{
  "abstract": "Early-warning signals are usually computed from one or a few observables, although the underlying dynamics are high dimensional. Projection creates a structural ambiguity: the same scalar record may reflect spectral loss of stability, hidden-state memory, nonnormal amplification, a turning point of a physical balance, or a combination of these. We develop a unified operator theory that makes these alternatives explicit. For a resolved-hidden block linearization we derive exact Volterra reductions, finite and diffusive Markovian liftings, and a Schur-resolvent factorization of the stability problem, distinguishing three notions of memory: realized pathwise, structural Schur return, and finite-horizon projected. A covariance theorem gives sufficient conditions for a scalar to inherit critical slowing down from a real multiplier approaching the unit circle: spectral separation, bounded eigenvector conditioning, observability of the critical mode, stochastic excitation of that mode, and quasi-stationary forcing. We then prove failure mechanisms: a critical mode can be invisible or unexcited; an oscillatory crossing need not raise lag-one correlation; a nonnormal family can produce diverging scalar variance with a fixed stable spectrum; a physics-based turning point need not coincide with a spectral boundary; and finite-horizon data cannot identify long memory uniquely. The theory is illustrated with a 4400-year CESM AMOC hysteresis experiment. EOF1 alone detects altered susceptibility, but EOF1-EOF2 is the smallest resolved space revealing changed Schur return and projected memory after recovery; F_ovS reaches its minimum 26 years before collapse while the modal margin does not decrease toward zero: the two diagnostics probe complementary mechanisms. The application is mechanistic and single-trajectory; no validated early-warning signal is claimed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16235",
  "author_total": 1,
  "authors": [
    {
      "name": "Mauricio Herrera-Marín"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d0vaj1ye48h89.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16235",
  "fulltext_url": null,
  "id": "d0vaj1ye48h89",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16235",
  "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.DS"
    ],
    [
      "arxiv",
      "physics.ao-ph"
    ],
    [
      "arxiv",
      "stat.AP"
    ],
    [
      "openalex",
      "2306|Ecosystem dynamics and resilience"
    ]
  ],
  "openalex": "W7213421601",
  "pdf_url": "https://arxiv.org/pdf/2609.16235",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d0vaj1ye48h89.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:41Z",
      "native_id": "2609.16235",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16235"
    }
  ],
  "subjects": [
    {
      "field": "earth",
      "origin": "arxiv:physics.ao-ph",
      "subfield": "atmospheric"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.AP",
      "subfield": "statistics"
    }
  ],
  "title": "Projection, Memory, and the Validity of Scalar Early-Warning Signals: A Schur--Volterra Theory with an AMOC Application",
  "title_text": "Projection, Memory, and the Validity of Scalar Early-Warning Signals: A Schur--Volterra Theory with an AMOC Application",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/d0vaj1ye48h89",
  "venue": "arXiv",
  "versions": []
}