{
  "abstract": "We propose a mathematical theory of pragmatic information that connects communication, control, and decision-making. Its central notion is the isoteleia mapping, which formalizes equifinality: distinct semantic paths that lead to the same optimal action are treated as pragmatically equivalent. This mapping yields a three-tier hierarchy of syntactic, semantic, and pragmatic information, in which each successive abstraction removes distinctions that are irrelevant to the task. We then define pragmatic entropy, up/down pragmatic mutual information, channel capacity, and rate-distortion, and prove lossless source coding, channel coding, and rate-distortion theorems that extend Shannon's results. These measures quantify decision uncertainty, reliable transmission, and task-oriented compression at the level of terminal actions. We further introduce pragmatic value of information (VoI) and pragmatic cost of information (CoI) as decision-theoretic duals to rate-distortion and capacity, and develop a Lagrangian dual framework for cross-layer optimization. The resulting pragmatic efficiency bound $\\mathcal{E}_p(λ)=\\sup_R[Φ_p(R)-λ\\mathrm{CoI}_p(R)]$ characterizes the maximum net utility attainable by a resource-constrained intelligent system under a given resource price, yielding a behavioral capacity that extends Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages provide closed-form expressions for Gaussian channels and sources, while dynamic settings are addressed through a Bellman equation for sequential decision-making. The framework supports task-oriented communication, networked control, autonomous systems, and embodied AI by shifting emphasis from symbol fidelity to the effectiveness of information in guiding actions. In this way, it offers a common language for systems that extract value from information under resource constraints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10986",
  "author_total": 2,
  "authors": [
    {
      "name": "Kai Niu"
    },
    {
      "name": "Ping Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74p8kp9g48cem.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10986",
  "fulltext_url": null,
  "id": "74p8kp9g48cem",
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  "landing_url": "https://arxiv.org/abs/2609.10986",
  "language": null,
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  "licence_label": "arXiv licence",
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  "metadata_licence": "CC0-1.0",
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    [
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      "cs.AI"
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  "openalex": "W7212275814",
  "pdf_url": "https://arxiv.org/pdf/2609.10986",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74p8kp9g48cem.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.10986",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10986"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "A Mathematical Theory of Pragmatic Information",
  "title_text": "A Mathematical Theory of Pragmatic Information",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/74p8kp9g48cem",
  "venue": "arXiv",
  "versions": []
}