{
  "abstract": "We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_ν(z)=κ(z)\\,A_ν(σ(z))$ is equivalent to a single symmetry of the resolvent $R(γ)=(I-(1-γ)P)^{-1}$: the intertwining relation $[Δ^2\\mathcal{R},R(γ)]=0$, where $\\mathcal{R}$ is the reflection operator of an involution $σ$ and $Δ=\\operatorname{diag}(\\sqrt{κ(z)})$; equivalently, $\\widetilde{\\mathcal{T}}=K^{-1/2}Δ^2\\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\\sqrt{K})$, with $K=κ(z)κ(σ(z))$, which depends only on the scalar $K$ --- not on the resetting rate $γ$, the reset distribution, or the particular chain. We characterize the class of $(σ,κ)$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^*$; a Doob $h$-transform realizes the duality $K\\mapsto1/K$, hence $C^*\\mapsto1-C^*$, with fixed point $C^*=1/2$. The orientation field admits the explicit resolvent representation $ψ(γ)=R(γ)(b^{(0)}-C^*b)$: its gauge-normalized form $Δ^{-1}ψ(γ)$ is antisymmetric under $σ$, it has an exact node at the fixed point of $σ$, and it governs the exact sign law $\\operatorname{sgn}(C(π,γ)-C^*) =\\operatorname{sgn}\\langleπ,ψ(γ)\\rangle$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15140",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Antonio Vega Coso"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9xcy63e325cvr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15140",
  "fulltext_url": null,
  "id": "9xcy63e325cvr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15140",
  "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.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7203676579",
  "pdf_url": "https://arxiv.org/pdf/2608.15140",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9xcy63e325cvr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2608.15140",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15140"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Resolvent intertwining and spectral duality in Markov chains with geometric resetting",
  "title_text": "Resolvent intertwining and spectral duality in Markov chains with geometric resetting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/9xcy63e325cvr",
  "venue": "arXiv",
  "versions": []
}