{
  "abstract": "A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator $L$, coarse channel $C$, and stochastic section $R$ satisfying $CR=I$, define $P=RC$, $Q=I-P$, and $K=CLR$. The exact coarse propagator $T_t=C\\mathrm{e}^{tL}R$ obeys a Volterra equation with memory kernel $\\mathcal{M}(t)=CLQ\\,\\mathrm{e}^{tQLQ}\\,QLR$. Every memory jet $\\mathcal{M}^{(k)}(0)$ is recursively recoverable from the short-time derivatives of $T_t$ alone. The rank of the block Hankel operator $\\mathbb{H}_{\\infty}=(\\mathcal{M}^{(i+j)}(0))_{i,j\\geq 0}$ is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho--Kalman factorization and is unique up to similarity. If $d=\\min\\{k:\\mathcal{M}^{(k)}(0)\\neq 0\\}$, then every smooth classical $f$-divergence generically scales as $t^{2d+4}$. Exact Markov examples realize exponents $4,6,8$ and closure ranks $1,2,3$. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13951",
  "author_total": 1,
  "authors": [
    {
      "name": "Maxim V. Churilov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8mf6f7p1ryx1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13951",
  "fulltext_url": null,
  "id": "e8mf6f7p1ryx1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13951",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221816957",
  "pdf_url": "https://arxiv.org/pdf/2610.13951",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8mf6f7p1ryx1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.13951",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13951"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Predictive Holonomy and Minimal Completion of Coarse Dynamics",
  "title_text": "Predictive Holonomy and Minimal Completion of Coarse Dynamics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/e8mf6f7p1ryx1",
  "venue": "arXiv",
  "versions": []
}