Fractal Signature Is a Property of the Generator, Not the Document: Split-Half Reliability Bounds for Scaling-Exponent AI-Text Detectors

Scaling exponents that summarise the fractal structure of language, the Hurst and H\"older exponents, separate human from machine-generated text at the corpus level, and a growing literature uses them as per-document detector features. Each exponent is defined by an asymptotic limit, so a per-document value is an estimate computed from roughly $400$ tokens, yet no reliability coefficient has been reported for such an estimate at document length. We decompose the per-document estimate into a population term, a document term, and estimator error, and measure the ratio of the latter two with a pre-registered interleaved split-half check on $223,834$ GAGLE documents scored under Gemma-2B and Mistral-7B. The decomposition yields four falsifiable predictions, which we test with a fixed gradient-boosted detector, a within-population classification task, an edit battery that separates token-order from content perturbations, and zero-shot transfer to RAID. At the population level, all $20$ cells (domain, generator, exponent) separate with non-overlapping $95\%$ intervals. At the document level, the H\"older estimate reaches $\rho =0.063$ and $0.050$ under the two scorers, the Hurst estimate at most $0.576$, and none of the twenty candidate quantities clears the bar. All four predictions hold. Deleting the $54$ fractal features from an $85$-feature detector leaves AUROC unchanged to four decimals; the family receives $5.52\%$ of split gain but $0.47\%$ of permutation importance; within machine text, where population factors are balanced by design, it reaches macro-F1 $0.198$ against $0.582$ for all features (chance $0.125$); and its $+0.0240$ AUROC gain on RAID is recovered ($149\%$) by replacing each document's fractal features with their group (generator, domain) mean. Under token-order edits, deleting the family raises recall by $1.96$ and $8.76$ points. Independently, a curvature detector's apparent robustness to paraphrase is an artifact of scorer–attacker lineage: its recall of $0.63$–$0.86$ falls to $1.2$–$4.5\%$ under attackers outside that lineage. The fractal signature is a property of the generating process, not of the document. We propose an admissibility condition: a scaling statistic should be used as a document-level representation only once its split-half reliability at the deployment length clears a declared bar, a check that costs one split and one correlation and does not depend on the downstream model.