{
  "abstract": "We study the geometry of reset distributions for absorbed Markov processes with geometric resetting, working at an abstract level that isolates the structural mechanism underlying reset-neutral invariance. We show that the spectral duality endows the simplex of reset distributions with a non-trivial spectral response geometry: the simplex $Δ_{m-1}$ carries a foliation by level sets of the coupling functional $C$, organized around a critical manifold $Σ$. Under four structural conditions (S1)-(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix $Σ$, derive the invariant value $C^* = 1/(1+\\sqrt{K})$, identify a projective structure in the spectral coefficients, and prove a global sign principle in the two-site case. In the canonical realization, the biased random walk with multi-site geometric resetting, numerical evidence shows that $\\mathrm{sgn}(\\partial_γC) = \\mathrm{sgn}\\langleπ-π^*,ψ(γ)\\rangle$, where $ψ(γ)$ is the linear response functional at the separatrix; this global orientation does not, however, follow from (S1)-(S4) alone, and is dynamical rather than algebraic information. This is the third paper in a program connecting stochastic resetting with spectral theory and information geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.10717",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Antonio Vega coso"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9d0akz9fx40jx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.10717",
  "fulltext_url": null,
  "id": "9d0akz9fx40jx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.10717",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7168276520",
  "pdf_url": "https://arxiv.org/pdf/2607.10717",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9d0akz9fx40jx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2607.10717",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.10717"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Separatrix structure and the geometry of reset distributions",
  "title_text": "Separatrix structure and the geometry of reset distributions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/9d0akz9fx40jx",
  "venue": "arXiv",
  "versions": []
}