Fractal Geometry and the Verifiability of Electronic Election Results

Fractal Geometry and the Verifiability of Electronic Election Results: Limits of Identification and a Multiscale Framework for Analyzing Tabulation This record contains an integrated research article (preprint; not peer reviewed) that asks a deliberately narrow question: under what conditions can fractal and multiscale methods applied to the published tabulation of a two-candidate election yield useful statistical evidence of irregularity, and under what conditions can no such method identify the origin of the data? The paper is addressed equally to those who read smoothness in a tabulation curve as a certificate of integrity and to those who read any irregularity as proof of manipulation; it argues that both readings mistake arithmetic and model specification for evidence. The analysis separates five stages of the electoral chain (voter choice, machine record, transmission, tabulation, publication) and three research questions that are routinely conflated: whether the order in which precincts are reported is atypical, whether precinct-level results are atypical relative to an explicit reference model, and whether published results agree with independently preserved records. Three elementary propositions formalize what tabulation data can and cannot show. The cumulative vote share is mechanically damped as the count grows, so late smoothness is a property of weighted averaging rather than a sign of control. Two scenarios that induce the same distribution over the observable data cannot be separated by any test confined to those data, with discriminating power bounded by their total variation distance. And a class of proportional vote transfers leaves every increment-based statistic (first differences, centered CUSUM, zero-mean wavelet contrasts) unchanged while shifting the cumulative margin by 2δN_k. A fully synthetic experiment models 256 precincts in 16 regions with correlated regional preferences and geographically dependent reporting times, and compares four two-sided detectors (level test, centered CUSUM, multiscale scan and Haar contrast over windows of 4 to 128 precincts), each calibrated separately on 2,000 null elections and evaluated on 1,500 independent elections per scenario. Under the correct null, alert rates lie between 4.1% and 5.7%. With a 4% transfer concentrated in a block of 51 precincts, the Haar contrast alerts in 94.0% of replicates (Wilson 95% interval 92.7%–95.1%); with a diffuse 0.8% transfer across all 256 precincts, moving approximately the same number of votes (608 versus 612 on average), it alerts in 4.5%, indistinguishable from the null. A legitimate regional preference shift omitted from the reference model produces Haar alerts in 57.7% of elections in which no vote was transferred, and a deliberately homogeneous reference that ignores geography drives three of the four detectors to 100% false alerts. Sensitivity depends on the shape of the alteration and on the specification of the reference model, not on the volume of votes moved. The limits of these findings are stated explicitly. The reference model is an oracle known to the simulator; all data are synthetic; the experiment was not preregistered; detection delay and localization accuracy were not systematically evaluated; no actual election is analyzed, and nothing in the paper certifies or refutes the integrity of any election. A p-value is not a probability of fraud, and the absence of an alert cannot certify the absence of manipulation. The paper closes with a protocol for future application to real tabulation records, covering data provenance and chronology, preregistration, a global multiscale statistic with Monte Carlo calibration, nonfractal comparators at equal false-alert rates, adversarial and benign alternatives, and robustness checks, and situates the discussion within software independence and the cast-as-intended, recorded-as-cast and tallied-as-recorded framework of election verifiability. The PDF comprises the article and two appendices. Appendix A is an English-language guide to the experiment; Appendix B reproduces verbatim the complete reproducibility archive: simulacao.py (full source code), resultados.csv, modelo_geografia_omitida.csv, exemplo_secoes_sinteticas.csv (the 256 precinct rows of the illustrative election), metadados.json (seeds, thresholds and parameters), LEIA_ME.md, AMBIENTE.txt, SHA256SUMS.txt and the three original figures. The original ZIP archive, SHA-256 d49e2ab756603a4511727e15d31b20ec5276bd88a74883622d7204c8ebcfd306, is embedded in the PDF as an attachment. Code, data, file names and figure labels remain in Portuguese to preserve the archive and its checksums. An independent rerun in the same environment reproduced the numerical results; this does not replace external validation or scientific peer review.