to the model, any more than whether it's "trivia", but it will learn those categories.
There's a pattern to "false" sentences. For example, even if there's no training data directly saying that "the capital of Connecticut is Moscow" is false, there are a lot of other sentences like "Moscow is in Russia" and "Moscow is really far from CT" and "people in Moscow speak Russian", that all together follow the statistical pattern of "false" sentences, so a model could categorize "Moscow is the capital of Connecticut" as "false" even if it's never directly told so.