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But you're probably right about gestures.
Wild chimps have a vocabulary of about 66 signs. We can also observe tribes with languages more primitive than ours (no pronouns, for example). But there's a missing link of several millions of years of evolution between both.
What are the (known) intricacies of the evolution of our ability to communicate?
There's no definitive proof for the statistical argument, but a growing amount of (neuro)scientific evidence points to it. What's (are) your alternative hypothese(s)?