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I was really hoping you'd go all in and use rare words in that sentence just for the joke.

agreed, he should have used "entropy" instead of information

I think the rarest words in your sentence are “information” and “sentence” yet you can drop both of those and figure out your meaning, especially if the sentence is part of a broader conversation.

“Rare words carry the most” basically conveys what you’re saying but just more vague. And I think that’s a really important distinction. Rare words reduce ambiguity and increase the specificity of your sentence, but they don’t necessarily carry the most meaning.

I’d disagree with that. For example, in many sentences, the most load-bearing (heh) word is “not”, which is also one of the most common words.

Disagree because it's one bit of information and in practice you can often guess the correct direction even if you're not sure if "not" was or wasn't said.