I looked briefly again at Newman's article whence I got the example of word lengths and distributions. Here are a couple more comments on it:
(1) The circularity of the argument that we discussed is present directly in this paper, and Newman brings it up explicitly. If you want to pursue where things have progressed on this, you should look at the references to see what people have done.
(2) The error that I did make beyond the inherited logical issues are that when I introduced the example, I mistakenly used specific words in my introduction rather than length of words. That was just me messing up and unfortunately causing some confusion in the process.