Where is the math for this?

What steven0461 said. Square rooting both sides of the Bienaymé formula gives the standard deviation of the mean going as 1/√n. Taking precision as the reciprocal of that "standard error" then gives a √n dependence.

I agree that methodology is important, but humans can often be good at inferring causality even without randomized controlled trials.

This is true, but we're also often wrong, and for small-to-medium effects it's often tough to say when we're right and when we're wrong without a technique that severs all possible links between confounders and outcome.