{
  "abstract": "A state may need to be compressed before its exact downstream task is known. For a finite family of linear task operators $\\{T_u\\}_{u\\in\\U}$, we characterize exactly how much state must survive when only one of $K$ coarse task messages is available at compression time: $$p^*(K)= \\min_{\\substack{\\Pcal\\text{ partition of }\\U\\\\|\\Pcal|\\le K}} \\max_{C\\in\\Pcal}\\rank(T_C),$$ where $T_C$ stacks the tasks that remain unresolved within cell $C$. Thus task information reduces retained state precisely by separating tasks whose joint task-visible subspace is expensive. We derive the dual bit frontier, an irreducible common-core floor, and a sharp singular-value characterization for nonzero tolerance. For overlapping task families, a cumulative overlap deficit places the exact frontier within an additive $Δ$ of weighted load balancing. Optimal partitioning is strongly NP-hard, yet direct-sum families admit classical approximation guarantees and a PTAS. We stress-test the theory on learned task operators from CIFAR-10, Fashion-MNIST, SVHN, and Burgers observables. For Burgers with $K=2$ and $\\eps=0.35$, a constructive partition attains $p=4$, while balanced random partitions require median $p=6$, with held-out worst standardized RMSE $0.0368$. The results distinguish uniform task sufficiency from distributional and decision-level performance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21523",
  "author_total": 1,
  "authors": [
    {
      "name": "Ronald Katende"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c8v0fmr3fwtqn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21523",
  "fulltext_url": null,
  "id": "c8v0fmr3fwtqn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21523",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7213939074",
  "pdf_url": "https://arxiv.org/pdf/2609.21523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c8v0fmr3fwtqn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.21523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21523"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3109|Model Reduction and Neural Networks",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "What Must Survive? Exact Task-Information--State Frontiers Under Partial Task Revelation",
  "title_text": "What Must Survive? Exact Task-Information--State Frontiers Under Partial Task Revelation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/c8v0fmr3fwtqn",
  "venue": "arXiv",
  "versions": []
}