{
  "abstract": "Linear probing is the standard instrument for detecting social biases in the hidden representations of large language models. Yet reported probe accuracies come almost exclusively from \\emph{counterfactual} evaluations in which every input carries an explicit demographic marker. Once only a fraction $α$ of inputs carries demographic information, performance degrades sharply, and a weak probe may reflect either an unbiased model or an underpowered detector. We develop a theory that resolves this ambiguity. Modeling representations as two class-conditional clusters with Mahalanobis separation $s$ on a manifold of curvature $\\kap$, we prove: (i) a finite-sample generalization bound governed by the manifold's extrinsic radius with a matching $\\smash{\\sqrt{\\dB/n}}$ minimax lower bound; (ii) an exact purity law for the maximum linear-probe AUC, strictly increasing in $α$; (iii) a curvature ceiling: ambient chordal separation on a space form cannot exceed $2/\\sqrt{\\kap}$; and (iv) a detectability threshold below which no audit can distinguish probe output from chance. Every theorem is validated on synthetic manifolds with known ground truth and on six open-weight models $\\times$ four bias dimensions, where the purity law predicts entire AUC--$α$ curves from a single cross-fitted $\\hat s$ measured at $α=1$, with no parameters fitted to those curves. The framework turns bias auditing into a power analysis: given a target purity and effect size, it prescribes the sample budget $n(α)$ for a conclusive audit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22337",
  "author_total": 2,
  "authors": [
    {
      "name": "Mo Hai"
    },
    {
      "name": "Haifeng Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7kgz90ppvgfx1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22337",
  "fulltext_url": null,
  "id": "7kgz90ppvgfx1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22337",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "3311|Ethics and Social Impacts of AI"
    ]
  ],
  "openalex": "W7214050872",
  "pdf_url": "https://arxiv.org/pdf/2609.22337",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7kgz90ppvgfx1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:33Z",
      "native_id": "2609.22337",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22337"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "When and Why Do Linear Bias Probes Fail? A Geometric and Statistical Theory of Bias Detectability in Large Language Model Representations",
  "title_text": "When and Why Do Linear Bias Probes Fail? A Geometric and Statistical Theory of Bias Detectability in Large Language Model Representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/7kgz90ppvgfx1",
  "venue": "arXiv",
  "versions": []
}