{
  "abstract": "LLM judges are often evaluated with a single prompt and only a few repeated calls. When their verdicts vary, it remains unclear whether the variation comes from sampling noise within a prompt or systematic differences across prompts. We formalize this distinction using a second-order response law: the distribution of prompt-conditioned verdict distributions induced by a declared prompt policy. For a quadratic measure of prompt instability, we show that the usual plug-in estimator is biased upward at finite repeat budgets because it confounds within-prompt noise with between-prompt variation. We derive unbiased estimators for both sampled prompts and declared fixed prompt censuses from the difference between within- and across-prompt agreement. Under a crossed prompt-by-answer-order design, the same framework separates prompt, order, interaction, and residual call variation, while retaining invalid completed outputs as outcomes. Known-law simulations and a byte-identical live null recover the predicted finite-$R$ inflation. In a matched Qwen study, corrected low-repeat estimates are closer to an independently acquired $R=16$ reference than plug-in estimates, with the largest gains at small repeat budgets. A matched panel across four frozen judge configurations exhibits configuration-specific inflation magnitudes and component profiles. Prompt robustness can therefore be estimated separately from finite-call noise.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.16253",
  "author_total": 6,
  "authors": [
    {
      "name": "Pengbin Feng"
    },
    {
      "name": "Chunlei Meng"
    },
    {
      "name": "Daozheng Qu"
    },
    {
      "name": "Zhilin Zhang"
    },
    {
      "name": "Haoran Liu"
    },
    {
      "name": "Jiekai Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cx3hxnjjmvxg6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16253",
  "fulltext_url": null,
  "id": "cx3hxnjjmvxg6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16253",
  "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",
      "stat.AP"
    ],
    [
      "openalex",
      "3320|Artificial Intelligence in Law"
    ]
  ],
  "openalex": "W7203670470",
  "pdf_url": "https://arxiv.org/pdf/2608.16253",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cx3hxnjjmvxg6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2608.16253",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16253"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.AP",
      "subfield": "statistics"
    }
  ],
  "title": "Second-Order Response Laws for LLM Judges: Debiased Estimation of Prompt Instability",
  "title_text": "Second-Order Response Laws for LLM Judges: Debiased Estimation of Prompt Instability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/cx3hxnjjmvxg6",
  "venue": "arXiv",
  "versions": []
}