{
  "abstract": "Learning independence is fundamental to statistical inference and machine learning, yet learning independence between entire time series by a test remains challenging when only a single observation record is available. Existing guarantees often rely on structural or mixing assumptions, while stationarity and ergodicity alone do not permit consistent testing. In this work, we identify a class-level boundary governed by shared nontrivial Koopman eigenvalues, which represent common persistent oscillatory frequencies of the two processes. When no such eigenvalues are shared, we extend shift-based rank calibration with finite-window dependence witnesses and a data-adaptive threshold to construct a deterministic, weakly consistent test with finite-sample false-positive control, without quantitative mixing assumptions. Conversely, we show that no possibly randomized test can achieve vanishing false-positive probabilities under every independent jointly totally ergodic law while consistently detecting every dependent jointly totally ergodic law with a shared nontrivial eigenvalue. This impossibility already holds for binary observations and persists within any prescribed nonempty open eigenvalue window, identifying a spectral obstruction to universal learning of full-process independence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38999",
  "author_total": 2,
  "authors": [
    {
      "name": "Senhan Yao"
    },
    {
      "name": "Yuxin Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9aj0mza8fhz4m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38999",
  "fulltext_url": null,
  "id": "9aj0mza8fhz4m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38999",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38999",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9aj0mza8fhz4m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38999",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38999"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A Spectral Boundary for Independence Learnability",
  "title_text": "A Spectral Boundary for Independence Learnability",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/9aj0mza8fhz4m",
  "venue": "arXiv",
  "versions": []
}