{
  "abstract": "We present DisCo algebra---a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a magma-operad (non-associative partial combination, restricted to the image of disintegration) on the bare family $\\mathcal{P}_n=\\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding on admissible inputs. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not---it defines a renormalization operator $R_{\\mathcal{I}}=\\text{Combine}\\circ\\text{Disintegrate}$. The information loss of this operator is exactly the probability curvature $\\mathcal{K}_{\\mathcal{I}}(μ)$---the conditional KL divergence between the true conditional distribution and the conditionally independent product. This curvature measures the obstruction to lossless decomposition and reveals a hard limit on how much cross-block dependence can be captured by CDF-based scalar block summaries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15006",
  "author_total": 1,
  "authors": [
    {
      "name": "Xuexing Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1a7hb5vz6n41b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15006",
  "fulltext_url": null,
  "id": "1a7hb5vz6n41b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15006",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CT"
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    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
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    [
      "arxiv",
      "math.ST"
    ],
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      "arxiv",
      "stat.TH"
    ],
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      "openalex",
      "1702|Bayesian Modeling and Causal Inference"
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  "openalex": "W7221823533",
  "pdf_url": "https://arxiv.org/pdf/2610.15006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1a7hb5vz6n41b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:18Z",
      "native_id": "2610.15006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "DisCo Algebra: Why Disintegration and Combination Are Not Inverses",
  "title_text": "DisCo Algebra: Why Disintegration and Combination Are Not Inverses",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/1a7hb5vz6n41b",
  "venue": "arXiv",
  "versions": []
}