Zipf's Exponent in a Law Book: The Third Text, and Whether the Spread Depends on the Kind of Text

Zipf's law says word frequencies fall off with rank as a power law, and the exponent of that fall-off is quoted as if a text simply has one. Two earlier papers in this series priced how much of the exponent is instead the fitting method's opinion: run three standard fits on one text — a straight-line fit to the rank–frequency curve, a maximum-likelihood power-law fit, and a straight-line fit to the frequency-of-frequencies curve — and compare their disagreement with the errors they quote. On a long novel the three fits disagreed violently and the disagreement beat the quoted errors. On a short story the disagreement was large but so were the errors, and the claim failed its own bar. Both texts were stories. This paper runs the identical protocol, byte for byte the same code, on a text of a different kind altogether: a legal instrument, the Constitution of the United States (the original articles, 4,531 words of running text). The result answers the question the first two papers could not: the spread is not a fixed tax on Zipf exponents. It changes its shape with the kind of text. On the legal text the two sound estimators agree with each other to three decimal places (0.957 and 0.955) — a hundred times closer than they managed on the short story — and the entire spread retreats into the third estimator, the one whose algebra is known to explode when its slope passes near 1, which it does here (its answer, 33.9 with an error of 109, is not a measurement of anything). By the series' registered bar the claim fails on this text, and the failure is printed at the same size as the successes were: spread 32.90 against a largest quoted error of 109.45. What the three texts together show is that "the estimator spread" is not one phenomenon. On fiction it lives between the honest estimators. On a law book it lives entirely inside the broken one.