{
  "abstract": "Probability plays two distinct roles in modern data-analytic practice: (i) as an internally coherent, filtration-relative language for sequential forecasting and, together with a stated loss or utility, decision making; and (ii) as a foundation for empirical claims such as stabilization, calibration, and repeated-sampling validity. The first role is relative to an assessment law $\\mathbb Q$, whereas the second is evaluated under a governing law $\\mathbb P$ and a declared repetition regime. In classical settings the roles are often aligned by conditional correctness under $\\mathbb P$ together with i.i.d.\\ or other stable-law assumptions under a fixed design, or by exchangeability. In adaptive and AI-mediated settings they can diverge because forecasts operate within feedback loops while reliability is evaluated under a potentially different law and regime. We synthesize relevant results from prequential forecasting, calibration, martingale and game-theoretic validity, adaptive data analysis, conformal prediction, and long-context AI reliability into a formal account of \\emph{role separation}. The resulting \\emph{Role Separation Principle} distinguishes internal coherence from conditional-mean correctness under the evaluation law and from target-specific reference and stability conditions. Two separation results show that a proper coarsening of the limiting-frequency distribution need not identify the governing law, and that even conditionally correct forecasts under that law need not accompany stabilization. We then develop a unified stabilization framework spanning filtration-based, shift-based, policy--environment, and design-based regimes and operationalize it through a regime-conditional audit checklist. Applied illustrations clarify calibration claims, synthetic-data validity, and structural mismatches relevant to some high-confidence unsupported AI outputs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04115",
  "author_total": 1,
  "authors": [
    {
      "name": "Yonggang Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8fn47j5s7hj7t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04115",
  "fulltext_url": null,
  "id": "8fn47j5s7hj7t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04115",
  "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",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7208727841",
  "pdf_url": "https://arxiv.org/pdf/2609.04115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8fn47j5s7hj7t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.04115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "One Probability, Two Roles: The Separation of Coherence and Frequency in Adaptive Regimes",
  "title_text": "One Probability, Two Roles: The Separation of Coherence and Frequency in Adaptive Regimes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:42Z",
  "url": "https://research.draconitic.com/paper/8fn47j5s7hj7t",
  "venue": "arXiv",
  "versions": []
}