{
  "abstract": "Benford's law specifies a marginal distribution for significant digits, whereas the usual first-digit Pearson $p$-value is calibrated under an independent multinomial sampling model. We separate these statements with four constructions that share the same one-time or pooled Benford target but have different joint structures. Under a wrapped-Gaussian circular Markov construction, the nominal 5% Pearson test rejects 21.6% of samples at a fixed persistence level even though every one-time marginal is exactly Benford; a sequence-level second-order moment-matching calibration reduces the rejection rate to 4.8%. The same strategy performs well in the randomized-rotation and random-composition designs. The effect persists across changes in sample size and sequence length, and a grid approximation to a continuous log-significand Cramér-von Mises discrepancy shows the same design dependence. In the wrapped-Gaussian design examined here, corrected procedures retain rejection probabilities that rise with the size of a smooth marginal departure. Finally, we distinguish calibration from an information boundary: significand-only methods cannot detect changes that leave the complete significand process unchanged, but they readily detect perturbations that alter it. Benford conformity is a marginal statement; a Benford $p$-value is valid only relative to a specified statistic, sampling law, and calibration procedure; an integrity claim requires substantive competing models.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18424",
  "author_total": 1,
  "authors": [
    {
      "name": "Arthur Charpentier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/856g4106ydjtm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18424",
  "fulltext_url": null,
  "id": "856g4106ydjtm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18424",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "2613|Benford’s Law and Fraud Detection"
    ]
  ],
  "openalex": "W7213516575",
  "pdf_url": "https://arxiv.org/pdf/2609.18424",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/856g4106ydjtm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:10Z",
      "native_id": "2609.18424",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18424"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "What Does a Benford Test Actually Test? Marginal Conformity, Sampling Structure, and Forensic Inference",
  "title_text": "What Does a Benford Test Actually Test? Marginal Conformity, Sampling Structure, and Forensic Inference",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:10Z",
  "url": "https://research.draconitic.com/paper/856g4106ydjtm",
  "venue": "arXiv",
  "versions": []
}