[andy_wrote]

I think I understand what you're trying to get at, but I don't think describing a p-value in terms of "pure noise" is quite right. The null hypothesis may well carry with it all sorts of specific assumptions: independence, finite variance, maybe even specific probability distributions (and not necessarily the normal distribution). Typically somewhere in there we define a θ parameter and then include something like θ = 0 that provides grounds for possible rejection. We think of "θ = 0" as the null hypothesis, but all the other assumptions are part of the null hypothesis as well; the statement "θ = 0" does not make sense without them.