{
  "abstract": "Entropy-based selection objectives suffer from a fundamental degeneracy: minimizing Shannon entropy $H(p_A)$ rewards confident selection regardless of whether the selected candidate is informative. We address this limitation with the residual entropy $D = H(p_A) - H(p_β)$, where $p_β$ is induced by candidate trust weights. We prove the exact identity $D = -\\mathrm{KL}(p_A\\Vert p_β) - Δ$, where $Δ$ measures whether the score-induced distribution and trust profile favor the same candidates. Boundary cases establish basic safety: under uniform trust, $D\\leq0$ automatically, so an equal-trust, non-starving state is never penalized, while at any one-hot limit, $D\\to0$ regardless of the selected candidate. For the intermediate regime where selection occurs, we prove that $D\\leq0$ when candidate ordering by trust agrees pairwise with ordering by informativeness, and derive a tighter certificate based on the leading candidate's margin over its competitors. These results are independent of the candidate-scoring function and apply to both stationary and dynamically changing information. Experiments with a gradient-based mixture-of-experts router confirm that the ordering conditions can hold during real optimization and show that correct ordering improves downstream performance when candidates are non-interchangeable and selections are used directly rather than averaged. Beyond routing, margin-based reweighting matches or outperforms fixed-strength baselines in a class-imbalance task, while informative selection in a production video-prediction system reduces MSE by approximately 20$\\%$ and transfers to a related species. Residual entropy, therefore, provides a safety criterion for selection and a usable signal for deciding when that selection is informative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05925",
  "author_total": 2,
  "authors": [
    {
      "name": "Alyssa H. Shin"
    },
    {
      "name": "Claire H. Shin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzfa6s7stdtpb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05925",
  "fulltext_url": null,
  "id": "dzfa6s7stdtpb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05925",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Machine Learning and Data Classification"
    ]
  ],
  "openalex": "W7220741417",
  "pdf_url": "https://arxiv.org/pdf/2610.05925",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzfa6s7stdtpb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05925",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05925"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Arbitrary-Placement Problem in Entropy-Minimizing Selection, and a Residual-Entropy Formulation",
  "title_text": "The Arbitrary-Placement Problem in Entropy-Minimizing Selection, and a Residual-Entropy Formulation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dzfa6s7stdtpb",
  "venue": "arXiv",
  "versions": []
}