Papers · William Alexander Aldred

Zipf's Exponent in a Sickroom: The Second-Text Arm

by William Alexander Aldred — Independent Researcher

Publicly deposited November 1, 2026 · Zenodo · 5 pp · DOI: 10.5281/zenodo.23869951

Abstract

Paper #147 ran three standard estimators of Zipf's exponent on one book, Moby Dick, and found that the exponent is partly a choice: the two methods that returned a usable number disagreed by more than their own error bars, the third returned no exponent at all, and inside the likelihood method the choice of where the fit starts was worth 0.316 on its own. This paper runs the identical protocol — same counting rule, same three estimators, same fit ranges, same verdict rule, sealed before the text was fetched — on a second book chosen to be as different as a book can be: Charlotte Perkins Gilman's "The Yellow Wallpaper" (1892), a first-person journal written from a sickroom, 6,251 tokens against Moby Dick's 219,054. The disagreement survives the change of book, and the verdict does not. The rank–frequency fit returns α = 1.009, almost Zipf's textbook 1. The likelihood fit, at its own chosen starting point, returns α = 0.725 — a gap of 0.284, nearly three times the larger of the two methods' errors — and moving that starting point to the full range moves the answer by 0.405, more than the same choice cost on Moby Dick, onto a fit the data plainly reject. The third estimator, which on Moby Dick returned no exponent at all, here returns one: α = 12.16, with a formal error of ±12.08 — a number whose uncertainty is as large as itself. And because the registered verdict rule compares the full spread of returned exponents against the largest formal error, that third number decides the verdict: the spread is 11.44, the largest error is 12.08, and the claim that the spread exceeds the errors falls on this text. The fall is printed at the same size as #147's stand, and it is the more instructive of the two results: the rule is decided by whichever estimator is noisiest, and on a short text the noisiest estimator is very noisy indeed.

The result, in plain language

Paper #147 ran three standard estimators of Zipf's exponent on one book, Moby Dick, and found that the exponent is partly a choice: the two methods that returned a usable number disagreed by more than their own error bars, the third returned no exponent at all, and inside the likelihood method the choice of where the fit starts was worth 0.316 on its own. This paper runs the identical protocol — same counting rule, same three estimators, same fit ranges, same verdict rule, sealed before the text was fetched — on a second book chosen to be as different as a book can be: Charlotte Perkins Gilman's "The Yellow Wallpaper" (1892), a first-person journal written from a sickroom, 6,251 tokens against Moby Dick's 219,054. The disagreement survives the change of book, and the verdict does not. The rank–frequency fit returns α = 1.009, almost Zipf's textbook 1. The likelihood fit, at its own chosen starting point, returns α = 0.725 — a gap of 0.284, nearly three times the larger of the two methods' errors — and moving that starting point to the full range moves the answer by 0.405, more than the same choice cost on Moby Dick, onto a fit the data plainly reject.

The third estimator, which on Moby Dick returned no exponent at all, here returns one: α = 12.16, with a formal error of ±12.08 — a number whose uncertainty is as large as itself. And because the registered verdict rule compares the full spread of returned exponents against the largest formal error, that third number decides the verdict: the spread is 11.44, the largest error is 12.08, and the claim that the spread exceeds the errors falls on this text. The fall is printed at the same size as #147's stand, and it is the more instructive of the two results: the rule is decided by whichever estimator is noisiest, and on a short text the noisiest estimator is very noisy indeed.

Program note

Paper 159 of the November 1, 2026 day charge — batch 3 of the writing bench, deposited the day it was written. Its verdict, as printed on its face: FALLS as registered — spread S = 11.440 < largest formal error E = 12.081 (Estimator C's self-exceeding error decides the rule, as #147's degeneracy warning predicted); the A-B gap of 0.284 stands as a descriptive fact

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