{
  "abstract": "For cumulative-count first-passage problems, the Laplace transform of the time $\\mathcal T_N$ of the $N$th count is naturally expressed through powers of a one-count kernel. A Jordan defect of that kernel does not, however, automatically survive the complete sum over terminal phases. We give an exact three-state Markov-renewal example in which it does. At $s=1$, the one-count kernel has spectrum $\\{1/2,1/8,1/8\\}$, with a one-dimensional eigenspace at $1/8$, and for initial phase $1$, \\[ \\mathbb E_1[e^{-\\mathcal T_N}] =\\frac56\\,2^{-N}+\\frac{N+1}{6}\\,8^{-N}. \\] Thus the ordinary, unconditioned cumulative-count transform contains the explicit Jordan contribution $N/(6\\,8^N)$ at every threshold $N\\ge1$. For the same fixed stochastic process, the two eigenvalues meeting at $1/8$ unfold as \\[ λ_\\pm(s)=\\frac18\\pm\\frac1{96}\\sqrt{s-1} -\\frac{131}{2304}(s-1)+O((s-1)^{3/2}), \\] and the characteristic discriminant has a simple zero at $s=1$. We place the example beside two structures that suppress such a marginal signature: fixed post-count reset, which gives scalar renewal, and a common total holding rate, for which terminal summation scalarizes the matrix power. The model also admits a monitored-Lindblad realization, but the mechanism is entirely classical. The result provides a small exact benchmark for how generalized spectral modes can remain visible in finite-threshold first-passage statistics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20148",
  "author_total": 1,
  "authors": [
    {
      "name": "Lachlan Bridges"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4566h19xawkt0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20148",
  "fulltext_url": null,
  "id": "4566h19xawkt0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20148",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7213578540",
  "pdf_url": "https://arxiv.org/pdf/2609.20148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4566h19xawkt0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.20148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "An exact Jordan signature in cumulative-count first-passage statistics",
  "title_text": "An exact Jordan signature in cumulative-count first-passage statistics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/4566h19xawkt0",
  "venue": "arXiv",
  "versions": []
}